iconEuler Reference

Maxima 5.26.0 Manual

Maxima is a computer algebra system, implemented in Lisp.

   Maxima is derived from the Macsyma system, developed at MIT in the
years 1968 through 1982 as part of Project MAC.  MIT turned over a copy
of the Macsyma source code to the Department of Energy in 1982; that
version is now known as DOE Macsyma.  A copy of DOE Macsyma was
maintained by Professor William F. Schelter of the University of Texas
from 1982 until his death in 2001.  In 1998, Schelter obtained
permission from the Department of Energy to release the DOE Macsyma
source code under the GNU Public License, and in 2000 he initiated the
Maxima project at SourceForge to maintain and develop DOE Macsyma, now
called Maxima.

Maxima infrastructure

Introduction to Maxima      Sample Maxima sessions.
Bug Detection and Reporting Finding and reporting bugs in Maxima.
Help                        Asking for help from within a Maxima session.
Command Line                Maxima command line syntax, Input, and Output.
Data Types and Structures   Numbers, Strings, Lists, Arrays, and Structures.
Expressions                 Expressions in Maxima.
Operators                   Operators used in Maxima expressions.
Evaluation                  Evaluating expressions.
Simplification              Simplifying expressions.
Mathematical Functions      Mathematical functions in Maxima.
Maximas Database            Declarations, Contexts, Facts, and Properties.
Plotting                    2D and 3D graphical output.
File Input and Output       File input and output.

Support for specific areas of mathematics
Polynomials                 Standard forms for polynomials, and
                                functions operating on them.
Special Functions           Special functions
Elliptic Functions          Elliptic Functions and Integrals
Limits                      Limits of expressions.
Differentiation             Differential calculus.
Integration                 Integral calculus.
Equations                   Defining and solving equations.
Differential Equations      Defining and solving differential equations.
Numerical                   Numerical integration, Fourier
                                transforms, etc.
Matrices and Linear Algebra Matrix operations.
Affine
itensor                     Indicial Tensor Manipulation.
ctensor                     Component Tensor Manipulation.
atensor                     Algebraic Tensor Manipulation.
* Sums Products and Series::    Sums, Products, Taylor and power series.
Number Theory               Number theory.
Symmetries
Groups                      Abstract algebra.

Advanced facilities and programming
Runtime Environment         Customization of the Maxima environment.
Miscellaneous Options       Options with a global effect on Maxima.
Rules and Patterns          User defined pattern matching and
                                simplification rules.
Sets                        Manipulation of sets.
Function Definition         Defining functions.
Program Flow                Defining Maxima programs.
Debugging                   Debugging Maxima programs.

Additional packages
asympa                      Asymptotic analysis package
augmented_lagrangian        augmented_lagrangian package.
Bernstein                   Bernstein polynomials.
bode                        Bode gain and phase plots.
cobyla                      Nonlinear optimization with inequality constraints.
contrib_ode                 Additional routines for ODEs
descriptive                 Descriptive statistics.
diag                        Jordan matrices.
distrib                     Probability distributions.
draw                        A Maxima-Gnuplot interface.
drawdf                      Direction fields with Gnuplot.
dynamics                    Graphics for dynamical systems and fractals.
ezunits                     Dimensional quantities.
f90                         Maxima to fortran translator.
finance                     Financial package.
fractals                    Fractals.
ggf                         Generating function of sequences.
graphs                      Graph theory package.
grobner                     Functions for working with Groebner bases.
impdiff                     Implicit derivatives.
interpol                    Interpolation package.
lapack                      LAPACK functions for linear algebra.
lbfgs                       L-BFGS unconstrained minimization package.
lindstedt                   Lindstedt package.
linearalgebra               Functions for linear algebra.
lsquares                    Least squares.
makeOrders                  Polynomial utility.
minpack                     MINPACK functions for minimization and roots
mnewton                     Newton's method.
numericalio                 Reading and writing files.
opsubst                     Substitutions utility.
orthopoly                   Orthogonal polynomials.
plotdf                      Direction fields plots.
romberg                     Romberg method for numerical integration.
simplex                     Linear programming.
simplification              Simplification rules and functions.
solve_rec                   Linear recurrences.
stats                       Statistical inference package.
stirling                    Stirling formula.
stringproc                  String processing.
to_poly_solve               to_poly_solve package.
unit                        Units and dimensions package.
zeilberger                  Functions for hypergeometric summation.

Index
* Function and Variable Index:: Index.

 --- The Detailed Node Listing ---

Introduction
Introduction to Maxima

Bugs
Bug Detection and Reporting

Help
Documentation
Functions and Variables for Help

Command Line
Introduction to Command Line
Functions and Variables for Command Line
Functions and Variables for Display

Data Types and Structures
Numbers
Strings
Constants
Lists
Arrays
Structures

Expressions
Introduction to Expressions
Nouns and Verbs
Identifiers
Inequality
Functions and Variables for Expressions

Operators
Introduction to operators
Arithmetic operators
Relational operators
Logical operators
Operators for Equations
Assignment operators
User defined operators

Evaluation
Functions and Variables for Evaluation

Simplification
Functions and Variables for Simplification

Mathematical Functions
Functions for Numbers
Functions for Complex Numbers
Combinatorial Functions
* Root Exponential and Logarithmic Functions::
Trigonometric Functions
Random Numbers

Maximas Database
Introduction to Maximas Database
Functions and Variables for Properties
Functions and Variables for Facts
Functions and Variables for Predicates

Plotting
Introduction to Plotting
Plotting Formats
Functions and Variables for Plotting
Plotting Options
Gnuplot Options
Gnuplot_pipes Format Functions

File Input and Output
Comments
Files
Functions and Variables for File Input and Output
Functions and Variables for TeX Output
Functions and Variables for Fortran Output

Polynomials
Introduction to Polynomials
Functions and Variables for Polynomials

Special Functions
Introduction to Special Functions
Bessel Functions
Airy Functions
Gamma and factorial Functions
Exponential Integrals
Error Function
Struve Functions
Hypergeometric Functions
Parabolic Cylinder Functions
Functions and Variables for Special Functions

Elliptic Functions
Introduction to Elliptic Functions and Integrals
Functions and Variables for Elliptic Functions
Functions and Variables for Elliptic Integrals

Limits
Functions and Variables for Limits

Differentiation
Functions and Variables for Differentiation

Integration
Introduction to Integration
Functions and Variables for Integration

Equations
Functions and Variables for Equations

Differential Equations
Introduction to Differential Equations
Functions and Variables for Differential Equations

Numerical
Introduction to fast Fourier transform
Functions and Variables for fast Fourier transform
Introduction to Fourier series
Functions and Variables for Fourier series

Matrices and Linear Algebra
Introduction to Matrices and Linear Algebra
* Dot::
* Vectors::
* eigen::
Functions and Variables for Matrices and Linear Algebra

Affine
Introduction to Affine
Functions and Variables for Affine

itensor
Introduction to itensor
Functions and Variables for itensor

ctensor
Introduction to ctensor
Functions and Variables for ctensor

atensor
Introduction to atensor
Functions and Variables for atensor

Sums, Products, and Series
Sums and Products
Introduction to Series
Functions and Variables for Series
Poisson series

Number Theory
Functions and Variables for Number Theory

Symmetries
Introduction to Symmetries
Functions and Variables for Symmetries

Groups
Functions and Variables for Groups

Runtime Environment
Introduction for Runtime Environment
Interrupts
Functions and Variables for Runtime Environment

Miscellaneous Options
Introduction to Miscellaneous Options
Share
Functions and Variables for Miscellaneous Options

Rules and Patterns
Introduction to Rules and Patterns
Functions and Variables for Rules and Patterns

Sets
Introduction to Sets
Functions and Variables for Sets

Function Definition
Introduction to Function Definition
Function
Macros
Functions and Variables for Function Definition

Program Flow
Lisp and Maxima
Garbage Collection
Introduction to Program Flow
Functions and Variables for Program Flow

Debugging
Functions and Variables for Debugging

asympa
Introduction to asympa
Functions and variables for asympa

augmented_lagrangian
Functions and Variables for augmented_lagrangian

Bernstein
Functions and Variables for Bernstein

bode
Functions and Variables for bode

cobyla
Introduction to cobyla
Functions and Variables for cobyla
Examples for cobyla

contrib_ode
Introduction to contrib_ode
Functions and Variables for contrib_ode
Possible improvements to contrib_ode
Test cases for contrib_ode
References for contrib_ode

descriptive
Introduction to descriptive
Functions and Variables for data manipulation
Functions and Variables for descriptive statistics
Functions and Variables for statistical graphs

diag
Functions and Variables for diag

distrib
Introduction to distrib
Functions and Variables for continuous distributions
Functions and Variables for discrete distributions

draw
Introduction to draw
Functions and Variables for draw
Functions and Variables for pictures
Functions and Variables for worldmap

drawdf
Introduction to drawdf
Functions and Variables for drawdf

dynamics
Introduction to dynamics
Functions and Variables for dynamics

ezunits
Introduction to ezunits
Introduction to physical_constants
Functions and Variables for ezunits

f90
Functions and Variables for f90

finance
Introduction to finance
Functions and Variables for finance

fractals
Introduction to fractals
Definitions for IFS fractals
Definitions for complex fractals
Definitions for Koch snowflakes
Definitions for Peano maps

ggf
Functions and Variables for ggf

graphs
Introduction to graphs
Functions and Variables for graphs

grobner
Introduction to grobner
Functions and Variables for grobner

impdiff
Functions and Variables for impdiff

interpol
Introduction to interpol
Functions and Variables for interpol

lapack
Introduction to lapack
Functions and Variables for lapack

lbfgs
Introduction to lbfgs
Functions and Variables for lbfgs

lindstedt
Functions and Variables for lindstedt

linearalgebra
Introduction to linearalgebra
Functions and Variables for linearalgebra

lsquares
Introduction to lsquares
Functions and Variables for lsquares

makeOrders
Functions and Variables for makeOrders

minpack
Introduction to minpack
Functions and Variables for minpack

mnewton
Introduction to mnewton
Functions and Variables for mnewton

numericalio
Introduction to numericalio
Functions and Variables for plain-text input and output
Functions and Variables for binary input and output

opsubst
Functions and Variables for opsubst

orthopoly
Introduction to orthogonal polynomials
Functions and Variables for orthogonal polynomials

plotdf
Introduction to plotdf
Functions and Variables for plotdf

romberg
Functions and Variables for romberg

simplex
Introduction to simplex
Functions and Variables for simplex

simplification
Introduction to simplification
Package absimp
Package facexp
Package functs
Package ineq
Package rducon
Package scifac
Package sqdnst

solve_rec
Introduction to solve_rec
Functions and Variables for solve_rec

stats
Introduction to stats
Functions and Variables for inference_result
Functions and Variables for stats
Functions and Variables for special distributions

stirling
Functions and Variables for stirling

stringproc
Introduction to string processing
Functions and Variables for input and output
Functions and Variables for characters
Functions and Variables for strings

to_poly_solve
Functions and Variables for to_poly_solve

unit
Introduction to Units
Functions and Variables for Units

zeilberger
Introduction to zeilberger
Functions and Variables for zeilberger

1 Introduction to Maxima

Start Maxima with the command "maxima".  Maxima will display version
information and a prompt.  End each Maxima command with a semicolon.
End the session with the command "quit();".  Here's a sample session:

     [wfs@chromium]$ maxima
     Maxima 5.9.1 http://maxima.sourceforge.net
     Using Lisp CMU Common Lisp 19a
     Distributed under the GNU Public License. See the file COPYING.
     Dedicated to the memory of William Schelter.
     This is a development version of Maxima. The function bug_report()
     provides bug reporting information.
     (%i1) factor(10!);
                                 8  4  2
     (%o1)                      2  3  5  7
     (%i2) expand ((x + y)^6);
            6        5       2  4       3  3       4  2      5      6
     (%o2) y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x
     (%i3) factor (x^6 - 1);
                                   2            2
     (%o3)       (x - 1) (x + 1) (x  - x + 1) (x  + x + 1)
     (%i4) quit();
     [wfs@chromium]$

   Maxima can search the info pages.  Use the `describe'  command to
show information about the command or all the commands and variables
containing a string.  The question mark `?'  (exact search) and double
question mark `??'

   (inexact search) are abbreviations for `describe':

     (%i1) ?? integ
      0: Functions and Variables for Elliptic Integrals
      1: Functions and Variables for Integration
      2: Introduction to Elliptic Functions and Integrals
      3: Introduction to Integration
      4: askinteger  (Functions and Variables for Simplification)
      5: integerp  (Functions and Variables for Miscellaneous Options)
      6: integer_partitions  (Functions and Variables for Sets)
      7: integrate  (Functions and Variables for Integration)
      8: integrate_use_rootsof  (Functions and Variables for Integration)
      9: integration_constant_counter  (Functions and Variables for
         Integration)
      10: nonnegintegerp  (Functions and Variables for linearalgebra)
     Enter space-separated numbers, `all' or `none': 5 4

      -- Function: integerp (<expr>)
          Returns `true' if <expr> is a literal numeric integer, otherwise
          `false'.

          `integerp' returns false if its argument is a symbol, even if the
          argument is declared integer.

          Examples:

               (%i1) integerp (0);
               (%o1)                         true
               (%i2) integerp (1);
               (%o2)                         true
               (%i3) integerp (-17);
               (%o3)                         true
               (%i4) integerp (0.0);
               (%o4)                         false
               (%i5) integerp (1.0);
               (%o5)                         false
               (%i6) integerp (%pi);
               (%o6)                         false
               (%i7) integerp (n);
               (%o7)                         false
               (%i8) declare (n, integer);
               (%o8)                         done
               (%i9) integerp (n);
               (%o9)                         false

      -- Function: askinteger (<expr>, integer)
      -- Function: askinteger (<expr>)
      -- Function: askinteger (<expr>, even)
      -- Function: askinteger (<expr>, odd)
          `askinteger (<expr>, integer)' attempts to determine from the
          `assume' database whether <expr> is an integer.  `askinteger'
          prompts the user if it cannot tell otherwise, and attempt to
          install the information in the database if possible.  `askinteger
          (<expr>)' is equivalent to `askinteger (<expr>, integer)'.

          `askinteger (<expr>, even)' and `askinteger (<expr>, odd)'
          likewise attempt to determine if <expr> is an even integer or odd
          integer, respectively.

     (%o1)                                true

   To use a result in later calculations, you can assign it to a
variable or refer to it by its automatically supplied label.  In
addition, `%'

   refers to the most recent calculated result:

     (%i1) u: expand ((x + y)^6);
            6        5       2  4       3  3       4  2      5      6
     (%o1) y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x
     (%i2) diff (u, x);
              5         4       2  3       3  2       4        5
     (%o2) 6 y  + 30 x y  + 60 x  y  + 60 x  y  + 30 x  y + 6 x
     (%i3) factor (%o2);
                                         5
     (%o3)                      6 (y + x)

   Maxima knows about complex numbers and numerical constants:

     (%i1) cos(%pi);
     (%o1)                          - 1
     (%i2) exp(%i*%pi);
     (%o2)                          - 1

   Maxima can do differential and integral calculus:

     (%i1) u: expand ((x + y)^6);
            6        5       2  4       3  3       4  2      5      6
     (%o1) y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x
     (%i2) diff (%, x);
              5         4       2  3       3  2       4        5
     (%o2) 6 y  + 30 x y  + 60 x  y  + 60 x  y  + 30 x  y + 6 x
     (%i3) integrate (1/(1 + x^3), x);
                                       2 x - 1
                     2            atan(-------)
                log(x  - x + 1)        sqrt(3)    log(x + 1)
     (%o3)    - --------------- + ------------- + ----------
                       6             sqrt(3)          3

   Maxima can solve linear systems and cubic equations:

     (%i1) linsolve ([3*x + 4*y = 7, 2*x + a*y = 13], [x, y]);
                             7 a - 52        25
     (%o1)              [x = --------, y = -------]
                             3 a - 8       3 a - 8
     (%i2) solve (x^3 - 3*x^2 + 5*x = 15, x);
     (%o2)       [x = - sqrt(5) %i, x = sqrt(5) %i, x = 3]

   Maxima can solve nonlinear sets of equations.  Note that if you don't
want a result printed, you can finish your command with `$' instead of
`;'.

     (%i1) eq_1: x^2 + 3*x*y + y^2 = 0$
     (%i2) eq_2: 3*x + y = 1$
     (%i3) solve ([eq_1, eq_2]);
                   3 sqrt(5) + 7      sqrt(5) + 3
     (%o3) [[y = - -------------, x = -----------],
                         2                 2

                                    3 sqrt(5) - 7        sqrt(5) - 3
                               [y = -------------, x = - -----------]]
                                          2                   2

   Maxima can generate plots of one or more functions:

     (%i1) plot2d (sin(x)/x, [x, -20, 20])$

     (%i2) plot2d ([atan(x), erf(x), tanh(x)], [x, -5, 5], [y, -1.5, 2])$

     (%i3) plot3d (sin(sqrt(x^2 + y^2))/sqrt(x^2 + y^2),
              [x, -12, 12], [y, -12, 12])$

2 Bug Detection and Reporting


Functions and Variables for Bug Detection and Reporting

2.1 Functions and Variables for Bug Detection and Reporting

Function: run_testsuite ([<options>])

     Run the Maxima test suite.  Tests producing the desired answer are
     considered "passes," as are tests that do not produce the desired
     answer, but are marked as known bugs.

     `run_testsuite' takes the following optional keyword arguments

    <display_all>
          Display all tests.  Normally, the tests are not displayed,
          unless the test fails.  (Defaults to `false').

    <display_known_bugs>
          Displays tests that are marked as known bugs.  (Default is
          `false').

    <tests>
          This is a list of tests that should be run.  Each test can be
          specified by either a string or a symbol.  By default, all
          tests are run.  The complete set of tests is specified by
          `testsuite_files'.

    <time>
          Display time information.  If `true', the time taken for each
          test file is displayed.  If `all', the time for each
          individual test is shown if `display_all' is `true'.  The
          default is `false', so no timing information is shown.

     For example `run_testsuite(display_known_bugs = true,
     tests=[rtest5])' runs just test `rtest5' and displays the test
     that are marked as known bugs.

     `run_testsuite(display_all = true, tests=["rtest1", rtest1a])' will
     run tests `rtest1' and `rtest2', and displays each test.

     `run_testsuite' changes the Maxima environment.  Typically a test
     script executes `kill'  to establish a known environment (namely
     one without user-defined functions and variables) and then defines
     functions and variables appropriate to the test.

     `run_testsuite' returns `done'.

Option variable: testsuite_files

     `testsuite_files' is the set of tests to be run by `run_testsuite'.
      It is a list of names of the files containing the tests to run.
     If some of the tests in a file are known to fail, then instead of
     listing the name of the file, a list containing the file name and
     the test numbers that fail is used.

     For example, this is a part of the default set of tests:

           ["rtest13s", ["rtest14", 57, 63]]

     This specifies the testsuite consists of the files "rtest13s" and
     "rtest14", but "rtest14" contains two tests that are known to
     fail: 57 and 63.

Function: bug_report ()

     Prints out Maxima and Lisp version numbers, and gives a link to
     the Maxima project bug report web page.  The version information
     is the same as reported by `build_info'.

     When a bug is reported, it is helpful to copy the Maxima and Lisp
     version information into the bug report.

     `bug_report' returns an empty string `""'.

Function: build_info ()

     Prints out a summary of the parameters of the Maxima build.

     `build_info' returns an empty string `""'.  See also `bug_report'.

3 Help


Documentation
Functions and Variables for Help

3.1 Documentation

The Maxima on-line user's manual can be viewed in different forms.
From the Maxima interactive prompt, the user's manual is viewed as
plain text by the `?'  command (i.e., the `describe'  function).  The
user's manual is viewed as `info' hypertext by the `info' viewer
program and as a web page by any ordinary web browser.

   `example'  displays examples for many Maxima functions.  For example,

     (%i1) example (integrate);

   yields

     (%i2) test(f):=block([u],u:integrate(f,x),ratsimp(f-diff(u,x)))
     (%o2) test(f) := block([u], u : integrate(f, x),
                                              ratsimp(f - diff(u, x)))
     (%i3) test(sin(x))
     (%o3)                           0
     (%i4) test(1/(x+1))
     (%o4)                           0
     (%i5) test(1/(x^2+1))
     (%o5)                           0

   and additional output.

3.2 Functions and Variables for Help

Function: apropos (<string>)

     Searches for Maxima names which have <string> appearing anywhere
     within them.  Thus, `apropos (exp)' returns a list of all the
     flags and functions which have `exp' as part of their names, such
     as `expand', `exp', and `exponentialize'.  Thus if you can only
     remember part of the name of something you can use this command to
     find the rest of the name.  Similarly, you could say `apropos
     (tr_)' to find a list of many of the switches relating to the
     translator, most of which begin with `tr_'.

     `apropos("")' returns a list with all Maxima names.

     `apropos' returns the empty list `[]', if no name is found.

     Example:

     Show all Maxima symbols which have `"gamma"' in the name:

          (%i1) apropos("gamma");
          (%o1) [%gamma, gamma, gammalim, gamma_expand, gamma_greek,
          gamma_incomplete, gamma_incomplete_generalized,
          gamma_incomplete_regularized, Gamma, log_gamma, makegamma,
          prefer_gamma_incomplete,
          gamma_incomplete_generalized_regularized]

Function: demo (<filename>)

     Evaluates Maxima expressions in <filename> and displays the
     results.  `demo' pauses after evaluating each expression and
     continues after the user enters a carriage return.  (If running in
     Xmaxima, `demo' may need to see a semicolon `;' followed by a
     carriage return.)

     `demo' searches the list of directories `file_search_demo'  to find
     `filename'.  If the file has the suffix `dem', the suffix may be
     omitted.  See also `file_search'.

     `demo' evaluates its argument.  `demo' returns the name of the
     demonstration file.

     Example:

          (%i1) demo ("disol");

          batching /home/wfs/maxima/share/simplification/disol.dem
           At the _ prompt, type ';' followed by enter to get next demo
          (%i2)                      load(disol)

          _
          (%i3)           exp1 : a (e (g + f) + b (d + c))
          (%o3)               a (e (g + f) + b (d + c))

          _
          (%i4)                disolate(exp1, a, b, e)
          (%t4)                         d + c

          (%t5)                         g + f

          (%o5)                   a (%t5 e + %t4 b)

          _

Function: describe (<string>)
Function: describe (<string>, exact)
Function: describe (<string>, inexact)

     `describe(<string>)' is equivalent to `describe(<string>, exact)'.

     `describe(<string>, exact)' finds an item with title equal
     (case-insensitive) to <string>, if there is any such item.

     `describe(<string>, inexact)' finds all documented items which
     contain <string> in their titles.  If there is more than one such
     item, Maxima asks the user to select an item or items to display.

     At the interactive prompt, `? foo' (with a space between `?' and
     `foo') is equivalent to `describe("foo", exact)', and `?? foo' is
     equivalent to `describe("foo", inexact)'.

     `describe("", inexact)' yields a list of all topics documented in
     the on-line manual.

     `describe' quotes its argument.  `describe' returns `true' if some
     documentation is found, otherwise `false'.

     See also *note Documentation::.

     Example:

          (%i1) ?? integ
           0: Functions and Variables for Elliptic Integrals
           1: Functions and Variables for Integration
           2: Introduction to Elliptic Functions and Integrals
           3: Introduction to Integration
           4: askinteger  (Functions and Variables for Simplification)
           5: integerp  (Functions and Variables for Miscellaneous Options)
           6: integer_partitions  (Functions and Variables for Sets)
           7: integrate  (Functions and Variables for Integration)
           8: integrate_use_rootsof  (Functions and Variables for
              Integration)
           9: integration_constant_counter  (Functions and Variables for
              Integration)
           10: nonnegintegerp  (Functions and Variables for linearalgebra)
          Enter space-separated numbers, `all' or `none': 7 8

           -- Function: integrate (<expr>, <x>)
           -- Function: integrate (<expr>, <x>, <a>, <b>)
               Attempts to symbolically compute the integral of <expr> with
               respect to <x>.  `integrate (<expr>, <x>)' is an indefinite
               integral, while `integrate (<expr>, <x>, <a>, <b>)' is a
               definite integral, [...]

           -- Option variable: integrate_use_rootsof
               Default value: `false'

               When `integrate_use_rootsof' is `true' and the denominator of
               a rational function cannot be factored, `integrate' returns
               the integral in a form which is a sum over the roots (not yet
               known) of the denominator.
               [...]

     In this example, items 7 and 8 were selected (output is shortened
     as indicated by `[...]'.  All or none of the items could have been
     selected by entering `all' or `none', which can be abbreviated `a'
     or `n', respectively.

Function: example (<topic>)
Function: example ()

     `example (<topic>)' displays some examples of <topic>, which is a
     symbol or a string.  To get examples for operators like `if', `do',
     or `lambda' the argument must be a string, e.g. `example ("do")'.
     `example' is not case sensitive.  Most topics are function names.

     `example ()' returns the list of all recognized topics.

     The name of the file containing the examples is given by the
     global option variable `manual_demo',  which defaults to
     `"manual.demo"'.

     `example' quotes its argument.  `example' returns `done' unless no
     examples are found or there is no argument, in which case `example'
     returns the list of all recognized topics.

     Examples:

          (%i1) example(append);
          (%i2) append([x+y,0,-3.2],[2.5e+20,x])
          (%o2)             [y + x, 0, - 3.2, 2.5e+20, x]
          (%o2)                         done
          (%i3) example("lambda");

          (%i4) lambda([x,y,z],z^2+y^2+x^2)
                                              2    2    2
          (%o4)            lambda([x, y, z], z  + y  + x )
          (%i5) %(1,2,a)
                                        2
          (%o5)                        a  + 5
          (%i6) a+2+1
          (%o6)                         a + 3
          (%o6)                         done

Option variable: manual_demo

     Default value: `"manual.demo"'

     `manual_demo' specifies the name of the file containing the
     examples for the function `example'.  See `example'.

4 Command Line


Introduction to Command Line
Functions and Variables for Command Line
Functions and Variables for Display

4.1 Introduction to Command Line

4.2 Functions and Variables for Command Line

System variable: __

     `__' is the input expression currently being evaluated.  That is,
     while an input expression <expr> is being evaluated, `__' is
     <expr>.

     `__' is assigned the input expression before the input is
     simplified or evaluated.  However, the value of `__' is simplified
     (but not evaluated) when it is displayed.

     `__' is recognized by `batch'  and `load'.    In a file processed
     by `batch', `__' has the same meaning as at the interactive prompt.
     In a file processed by `load', `__' is bound to the input
     expression most recently entered at the interactive prompt or in a
     batch file; `__' is not bound to the input expressions in the file
     being processed.  In particular, when `load (<filename>)' is
     called from the interactive prompt, `__' is bound to `load
     (<filename>)' while the file is being processed.

     See also `_'  and `%'.

     Examples:

          (%i1) print ("I was called as", __);
          I was called as print(I was called as, __)
          (%o1)              print(I was called as, __)
          (%i2) foo (__);
          (%o2)                     foo(foo(__))
          (%i3) g (x) := (print ("Current input expression =", __), 0);
          (%o3) g(x) := (print("Current input expression =", __), 0)
          (%i4) [aa : 1, bb : 2, cc : 3];
          (%o4)                       [1, 2, 3]
          (%i5) (aa + bb + cc)/(dd + ee + g(x));
                                      cc + bb + aa
          Current input expression = --------------
                                     g(x) + ee + dd
                                          6
          (%o5)                        -------
                                       ee + dd

System variable: _

     `_' is the most recent input expression (e.g., `%i1', `%i2',
     `%i3', ...).

     `_' is assigned the input expression before the input is
     simplified or evaluated.  However, the value of `_' is simplified
     (but not evaluated) when it is displayed.

     `_' is recognized by `batch'  and `load'.    In a file processed
     by `batch', `_' has the same meaning as at the interactive prompt.
     In a file processed by `load', `_' is bound to the input expression
     most recently evaluated at the interactive prompt or in a batch
     file; `_' is not bound to the input expressions in the file being
     processed.

     See also `__'  and `%'.

     Examples:

          (%i1) 13 + 29;
          (%o1)                          42
          (%i2) :lisp $_
          ((MPLUS) 13 29)
          (%i2) _;
          (%o2)                          42
          (%i3) sin (%pi/2);
          (%o3)                           1
          (%i4) :lisp $_
          ((%SIN) ((MQUOTIENT) $%PI 2))
          (%i4) _;
          (%o4)                           1
          (%i5) a: 13$
          (%i6) b: 29$
          (%i7) a + b;
          (%o7)                          42
          (%i8) :lisp $_
          ((MPLUS) $A $B)
          (%i8) _;
          (%o8)                         b + a
          (%i9) a + b;
          (%o9)                          42
          (%i10) ev (_);
          (%o10)                         42

System variable: %

     `%' is the output expression (e.g., `%o1', `%o2', `%o3', ...) most
     recently computed by Maxima, whether or not it was displayed.

     `%' is recognized by `batch'  and `load'.    In a file processed
     by `batch', `%' has the same meaning as at the interactive prompt.
     In a file processed by `load', `%' is bound to the output
     expression most recently computed at the interactive prompt or in
     a batch file; `%' is not bound to output expressions in the file
     being processed.

     See also `_',  `%%',  and `%th'.

System variable: %%

     In compound statements, namely `block',  `lambda',  or `(<s_1>,
     ..., <s_n>)', `%%' is the value of the previous statement.

     At the first statement in a compound statement, or outside of a
     compound statement, `%%' is undefined.

     `%%' is recognized by `batch'  and `load',  and it has the same
     meaning as at the interactive prompt.

     See also `%'.

     Examples:

     The following two examples yield the same result.

          (%i1) block (integrate (x^5, x), ev (%%, x=2) - ev (%%, x=1));
                                         21
          (%o1)                          --
                                         2
          (%i2) block ([prev], prev: integrate (x^5, x),
                         ev (prev, x=2) - ev (prev, x=1));
                                         21
          (%o2)                          --
                                         2

     A compound statement may comprise other compound statements.
     Whether a statement be simple or compound, `%%' is the value of
     the previous statement.

          (%i3) block (block (a^n, %%*42), %%/6);
                                           n
          (%o3)                         7 a

     Within a compound statement, the value of `%%' may be inspected at
     a break prompt, which is opened by executing the `break'
     function.  For example, entering `%%;' in the following example
     yields `42'.

          (%i4) block (a: 42, break ())$

          Entering a Maxima break point. Type 'exit;' to resume.
          _%%;
          42
          _

Function: %th (<i>)

     The value of the <i>'th previous output expression.  That is, if
     the next expression to be computed is the <n>'th output, `%th
     (<m>)' is the (<n> - <m>)'th output.

     `%th' is recognized by `batch'  and `load'.    In a file processed
     by `batch', `%th' has the same meaning as at the interactive
     prompt.  In a file processed by `load', `%th' refers to output
     expressions most recently computed at the interactive prompt or in
     a batch file; `%th' does not refer to output expressions in the
     file being processed.

     See also `%'  and `%%'.

     Example:

     `%th' is useful in `batch' files or for referring to a group of
     output expressions.  This example sets `s' to the sum of the last
     five output expressions.

          (%i1) 1;2;3;4;5;
          (%o1)                           1
          (%o2)                           2
          (%o3)                           3
          (%o4)                           4
          (%o5)                           5
          (%i6) block (s: 0, for i:1 thru 5 do s: s + %th(i), s);
          (%o6)                          15

Special symbol: ?

     As prefix to a function or variable name, `?' signifies that the
     name is a Lisp name, not a Maxima name.  For example, `?round'
     signifies the Lisp function `ROUND'.  See *note Lisp and Maxima::
     for more on this point.

     The notation `? word' (a question mark followed a word, separated
     by whitespace) is equivalent to `describe("word")'.  The question
     mark must occur at the beginning of an input line; otherwise it is
     not recognized as a request for documentation.  See also
     `describe'.

Special symbol: ??

     The notation `?? word' (`??' followed a word, separated by
     whitespace) is equivalent to `describe("word", inexact)'.  The
     question mark must occur at the beginning of an input line;
     otherwise it is not recognized as a request for documentation.
     See also `describe'.

Option variable: inchar

     Default value: `%i'

     `inchar' is the prefix of the labels of expressions entered by the
     user.  Maxima automatically constructs a label for each input
     expression by concatenating `inchar' and `linenum'.

     `inchar' may be assigned any string or symbol, not necessarily a
     single character.  Because Maxima internally takes into account
     only the first char of the prefix, the prefixes `inchar',
     `outchar',  and `linechar'  should have a different first char.
     Otherwise some commands like `kill(inlables)' do not work as
     expected.

     See also `labels'.

     Example:

          (%i1) inchar: "input";
          (%o1)                         input
          (input2) expand((a+b)^3);
                               3        2      2      3
          (%o2)               b  + 3 a b  + 3 a  b + a
          (input3)

System variable: infolists

     Default value: `[]'

     `infolists' is a list of the names of all of the information lists
     in Maxima.  These are:

    `labels'
          All bound `%i', `%o', and `%t' labels.

    `values'
          All bound atoms which are user variables, not Maxima options
          or switches, created by `:'  or `::'  or functional binding.

    `functions'
          All user-defined functions, created by `:='  or `define'.

    `arrays'
          All declared and undeclared arrays, created by `:', `::', or
          `:='.

    `macros'
          All user-defined macro functions, created by `::='.

    `myoptions'
          All options ever reset by the user (whether or not they are
          later reset to their default values).

    `rules'
          All user-defined pattern matching and simplification rules,
          created by `tellsimp',  `tellsimpafter',  `defmatch',  or
          `defrule'.

    `aliases'
          All atoms which have a user-defined alias, created by the
          `alias',

          `ordergreat',  `orderless'  functions or by declaring the
          atom as a `noun'  with `declare'.

    `dependencies'
          All atoms which have functional dependencies, created by the
          `depends'  or `gradef'  functions.

    `gradefs'
          All functions which have user-defined derivatives, created by
          the `gradef'  function.

    `props'
          All atoms which have any property other than those mentioned
          above, such as properties established by `atvalue'  or
          `matchdeclare' , etc., as well as properties established in
          the `declare'  function.

    `let_rule_packages'
          All user-defined `let'  rule packages plus the special
          package `default_let_rule_package'.

          (`default_let_rule_package' is the name of the rule package
          used when one is not explicitly set by the user.)

Function: kill (<a_1>, ..., <a_n>)
Function: kill (labels)
Function: kill (inlabels, outlabels, linelabels)
Function: kill (<n>)
Function: kill ([<m>, <n>])
Function: kill (values, functions, arrays, ...)
Function: kill (all)
Function: kill (allbut (<a_1>, ..., <a_n>))

     Removes all bindings (value, function, array, or rule) from the
     arguments <a_1>, ..., <a_n>.  An argument <a_k> may be a symbol or
     a single array element.  When <a_k> is a single array element,
     `kill' unbinds that element without affecting any other elements
     of the array.

     Several special arguments are recognized.  Different kinds of
     arguments may be combined, e.g., `kill (inlabels, functions,
     allbut (foo, bar))'.

     `kill (labels)' unbinds all input, output, and intermediate
     expression labels created so far.  `kill (inlabels)' unbinds only
     input labels which begin with the current value of `inchar'.
     Likewise, `kill (outlabels)' unbinds only output labels which
     begin with the current value of `outchar',  and `kill
     (linelabels)' unbinds only intermediate expression labels which
     begin with the current value of `linechar'.

     `kill (<n>)', where <n> is an integer, unbinds the <n> most recent
     input and output labels.

     `kill ([<m>, <n>])' unbinds input and output labels <m> through
     <n>.

     `kill (<infolist>)', where <infolist> is any item in `infolists'
     (such as `values',  `functions',  or `arrays' ) unbinds all items
     in <infolist>.  See also `infolists'.

     `kill (all)' unbinds all items on all infolists.  `kill (all)' does
     not reset global variables to their default values; see `reset'
     on this point.

     `kill (allbut (<a_1>, ..., <a_n>))' unbinds all items on all
     infolists except for <a_1>, ..., <a_n>.  `kill (allbut
     (<infolist>))' unbinds all items except for the ones on
     <infolist>, where <infolist> is `values',

     `functions',  `arrays',  etc.

     The memory taken up by a bound property is not released until all
     symbols are unbound from it.  In particular, to release the memory
     taken up by the value of a symbol, one unbinds the output label
     which shows the bound value, as well as unbinding the symbol
     itself.

     `kill' quotes its arguments.  The quote-quote operator `'''
     defeats quotation.

     `kill (<symbol>)' unbinds all properties of <symbol>.  In contrast,
     the functions `remvalue',  `remfunction',

     `remarray',  and `remrule'  unbind a specific property.

     `kill' always returns `done', even if an argument has no binding.

Function: labels (<symbol>)
System variable: labels

     Returns the list of input, output, or intermediate expression
     labels which begin with <symbol>.  Typically <symbol> is the value
     of `inchar',  `outchar',  or `linechar'.    The label character
     may be given with or without a percent sign, so, for example, `i'
     and `%i' yield the same result.

     If no labels begin with <symbol>, `labels' returns an empty list.

     The function `labels' quotes its argument.  The quote-quote
     operator `''' defeats quotation.  For example, `labels (''inchar)'
     returns the input labels which begin with the current input label
     character.

     The variable `labels' is the list of input, output, and
     intermediate expression labels, including all previous labels if
     `inchar', `outchar', or `linechar' were redefined.

     By default, Maxima displays the result of each user input
     expression, giving the result an output label.  The output display
     is suppressed by terminating the input with `$' (dollar sign)
     instead of `;' (semicolon).  An output label is constructed and
     bound to the result, but not displayed, and the label may be
     referenced in the same way as displayed output labels.  See also
     `%',  `%%',  and `%th'.

     Intermediate expression labels can be generated by some functions.
     The option variable `programmode'  controls whether `solve'  and
     some other functions generate intermediate expression labels
     instead of returning a list of expressions.  Some other functions,
     such as `ldisplay',  always generate intermediate expression
     labels.

     See also `inchar',  `outchar',  `linechar',  and `infolists'.

Option variable: linechar

     Default value: `%t'

     `linechar' is the prefix of the labels of intermediate expressions
     generated by Maxima.  Maxima constructs a label for each
     intermediate expression (if displayed) by concatenating `linechar'
     and `linenum'.

     `linechar' may be assigned any string or symbol, not necessarily a
     single character.  Because Maxima internally takes into account
     only the first char of the prefix, the prefixes `inchar',
     `outchar',  and `linechar' should have a different first char.
     Otherwise some commands like `kill(inlables)' do not work as
     expected.

     Intermediate expressions might or might not be displayed.  See
     `programmode'  and `labels'.

System variable: linenum

     The line number of the current pair of input and output
     expressions.

System variable: myoptions

     Default value: `[]'

     `myoptions' is the list of all options ever reset by the user,
     whether or not they get reset to their default value.

Option variable: nolabels

     Default value: `false'

     When `nolabels' is `true', input and output result labels (`%i'
     and `%o', respectively) are displayed, but the labels are not
     bound to results, and the labels are not appended to the `labels'
     list.  Since labels are not bound to results, garbage collection
     can recover the memory taken up by the results.

     Otherwise input and output result labels are bound to results, and
     the labels are appended to the `labels' list.

     Intermediate expression labels (`%t') are not affected by
     `nolabels'; whether `nolabels' is `true' or `false', intermediate
     expression labels are bound and appended to the `labels' list.

     See also `batch',  `load',  and `labels'.

Option variable: optionset

     Default value: `false'

     When `optionset' is `true', Maxima prints out a message whenever a
     Maxima option is reset.  This is useful if the user is doubtful of
     the spelling of some option and wants to make sure that the
     variable he assigned a value to was truly an option variable.

     Example:

          (%i1) optionset:true;
          assignment: assigning to option optionset
          (%o1)                         true
          (%i2) gamma_expand:true;
          assignment: assigning to option gamma_expand
          (%o2)                         true

Option variable: outchar

     Default value: `%o'

     `outchar' is the prefix of the labels of expressions computed by
     Maxima.  Maxima automatically constructs a label for each computed
     expression by concatenating `outchar' and `linenum'.

     `outchar' may be assigned any string or symbol, not necessarily a
     single character.  Because Maxima internally takes into account
     only the first char of the prefix, the prefixes `inchar',
     `outchar' and `linechar'  should have a different first char.
     Otherwise some commands like `kill(inlables)' do not work as
     expected.

     See also `labels'.

     Example:

          (%i1) outchar: "output";
          (output1)                    output
          (%i2) expand((a+b)^3);
                               3        2      2      3
          (output2)           b  + 3 a b  + 3 a  b + a
          (%i3)

Function: playback ()
Function: playback (<n>)
Function: playback ([<m>, <n>])
Function: playback ([<m>])
Function: playback (input)
Function: playback (slow)
Function: playback (time)
Function: playback (grind)

     Displays input, output, and intermediate expressions, without
     recomputing them.  `playback' only displays the expressions bound
     to labels; any other output (such as text printed by `print'  or
     `describe',  or error messages) is not displayed.  See also
     `labels'.

     `playback' quotes its arguments.  The quote-quote operator `'''
     defeats quotation.  `playback' always returns `done'.

     `playback ()' (with no arguments) displays all input, output, and
     intermediate expressions generated so far.  An output expression
     is displayed even if it was suppressed by the `$' terminator when
     it was originally computed.

     `playback (<n>)' displays the most recent <n> expressions.  Each
     input, output, and intermediate expression counts as one.

     `playback ([<m>, <n>])' displays input, output, and intermediate
     expressions with numbers from <m> through <n>, inclusive.

     `playback ([<m>])' is equivalent to `playback ([<m>, <m>])'; this
     usually prints one pair of input and output expressions.

     `playback (input)' displays all input expressions generated so far.

     `playback (slow)' pauses between expressions and waits for the
     user to press `enter'.  This behavior is similar to `demo'.

     `playback (slow)' is useful in conjunction with `save' or
     `stringout'  when creating a secondary-storage file in order to
     pick out useful expressions.

     `playback (time)' displays the computation time for each
     expression.

     `playback (grind)' displays input expressions in the same format
     as the `grind' function.  Output expressions are not affected by
     the `grind' option.  See `grind'.

     Arguments may be combined, e.g., `playback ([5, 10], grind, time,
     slow)'.

Option variable: prompt

     Default value: `_'

     `prompt' is the prompt symbol of the `demo'  function, `playback
     (slow)' mode, and the Maxima break loop (as invoked by `break' ).

Function: quit ()

     Terminates the Maxima session.  Note that the function must be
     invoked as `quit();' or `quit()$', not `quit' by itself.

     To stop a lengthy computation, type `control-C'.  The default
     action is to return to the Maxima prompt.  If `*debugger-hook*' is
     `nil', `control-C' opens the Lisp debugger.  See also *note
     Debugging::.

Function: read (<expr_1>, ..., <expr_n>)

     Prints <expr_1>, ..., <expr_n>, then reads one expression from the
     console and returns the evaluated expression.  The expression is
     terminated with a semicolon `;' or dollar sign `$'.

     See also `readonly'

     Example:

          (%i1) foo: 42$
          (%i2) foo: read ("foo is", foo, " -- enter new value.")$
          foo is 42  -- enter new value.
          (a+b)^3;
          (%i3) foo;
                                               3
          (%o3)                         (b + a)

Function: readonly (<expr_1>, ..., <expr_n>)

     Prints <expr_1>, ..., <expr_n>, then reads one expression from the
     console and returns the expression (without evaluation).  The
     expression is terminated with a `;' (semicolon) or `$' (dollar
     sign).

     See also `read'.

     Examples:

          (%i1) aa: 7$
          (%i2) foo: readonly ("Enter an expression:");
          Enter an expression:
          2^aa;
                                            aa
          (%o2)                            2
          (%i3) foo: read ("Enter an expression:");
          Enter an expression:
          2^aa;
          (%o3)                            128

Function: reset ()

     Resets many global variables and options, and some other
     variables, to their default values.

     `reset' processes the variables on the Lisp list
     `*variable-initial-values*'.  The Lisp macro `defmvar' puts
     variables on this list (among other actions).  Many, but not all,
     global variables and options are defined by `defmvar', and some
     variables defined by `defmvar' are not global variables or options.

Option variable: showtime

     Default value: `false'

     When `showtime' is `true', the computation time and elapsed time is
     printed with each output expression.

     The computation time is always recorded, so `time'  and `playback'
     can display the computation time even when `showtime' is `false'.

     See also `timer'.

Function: to_lisp ()

     Enters the Lisp system under Maxima.  `(to-maxima)' returns to
     Maxima.

     Example:

     Define a function and enter the Lisp system under Maxima.  The
     definition is inspected on the property list, then the function
     definition is extracted, factored and stored in the variable
     $result.  The variable can be used in Maxima after returning to
     Maxima.

          (%i1) f(x):=x^2+x;
                                            2
          (%o1)                    f(x) := x  + x
          (%i2) to_lisp();
          Type (to-maxima) to restart, ($quit) to quit Maxima.
          MAXIMA> (symbol-plist '$f)
          (MPROPS (NIL MEXPR ((LAMBDA) ((MLIST) $X)
                                       ((MPLUS) ((MEXPT) $X 2) $X))))
          MAXIMA> (setq $result ($factor (caddr (mget '$f 'mexpr))))
          ((MTIMES SIMP FACTORED) $X ((MPLUS SIMP IRREDUCIBLE) 1 $X))
          MAXIMA> (to-maxima)
          Returning to Maxima
          (%o2)                         true
          (%i3) result;
          (%o3)                       x (x + 1)

System variable: values

     Initial value: `[]'

     `values' is a list of all bound user variables (not Maxima options
     or switches).  The list comprises symbols bound by `:',  or `::'.

     If the value of a variable is removed with the commands `kill',
     `remove',  or `remvalue'  the variable is deleted from `values'.

     See `functions'  for a list of user defined functions.

     Examples:

     First, `values' shows the symbols `a', `b', and `c', but not `d',
     it is not bound to a value, and not the user function `f'.  The
     values are removed from the variables.  `values' is the empty list.

          (%i1) [a:99, b::a-90, c:a-b, d, f(x):= x^2];
                                                     2
          (%o1)              [99, 9, 90, d, f(x) := x ]
          (%i2) values;
          (%o2)                       [a, b, c]
          (%i3) [kill(a), remove(b,value), remvalue(c)];
          (%o3)                   [done, done, [c]]
          (%i4) values;
          (%o4)                          []

4.3 Functions and Variables for Display

Option variable: %edispflag

     Default value: `false'

     When `%edispflag' is `true', Maxima displays `%e' to a negative
     exponent as a quotient.  For example, `%e^-x' is displayed as
     `1/%e^x'.  See also `exptdispflag'.

     Example:

          (%i1) %e^-10;
                                         - 10
          (%o1)                        %e
          (%i2) %edispflag:true$
          (%i3) %e^-10;
                                         1
          (%o3)                         ----
                                          10
                                        %e

Option variable: absboxchar

     Default value: `!'

     `absboxchar' is the character used to draw absolute value signs
     around expressions which are more than one line tall.

     Example:

          (%i1) abs((x^3+1));
                                      ! 3    !
          (%o1)                       !x  + 1!

Function: disp (<expr_1>, <expr_2>, ...)

     is like `display'  but only the value of the arguments are
     displayed rather than equations.  This is useful for complicated
     arguments which don't have names or where only the value of the
     argument is of interest and not the name.

     See also `ldisp'  and `print'.

     Example:

          (%i1) b[1,2]:x-x^2$
          (%i2) x:123$
          (%i3) disp(x, b[1,2], sin(1.0));
                                         123

                                            2
                                       x - x

                                  .8414709848078965

          (%o3)                         done

Function: display (<expr_1>, <expr_2>, ...)

     Displays equations whose left side is <expr_i> unevaluated, and
     whose right side is the value of the expression centered on the
     line.  This function is useful in blocks and `for'  statements in
     order to have intermediate results displayed.  The arguments to
     `display' are usually atoms, subscripted variables, or function
     calls.

     See also `ldisplay',  `disp',  and `ldisp'.

     Example:

          (%i1) b[1,2]:x-x^2$
          (%i2) x:123$
          (%i3) display(x, b[1,2], sin(1.0));
                                       x = 123

                                                2
                                   b     = x - x
                                    1, 2

                            sin(1.0) = .8414709848078965

          (%o3)                         done

Option variable: display2d

     Default value: `true'

     When `display2d' is `false', the console display is a string
     (1-dimensional) form rather than a display (2-dimensional) form.

     See also `leftjust'  to switch between a left justified and a
     centered display of equations.

     Example:

          (%i1) x/(x^2+1);
                                         x
          (%o1)                        ------
                                        2
                                       x  + 1
          (%i2) display2d:false$
          (%i3) x/(x^2+1);
          (%o3) x/(x^2+1)

Option variable: display_format_internal

     Default value: `false'

     When `display_format_internal' is `true', expressions are displayed
     without being transformed in ways that hide the internal
     mathematical representation.  The display then corresponds to what
     `inpart'  returns rather than `part'.

     Examples:

          User     part       inpart
          a-b;      a - b     a + (- 1) b

                     a            - 1
          a/b;       -         a b
                     b
                                 1/2
          sqrt(x);   sqrt(x)    x

                    4 X        4
          X*4/3;    ---        - X
                     3         3

Function: dispterms (<expr>)

     Displays <expr> in parts one below the other.  That is, first the
     operator of <expr> is displayed, then each term in a sum, or
     factor in a product, or part of a more general expression is
     displayed separately.  This is useful if <expr> is too large to be
     otherwise displayed.  For example if `P1', `P2', ...  are very
     large expressions then the display program may run out of storage
     space in trying to display `P1 + P2 + ...'  all at once.  However,
     `dispterms (P1 + P2 + ...)' displays `P1', then below it `P2',
     etc.  When not using `dispterms', if an exponential expression is
     too wide to be displayed as `A^B' it appears as `expt (A, B)' (or
     as `ncexpt (A, B)' in the case of `A^^B').

     Example:

          (%i1) dispterms(2*a*sin(x)+%e^x);

          +

          2 a sin(x)

            x
          %e

          (%o1)                         done

Special symbol: expt (<a>, <b>)
Special symbol: ncexpt (<a>, <b>)

     If an exponential expression is too wide to be displayed as
     `<a>^<b>' it appears as `expt (<a>, <b>)' (or as `ncexpt (<a>,
     <b>)' in the case of `<a>^^<b>').

     `expt' and `ncexpt' are not recognized in input.

Option variable: exptdispflag

     Default value: `true'

     When `exptdispflag' is `true', Maxima displays expressions with
     negative exponents using quotients.  See also `%edispflag'.

     Example:

          (%i1) exptdispflag:true;
          (%o1)                         true
          (%i2) 10^-x;
                                          1
          (%o2)                          ---
                                           x
                                         10
          (%i3) exptdispflag:false;
          (%o3)                         false
          (%i4) 10^-x;
                                          - x
          (%o4)                         10

Function: grind (<expr>)
Option variable: grind

     The function `grind' prints <expr> to the console in a form
     suitable for input to Maxima.  `grind' always returns `done'.

     When <expr> is the name of a function or macro, `grind' prints the
     function or macro definition instead of just the name.

     See also `string',  which returns a string instead of printing its
     output.  `grind' attempts to print the expression in a manner
     which makes it slightly easier to read than the output of `string'.

     When the variable `grind' is `true', the output of `string' and
     `stringout'  has the same format as that of `grind'; otherwise no
     attempt is made to specially format the output of those functions.
     The default value of the variable `grind' is `false'.

     `grind' can also be specified as an argument of `playback'.    When
     `grind' is present, `playback' prints input expressions in the same
     format as the `grind' function.  Otherwise, no attempt is made to
     specially format input expressions.

     `grind' evaluates its argument.

     Examples:

          (%i1) aa + 1729;
          (%o1)                       aa + 1729
          (%i2) grind (%);
          aa+1729$
          (%o2)                         done
          (%i3) [aa, 1729, aa + 1729];
          (%o3)                 [aa, 1729, aa + 1729]
          (%i4) grind (%);
          [aa,1729,aa+1729]$
          (%o4)                         done
          (%i5) matrix ([aa, 17], [29, bb]);
                                     [ aa  17 ]
          (%o5)                      [        ]
                                     [ 29  bb ]
          (%i6) grind (%);
          matrix([aa,17],[29,bb])$
          (%o6)                         done
          (%i7) set (aa, 17, 29, bb);
          (%o7)                   {17, 29, aa, bb}
          (%i8) grind (%);
          {17,29,aa,bb}$
          (%o8)                         done
          (%i9) exp (aa / (bb + 17)^29);
                                          aa
                                      -----------
                                               29
                                      (bb + 17)
          (%o9)                     %e
          (%i10) grind (%);
          %e^(aa/(bb+17)^29)$
          (%o10)                        done
          (%i11) expr: expand ((aa + bb)^10);
                   10           9        2   8         3   7         4   6
          (%o11) bb   + 10 aa bb  + 45 aa  bb  + 120 aa  bb  + 210 aa  bb
                   5   5         6   4         7   3        8   2
           + 252 aa  bb  + 210 aa  bb  + 120 aa  bb  + 45 aa  bb
                  9        10
           + 10 aa  bb + aa
          (%i12) grind (expr);
          bb^10+10*aa*bb^9+45*aa^2*bb^8+120*aa^3*bb^7+210*aa^4*bb^6
               +252*aa^5*bb^5+210*aa^6*bb^4+120*aa^7*bb^3+45*aa^8*bb^2
               +10*aa^9*bb+aa^10$
          (%o12)                        done
          (%i13) string (expr);
          (%o13) bb^10+10*aa*bb^9+45*aa^2*bb^8+120*aa^3*bb^7+210*aa^4*bb^6\
          +252*aa^5*bb^5+210*aa^6*bb^4+120*aa^7*bb^3+45*aa^8*bb^2+10*aa^9*\
          bb+aa^10
          (%i14) cholesky (A):= block ([n : length (A), L : copymatrix (A),
            p : makelist (0, i, 1, length (A))], for i thru n do
            for j : i thru n do
            (x : L[i, j], x : x - sum (L[j, k] * L[i, k], k, 1, i - 1),
            if i = j then p[i] : 1 / sqrt(x) else L[j, i] : x * p[i]),
            for i thru n do L[i, i] : 1 / p[i],
            for i thru n do for j : i + 1 thru n do L[i, j] : 0, L)$
          (%i15) grind (cholesky);
          cholesky(A):=block(
                   [n:length(A),L:copymatrix(A),
                    p:makelist(0,i,1,length(A))],
                   for i thru n do
                       (for j from i thru n do
                            (x:L[i,j],x:x-sum(L[j,k]*L[i,k],k,1,i-1),
                             if i = j then p[i]:1/sqrt(x)
                                 else L[j,i]:x*p[i])),
                   for i thru n do L[i,i]:1/p[i],
                   for i thru n do (for j from i+1 thru n do L[i,j]:0),L)$
          (%o15)                        done
          (%i16) string (fundef (cholesky));
          (%o16) cholesky(A):=block([n:length(A),L:copymatrix(A),p:makelis\
          t(0,i,1,length(A))],for i thru n do (for j from i thru n do (x:L\
          [i,j],x:x-sum(L[j,k]*L[i,k],k,1,i-1),if i = j then p[i]:1/sqrt(x\
          ) else L[j,i]:x*p[i])),for i thru n do L[i,i]:1/p[i],for i thru \
          n do (for j from i+1 thru n do L[i,j]:0),L)

Option variable: ibase

     Default value: `10'

     `ibase' is the base for integers read by Maxima.

     `ibase' may be assigned any integer between 2 and 36 (decimal),
     inclusive.  When `ibase' is greater than 10, the numerals comprise
     the decimal numerals 0 through 9 plus letters of the alphabet A,
     B, C, ..., as needed to make `ibase' digits in all.  Letters are
     interpreted as digits only if the first digit is 0 through 9.
     Uppercase and lowercase letters are not distinguished.  The
     numerals for base 36, the largest acceptable base, comprise 0
     through 9 and A through Z.

     Whatever the value of `ibase', when an integer is terminated by a
     decimal point, it is interpreted in base 10.

     See also `obase'.

     Examples:

     `ibase' less than 10.

          (%i1) ibase : 2 $
          (%i2) obase;
          (%o2)                          10
          (%i3) 1111111111111111;
          (%o3)                         65535

     `ibase' greater than 10.  Letters are interpreted as digits only
     if the first digit is 0 through 9.

          (%i1) ibase : 16 $
          (%i2) obase;
          (%o2)                          10
          (%i3) 1000;
          (%o3)                         4096
          (%i4) abcd;
          (%o4)                         abcd
          (%i5) symbolp (abcd);
          (%o5)                         true
          (%i6) 0abcd;
          (%o6)                         43981
          (%i7) symbolp (0abcd);
          (%o7)                         false

     When an integer is terminated by a decimal point, it is
     interpreted in base 10.

          (%i1) ibase : 36 $
          (%i2) obase;
          (%o2)                          10
          (%i3) 1234;
          (%o3)                         49360
          (%i4) 1234.;
          (%o4)                         1234

Function: ldisp (<expr_1>, ..., <expr_n>)

     Displays expressions <expr_1>, ..., <expr_n> to the console as
     printed output.  `ldisp' assigns an intermediate expression label
     to each argument and returns the list of labels.

     See also `disp',  `display',  and `ldisplay'.

     Examples:

          (%i1) e: (a+b)^3;
                                             3
          (%o1)                       (b + a)
          (%i2) f: expand (e);
                               3        2      2      3
          (%o2)               b  + 3 a b  + 3 a  b + a
          (%i3) ldisp (e, f);
                                             3
          (%t3)                       (b + a)

                               3        2      2      3
          (%t4)               b  + 3 a b  + 3 a  b + a

          (%o4)                      [%t3, %t4]
          (%i4) %t3;
                                             3
          (%o4)                       (b + a)
          (%i5) %t4;
                               3        2      2      3
          (%o5)               b  + 3 a b  + 3 a  b + a

Function: ldisplay (<expr_1>, ..., <expr_n>)

     Displays expressions <expr_1>, ..., <expr_n> to the console as
     printed output.  Each expression is printed as an equation of the
     form `lhs = rhs' in which `lhs' is one of the arguments of
     `ldisplay' and `rhs' is its value.  Typically each argument is a
     variable.  `ldisp'  assigns an intermediate expression label to
     each equation and returns the list of labels.

     See also `display',  `disp',  and `ldisp'.

     Examples:

          (%i1) e: (a+b)^3;
                                             3
          (%o1)                       (b + a)
          (%i2) f: expand (e);
                               3        2      2      3
          (%o2)               b  + 3 a b  + 3 a  b + a
          (%i3) ldisplay (e, f);
                                               3
          (%t3)                     e = (b + a)

                                 3        2      2      3
          (%t4)             f = b  + 3 a b  + 3 a  b + a

          (%o4)                      [%t3, %t4]
          (%i4) %t3;
                                               3
          (%o4)                     e = (b + a)
          (%i5) %t4;
                                 3        2      2      3
          (%o5)             f = b  + 3 a b  + 3 a  b + a

Option variable: leftjust

     Default value: `false'

     When `leftjust' is `true', equations in 2D-display are drawn left
     justified rather than centered.

     See also `display2d'  to switch between 1D- and 2D-display.

     Example:

          (%i1) expand((x+1)^3);
                                  3      2
          (%o1)                  x  + 3 x  + 3 x + 1
          (%i2) leftjust:true$
          (%i3) expand((x+1)^3);
                 3      2
          (%o3) x  + 3 x  + 3 x + 1

Option variable: linel

     Default value: `79'

     `linel' is the assumed width (in characters) of the console
     display for the purpose of displaying expressions.  `linel' may be
     assigned any value by the user, although very small or very large
     values may be impractical.  Text printed by built-in Maxima
     functions, such as error messages and the output of `describe',
     is not affected by `linel'.

Option variable: lispdisp

     Default value: `false'

     When `lispdisp' is `true', Lisp symbols are displayed with a
     leading question mark `?'.  Otherwise, Lisp symbols are displayed
     with no leading mark.

     Examples:

          (%i1) lispdisp: false$
          (%i2) ?foo + ?bar;
          (%o2)                       foo + bar
          (%i3) lispdisp: true$
          (%i4) ?foo + ?bar;
          (%o4)                      ?foo + ?bar

Option variable: negsumdispflag

     Default value: `true'

     When `negsumdispflag' is `true', `x - y' displays as `x - y'
     instead of as `- y + x'.  Setting it to `false' causes the special
     check in display for the difference of two expressions to not be
     done.  One application is that thus `a + %i*b' and `a - %i*b' may
     both be displayed the same way.

Option variable: obase

     Default value: `10'

     `obase' is the base for integers displayed by Maxima.

     `obase' may be assigned any integer between 2 and 36 (decimal),
     inclusive.  When `obase' is greater than 10, the numerals comprise
     the decimal numerals 0 through 9 plus capital letters of the
     alphabet A, B, C, ..., as needed.  A leading 0 digit is displayed
     if the leading digit is otherwise a letter.  The numerals for base
     36, the largest acceptable base, comprise 0 through 9, and A
     through Z.

     See also `ibase'.

     Examples:

          (%i1) obase : 2;
          (%o1)                          10
          (%i2) 2^8 - 1;
          (%o10)                      11111111
          (%i3) obase : 8;
          (%o3)                          10
          (%i4) 8^8 - 1;
          (%o4)                       77777777
          (%i5) obase : 16;
          (%o5)                          10
          (%i6) 16^8 - 1;
          (%o6)                       0FFFFFFFF
          (%i7) obase : 36;
          (%o7)                          10
          (%i8) 36^8 - 1;
          (%o8)                       0ZZZZZZZZ

Option variable: pfeformat

     Default value: `false'

     When `pfeformat' is `true', a ratio of integers is displayed with
     the solidus (forward slash) character, and an integer denominator
     `n' is displayed as a leading multiplicative term `1/n'.

     Examples:

          (%i1) pfeformat: false$
          (%i2) 2^16/7^3;
                                        65536
          (%o2)                         -----
                                         343
          (%i3) (a+b)/8;
                                        b + a
          (%o3)                         -----
                                          8
          (%i4) pfeformat: true$
          (%i5) 2^16/7^3;
          (%o5)                       65536/343
          (%i6) (a+b)/8;
          (%o6)                      1/8 (b + a)

Option variable: powerdisp

     Default value: `false'

     When `powerdisp' is `true', a sum is displayed with its terms in
     order of increasing power.  Thus a polynomial is displayed as a
     truncated power series, with the constant term first and the
     highest power last.

     By default, terms of a sum are displayed in order of decreasing
     power.

     Example:

          (%i1) powerdisp:true;
          (%o1)                         true
          (%i2) x^2+x^3+x^4;
                                     2    3    4
          (%o2)                     x  + x  + x
          (%i3) powerdisp:false;
          (%o3)                         false
          (%i4) x^2+x^3+x^4;
                                     4    3    2
          (%o4)                     x  + x  + x

Function: print (<expr_1>, ..., <expr_n>)

     Evaluates and displays <expr_1>, ..., <expr_n> one after another,
     from left to right, starting at the left edge of the console
     display.

     The value returned by `print' is the value of its last argument.
     `print' does not generate intermediate expression labels.

     See also `display',  `disp',  `ldisplay',  and `ldisp'.    Those
     functions display one expression per line, while `print' attempts
     to display two or more expressions per line.

     To display the contents of a file, see `printfile'.

     Examples:

          (%i1) r: print ("(a+b)^3 is", expand ((a+b)^3), "log (a^10/b) is",
                radcan (log (a^10/b)))$
                      3        2      2      3
          (a+b)^3 is b  + 3 a b  + 3 a  b + a  log (a^10/b) is

                                                        10 log(a) - log(b)
          (%i2) r;
          (%o2)                  10 log(a) - log(b)
          (%i3) disp ("(a+b)^3 is", expand ((a+b)^3), "log (a^10/b) is",
                radcan (log (a^10/b)))$
                                     (a+b)^3 is

                               3        2      2      3
                              b  + 3 a b  + 3 a  b + a

                                   log (a^10/b) is

                                 10 log(a) - log(b)

Option variable: sqrtdispflag

     Default value: `true'

     When `sqrtdispflag' is `false', causes `sqrt' to display with
     exponent 1/2.

Option variable: stardisp

     Default value: `false'

     When `stardisp' is `true', multiplication is displayed with an
     asterisk `*' between operands.

Option variable: ttyoff

     Default value: `false'

     When `ttyoff' is `true', output expressions are not displayed.
     Output expressions are still computed and assigned labels.  See
     `labels'.

     Text printed by built-in Maxima functions, such as error messages
     and the output of `describe',  is not affected by `ttyoff'.

5 Data Types and Structures


Numbers
Strings
Constants
Lists
Arrays
Structures

5.1 Numbers


* Introduction to Numbers::
* Functions and Variables for Numbers::
Function: bfloat (<expr>)

     Converts all numbers and functions of numbers in <expr> to
     bigfloat numbers.  The number of significant digits in the
     resulting bigfloats is specified by the global variable `fpprec'.

     When `float2bf'  is `false' a warning message is printed when a
     floating point number is converted into a bigfloat number (since
     this may lead to loss of precision).

Function: bfloatp (<expr>)

     Returns `true' if <expr> is a bigfloat number, otherwise `false'.

Option variable: bftorat

     Default value: `false'

     `bftorat' controls the conversion of bfloats to rational numbers.
     When `bftorat' is `false', `ratepsilon'  will be used to control
     the conversion (this results in relatively small rational
     numbers).  When `bftorat' is `true', the rational number generated
     will accurately represent the bfloat.

     Note: `bftorat' has no effect on the transformation to rational
     numbers with the function `rationalize'.

     Example:

          (%i1) ratepsilon:1e-4;
          (%o1)                         1.e-4
          (%i2) rat(bfloat(11111/111111)), bftorat:false;
          `rat' replaced 9.99990999991B-2 by 1/10 = 1.0B-1
                                         1
          (%o2)/R/                       --
                                         10
          (%i3) rat(bfloat(11111/111111)), bftorat:true;
          `rat' replaced 9.99990999991B-2 by 11111/111111 = 9.99990999991B-2
                                       11111
          (%o3)/R/                     ------
                                       111111

Option variable: bftrunc

     Default value: `true'

     `bftrunc' causes trailing zeroes in non-zero bigfloat numbers not
     to be displayed.  Thus, if `bftrunc' is `false', `bfloat (1)'
     displays as `1.000000000000000B0'.  Otherwise, this is displayed as
     `1.0B0'.

Function: evenp (<expr>)

     Returns `true' if <expr> is an even integer.  `false' is returned
     in all other cases.

Function: float (<expr>)

     Converts integers, rational numbers and bigfloats in <expr> to
     floating point numbers.  It is also an `evflag',  `float' causes
     non-integral rational numbers and bigfloat numbers to be converted
     to floating point.

Option variable: float2bf

     Default value: `true'

     When `float2bf' is `false', a warning message is printed when a
     floating point number is converted into a bigfloat number (since
     this may lead to loss of precision).  The default value is `true'.

Function: floatnump (<expr>)

     Returns `true' if <expr> is a floating point number, otherwise
     `false'.

Option variable: fpprec

     Default value: 16

     `fpprec' is the number of significant digits for arithmetic on
     bigfloat numbers.  `fpprec' does not affect computations on
     ordinary floating point numbers.

     See also `bfloat'  and `fpprintprec'.

Option variable: fpprintprec

     Default value: 0

     `fpprintprec' is the number of digits to print when printing an
     ordinary float or bigfloat number.

     For ordinary floating point numbers, when `fpprintprec' has a
     value between 2 and 16 (inclusive), the number of digits printed
     is equal to `fpprintprec'.  Otherwise, `fpprintprec' is 0, or
     greater than 16, and the number of digits printed is 16.

     For bigfloat numbers, when `fpprintprec' has a value between 2 and
     `fpprec' (inclusive), the number of digits printed is equal to
     `fpprintprec'.  Otherwise, `fpprintprec' is 0, or greater than
     `fpprec', and the number of digits printed is equal to `fpprec'.

     `fpprintprec' cannot be 1.

Function: integerp (<expr>)

     Returns `true' if <expr> is a literal numeric integer, otherwise
     `false'.

     `integerp' returns `false' if its argument is a symbol, even if
     the argument is declared integer.

     Examples:

          (%i1) integerp (0);
          (%o1)                         true
          (%i2) integerp (1);
          (%o2)                         true
          (%i3) integerp (-17);
          (%o3)                         true
          (%i4) integerp (0.0);
          (%o4)                         false
          (%i5) integerp (1.0);
          (%o5)                         false
          (%i6) integerp (%pi);
          (%o6)                         false
          (%i7) integerp (n);
          (%o7)                         false
          (%i8) declare (n, integer);
          (%o8)                         done
          (%i9) integerp (n);
          (%o9)                         false

Option variable: m1pbranch

     Default value: `false'

     `m1pbranch' is the principal branch for `-1' to a power.
     Quantities such as `(-1)^(1/3)' (that is, an "odd" rational
     exponent) and `(-1)^(1/4)' (that is, an "even" rational exponent)
     are handled as follows:

                        domain:real

          (-1)^(1/3):      -1
          (-1)^(1/4):   (-1)^(1/4)

                       domain:complex
          m1pbranch:false          m1pbranch:true
          (-1)^(1/3)               1/2+%i*sqrt(3)/2
          (-1)^(1/4)              sqrt(2)/2+%i*sqrt(2)/2

Function: nonnegintegerp (<n>)

     Return `true' if and only if `<n> >= 0' and <n> is an integer.

Function: numberp (<expr>)

     Returns `true' if <expr> is a literal integer, rational number,
     floating point number, or bigfloat, otherwise `false'.

     `numberp' returns `false' if its argument is a symbol, even if the
     argument is a symbolic number such as `%pi' or `%i', or declared
     to be `even', `odd', `integer', `rational', `irrational', `real',
     `imaginary', or `complex'.

     Examples:

          (%i1) numberp (42);
          (%o1)                         true
          (%i2) numberp (-13/19);
          (%o2)                         true
          (%i3) numberp (3.14159);
          (%o3)                         true
          (%i4) numberp (-1729b-4);
          (%o4)                         true
          (%i5) map (numberp, [%e, %pi, %i, %phi, inf, minf]);
          (%o5)      [false, false, false, false, false, false]
          (%i6) declare (a, even, b, odd, c, integer, d, rational,
               e, irrational, f, real, g, imaginary, h, complex);
          (%o6)                         done
          (%i7) map (numberp, [a, b, c, d, e, f, g, h]);
          (%o7) [false, false, false, false, false, false, false, false]

Option variable: numer

     `numer' causes some mathematical functions (including
     exponentiation) with numerical arguments to be evaluated in
     floating point.  It causes variables in `expr' which have been
     given numerals to be replaced by their values.  It also sets the
     `float'  switch on.

     See also `%enumer'.

     Examples:

          (%i1) [sqrt(2), sin(1), 1/(1+sqrt(3))];
                                                  1
          (%o1)            [sqrt(2), sin(1), -----------]
                                             sqrt(3) + 1
          (%i2) [sqrt(2), sin(1), 1/(1+sqrt(3))],numer;
          (%o2) [1.414213562373095, .8414709848078965, .3660254037844387]

Option variable: numer_pbranch

     Default value: `false'

     The option variable `numer_pbranch' controls the numerical
     evaluation of the power of a negative integer, rational, or
     floating point number.  When `numer_pbranch' is `true' and the
     exponent is a floating point number or the option variable `numer'
     is `true' too, Maxima evaluates the numerical result using the
     principal branch.  Otherwise a simplified, but not an evaluated
     result is returned.

     Examples:

          (%i1) (-2)^0.75;
          (%o1) (-2)^0.75

          (%i2) (-2)^0.75,numer_pbranch:true;
          (%o2) 1.189207115002721*%i-1.189207115002721

          (%i3) (-2)^(3/4);
          (%o3) (-1)^(3/4)*2^(3/4)

          (%i4) (-2)^(3/4),numer;
          (%o4) 1.681792830507429*(-1)^0.75

          (%i5) (-2)^(3/4),numer,numer_pbranch:true;
          (%o5) 1.189207115002721*%i-1.189207115002721

Function: numerval (<x_1>, <expr_1>, ..., <var_n>, <expr_n>)

     Declares the variables `x_1', ..., <x_n> to have numeric values
     equal to `expr_1', ..., `expr_n'.  The numeric value is evaluated
     and substituted for the variable in any expressions in which the
     variable occurs if the `numer' flag is `true'.  See also `ev'.

     The expressions `expr_1', ..., `expr_n' can be any expressions,
     not necessarily numeric.

Function: oddp (<expr>)

     is `true' if <expr> is an odd integer.  `false' is returned in all
     other cases.

Option variable: ratepsilon

     Default value: `2.0e-15'

     `ratepsilon' is the tolerance used in the conversion of floating
     point numbers to rational numbers, when the option variable
     `bftorat'  has the value `false'.  See `bftorat' for an example.

Function: rationalize (<expr>)

     Convert all double floats and big floats in the Maxima expression
     <expr> to their exact rational equivalents.  If you are not
     familiar with the binary representation of floating point numbers,
     you might be surprised that `rationalize (0.1)' does not equal
     1/10.  This behavior isn't special to Maxima - the number 1/10 has
     a repeating, not a terminating, binary representation.

          (%i1) rationalize (0.5);
                                          1
          (%o1)                           -
                                          2
          (%i2) rationalize (0.1);
                                         1
          (%o2)                          --
                                         10
          (%i3) fpprec : 5$
          (%i4) rationalize (0.1b0);
                                       209715
          (%o4)                        -------
                                       2097152
          (%i5) fpprec : 20$
          (%i6) rationalize (0.1b0);
                               236118324143482260685
          (%o6)                ----------------------
                               2361183241434822606848
          (%i7) rationalize (sin (0.1*x + 5.6));
                                        x    28
          (%o7)                     sin(-- + --)
                                        10   5

     Example use:

          (%i1) unitfrac(r) := block([uf : [], q],
              if not(ratnump(r)) then
                 error("The input to 'unitfrac' must be a rational number"),
              while r # 0 do (
                  uf : cons(q : 1/ceiling(1/r), uf),
                  r : r - q),
              reverse(uf))$
          (%i2) unitfrac (9/10);
                                      1  1  1
          (%o2)                      [-, -, --]
                                      2  3  15
          (%i3) apply ("+", %);
                                         9
          (%o3)                          --
                                         10
          (%i4) unitfrac (-9/10);
                                            1
          (%o4)                       [- 1, --]
                                            10
          (%i5) apply ("+", %);
                                          9
          (%o5)                         - --
                                          10
          (%i6) unitfrac (36/37);
                                  1  1  1  1    1
          (%o6)                  [-, -, -, --, ----]
                                  2  3  8  69  6808
          (%i7) apply ("+", %);
                                         36
          (%o7)                          --
                                         37

Function: ratnump (<expr>)

     Returns `true' if <expr> is a literal integer or ratio of literal
     integers, otherwise `false'.

5.2 Strings


* Introduction to Strings::
* Functions and Variables for Strings::
Function: concat (<arg_1>, <arg_2>, ...)

     Concatenates its arguments.  The arguments must evaluate to atoms.
     The return value is a symbol if the first argument is a symbol and
     a string otherwise.

     `concat' evaluates its arguments.  The single quote `'' prevents
     evaluation.

          (%i1) y: 7$
          (%i2) z: 88$
          (%i3) concat (y, z/2);
          (%o3)                          744
          (%i4) concat ('y, z/2);
          (%o4)                          y44

     A symbol constructed by `concat' may be assigned a value and
     appear in expressions.  The `::'  (double colon) assignment
     operator evaluates its left-hand side.

          (%i5) a: concat ('y, z/2);
          (%o5)                          y44
          (%i6) a:: 123;
          (%o6)                          123
          (%i7) y44;
          (%o7)                          123
          (%i8) b^a;
                                         y44
          (%o8)                         b
          (%i9) %, numer;
                                         123
          (%o9)                         b

     Note that although `concat (1, 2)' looks like a number, it is a
     string.

          (%i10) concat (1, 2) + 3;
          (%o10)                       12 + 3

Function: sconcat (<arg_1>, <arg_2>, ...)

     Concatenates its arguments into a string.  Unlike `concat',  the
     arguments do not need to be atoms.

          (%i1) sconcat ("xx[", 3, "]:", expand ((x+y)^3));
          (%o1)               xx[3]:y^3+3*x*y^2+3*x^2*y+x^3

Function: string (<expr>)

     Converts `expr' to Maxima's linear notation just as if it had been
     typed in.

     The return value of `string' is a string, and thus it cannot be
     used in a computation.

Option variable: stringdisp

     Default value: `false'

     When `stringdisp' is `true', strings are displayed enclosed in
     double quote marks.  Otherwise, quote marks are not displayed.

     `stringdisp' is always `true' when displaying a function
     definition.

     Examples:

          (%i1) stringdisp: false$
          (%i2) "This is an example string.";
          (%o2)              This is an example string.
          (%i3) foo () :=
                print ("This is a string in a function definition.");
          (%o3) foo() :=
                        print("This is a string in a function definition.")
          (%i4) stringdisp: true$
          (%i5) "This is an example string.";
          (%o5)             "This is an example string."

5.3 Constants


* Functions and Variables for Constants::
Constant: %e

     `%e' represents the base of the natural logarithm, also known as
     Euler's number.  The numeric value of `%e' is the double-precision
     floating-point value 2.718281828459045d0.

Constant: %i

     `%i' represents the imaginary unit, sqrt(- 1).

Constant: false

     `false' represents the Boolean constant of the same name.  Maxima
     implements `false' by the value `NIL' in Lisp.

Constant: %gamma

     The Euler-Mascheroni constant, 0.5772156649015329 ....

Constant: ind

     `ind' represents a bounded, indefinite result.

     See also `limit'.

     Example:

          (%i1) limit (sin(1/x), x, 0);
          (%o1)                          ind

Constant: inf

     `inf' represents real positive infinity.

Constant: infinity

     `infinity' represents complex infinity.

Constant: minf

     `minf' represents real minus (i.e., negative) infinity.

Constant: %phi

     `%phi' represents the so-called golden mean, (1 + sqrt(5))/2.  The
     numeric value of `%phi' is the double-precision floating-point
     value 1.618033988749895d0.

     `fibtophi'  expresses Fibonacci numbers `fib(n)' in terms of
     `%phi'.

     By default, Maxima does not know the algebraic properties of
     `%phi'.  After evaluating `tellrat(%phi^2 - %phi - 1)' and
     `algebraic: true', `ratsimp'  can simplify some expressions
     containing `%phi'.

     Examples:

     `fibtophi' expresses Fibonacci numbers `fib(n)' in terms of `%phi'.

          (%i1) fibtophi (fib (n));
                                     n             n
                                 %phi  - (1 - %phi)
          (%o1)                  -------------------
                                     2 %phi - 1
          (%i2) fib (n-1) + fib (n) - fib (n+1);
          (%o2)          - fib(n + 1) + fib(n) + fib(n - 1)
          (%i3) fibtophi (%);
                      n + 1             n + 1       n             n
                  %phi      - (1 - %phi)        %phi  - (1 - %phi)
          (%o3) - --------------------------- + -------------------
                          2 %phi - 1                2 %phi - 1
                                                    n - 1             n - 1
                                                %phi      - (1 - %phi)
                                              + ---------------------------
                                                        2 %phi - 1
          (%i4) ratsimp (%);
          (%o4)                           0

     By default, Maxima does not know the algebraic properties of
     `%phi'.  After evaluating `tellrat (%phi^2 - %phi - 1)' and
     `algebraic: true', `ratsimp' can simplify some expressions
     containing `%phi'.

          (%i1) e : expand ((%phi^2 - %phi - 1) * (A + 1));
                           2                      2
          (%o1)        %phi  A - %phi A - A + %phi  - %phi - 1
          (%i2) ratsimp (e);
                            2                     2
          (%o2)        (%phi  - %phi - 1) A + %phi  - %phi - 1
          (%i3) tellrat (%phi^2 - %phi - 1);
                                      2
          (%o3)                  [%phi  - %phi - 1]
          (%i4) algebraic : true;
          (%o4)                         true
          (%i5) ratsimp (e);
          (%o5)                           0

Constant: %pi

     `%pi' represents the ratio of the perimeter of a circle to its
     diameter.  The numeric value of `%pi' is the double-precision
     floating-point value 3.141592653589793d0.

Constant: true

     `true' represents the Boolean constant of the same name.  Maxima
     implements `true' by the value `T' in Lisp.

Constant: und

     `und' represents an undefined result.

     See also `limit'.

     Example:

          (%i1) limit (x*sin(x), x, inf);
          (%o1)                          und

Constant: zeroa

     `zeroa' represents an infinitesimal above zero.  `zeroa' can be
     used in expressions.  `limit' simplifies expressions which contain
     infinitesimals.

     See also `zerob'  and `limit'.

     Example:

     `limit' simplifies expressions which contain infinitesimals:

          (%i1) limit(zeroa);
          (%o1)                           0
          (%i2) limit(x+zeroa);
          (%o2)                           x

Constant: zerob

     `zerob' represents an infinitesimal below zero.  `zerob' can be
     used in expressions.  `limit' simplifies expressions which contain
     infinitesimals.

     See also `zeroa'  and `limit'.

5.4 Lists


* Introduction to Lists::
* Functions and Variables for Lists::
Operator: [
Operator: ]

     `[' and `]' mark the beginning and end, respectively, of a list.

     `[' and `]' also enclose the subscripts of a list, array, hash
     array, or array function.

     Examples:

          (%i1) x: [a, b, c];
          (%o1)                       [a, b, c]
          (%i2) x[3];
          (%o2)                           c
          (%i3) array (y, fixnum, 3);
          (%o3)                           y
          (%i4) y[2]: %pi;
          (%o4)                          %pi
          (%i5) y[2];
          (%o5)                          %pi
          (%i6) z['foo]: 'bar;
          (%o6)                          bar
          (%i7) z['foo];
          (%o7)                          bar
          (%i8) g[k] := 1/(k^2+1);
                                            1
          (%o8)                     g  := ------
                                     k     2
                                          k  + 1
          (%i9) g[10];
                                          1
          (%o9)                          ---
                                         101

Function: append (<list_1>, ..., <list_n>)

     Returns a single list of the elements of <list_1> followed by the
     elements of <list_2>, ...  `append' also works on general
     expressions, e.g. `append (f(a,b), f(c,d,e));' yields
     `f(a,b,c,d,e)'.

     Do `example(append);' for an example.

Function: assoc (<key>, <list>, <default>)
Function: assoc (<key>, <list>)

     This function searches for the <key> in the left hand side of the
     input <list> of the form `[x,y,z,...]' where each of the <list>
     elements is an expression of a binary operand and 2 elements.  For
     example `x=1', `2^3', `[a,b]' etc.  The <key> is checked against
     the first operand.  `assoc' returns the second operand if the
     `key' is found.  If the `key' is not found it either returns the
     <default> value.  <default> is optional and defaults to `false'.

Function: cons (<expr>, <list>)

     Returns a new list constructed of the element <expr> as its first
     element, followed by the elements of <list>.  `cons' also works on
     other expressions, e.g. `cons(x, f(a,b,c));'  ->  `f(x,a,b,c)'.

Function: copylist (<list>)

     Returns a copy of the list <list>.

Function: create_list (<form>, <x_1>, <list_1>, ..., <x_n>, <list_n>)

     Create a list by evaluating <form> with <x_1> bound to each
     element of <list_1>, and for each such binding bind <x_2> to each
     element of <list_2>, ...  The number of elements in the result
     will be the product of the number of elements in each list.  Each
     variable <x_i> must actually be a symbol - it will not be
     evaluated.  The list arguments will be evaluated once at the
     beginning of the iteration.

          (%i1) create_list (x^i, i, [1, 3, 7]);
                                          3   7
          (%o1)                      [x, x , x ]

     With a double iteration:

          (%i1) create_list ([i, j], i, [a, b], j, [e, f, h]);
          (%o1)   [[a, e], [a, f], [a, h], [b, e], [b, f], [b, h]]

     Instead of <list_i> two args may be supplied each of which should
     evaluate to a number.  These will be the inclusive lower and upper
     bounds for the iteration.

          (%i1) create_list ([i, j], i, [1, 2, 3], j, 1, i);
          (%o1)   [[1, 1], [2, 1], [2, 2], [3, 1], [3, 2], [3, 3]]

     Note that the limits or list for the `j' variable can depend on
     the current value of `i'.

Function: delete (<expr_1>, <expr_2>)
Function: delete (<expr_1>, <expr_2>, <n>)

     `delete(<expr_1>, <expr_2>)' removes from <expr_2> any arguments
     of its top-level operator which are the same (as determined by
     "=") as <expr_1>.  Note that "=" tests for formal equality, not
     equivalence.  Note also that arguments of subexpressions are not
     affected.

     <expr_1> may be an atom or a non-atomic expression.  <expr_2> may
     be any non-atomic expression.  `delete' returns a new expression;
     it does not modify <expr_2>.

     `delete(<expr_1>, <expr_2>, <n>)' removes from <expr_2> the first
     <n> arguments of the top-level operator which are the same as
     <expr_1>.  If there are fewer than <n> such arguments, then all
     such arguments are removed.

     Examples:

     Removing elements from a list.

          (%i1) delete (y, [w, x, y, z, z, y, x, w]);
          (%o1)                  [w, x, z, z, x, w]

     Removing terms from a sum.

          (%i1) delete (sin(x), x + sin(x) + y);
          (%o1)                         y + x

     Removing factors from a product.

          (%i1) delete (u - x, (u - w)*(u - x)*(u - y)*(u - z));
          (%o1)                (u - w) (u - y) (u - z)

     Removing arguments from an arbitrary expression.

          (%i1) delete (a, foo (a, b, c, d, a));
          (%o1)                     foo(b, c, d)

     Limit the number of removed arguments.

          (%i1) delete (a, foo (a, b, a, c, d, a), 2);
          (%o1)                    foo(b, c, d, a)

     Whether arguments are the same as <expr_1> is determined by "=".
     Arguments which are `equal' but not "=" are not removed.

          (%i1) [is (equal (0, 0)), is (equal (0, 0.0)), is (equal (0, 0b0))];
          rat: replaced 0.0 by 0/1 = 0.0
          `rat' replaced 0.0B0 by 0/1 = 0.0B0
          (%o1)                  [true, true, true]
          (%i2) [is (0 = 0), is (0 = 0.0), is (0 = 0b0)];
          (%o2)                 [true, false, false]
          (%i3) delete (0, [0, 0.0, 0b0]);
          (%o3)                     [0.0, 0.0b0]
          (%i4) is (equal ((x + y)*(x - y), x^2 - y^2));
          (%o4)                         true
          (%i5) is ((x + y)*(x - y) = x^2 - y^2);
          (%o5)                         false
          (%i6) delete ((x + y)*(x - y), [(x + y)*(x - y), x^2 - y^2]);
                                        2    2
          (%o6)                       [x  - y ]

Function: eighth (<expr>)

     Returns the 8'th item of expression or list <expr>.  See `first'
     for more details.

Function: endcons (<expr>, <list>)

     Returns a new list consisting of the elements of `list' followed by
     <expr>.  `endcons' also works on general expressions, e.g.
     `endcons(x, f(a,b,c));'  ->  `f(a,b,c,x)'.

Function: fifth (<expr>)

     Returns the 5'th item of expression or list <expr>.  See `first'
     for more details.

Function: first (<expr>)

     Returns the first part of <expr> which may result in the first
     element of a list, the first row of a matrix, the first term of a
     sum, etc.  Note that `first' and its related functions, `rest' and
     `last', work on the form of <expr> which is displayed not the form
     which is typed on input.  If the variable `inflag'  is set to
     `true' however, these functions will look at the internal form of
     <expr>.  Note that the simplifier re-orders expressions.  Thus
     `first(x+y)' will be `x' if `inflag' is `true' and `y' if `inflag'
     is `false' (`first(y+x)' gives the same results).  The functions
     `second' ...  `tenth' yield the second through the tenth part of
     their input argument.

Function: fourth (<expr>)

     Returns the 4'th item of expression or list <expr>.  See `first'
     for more details.

Function: join (<l>, <m>)

     Creates a new list containing the elements of lists <l> and <m>,
     interspersed.  The result has elements `[<l>[1], <m>[1], <l>[2],
     <m>[2], ...]'.  The lists <l> and <m> may contain any type of
     elements.

     If the lists are different lengths, `join' ignores elements of the
     longer list.

     Maxima complains if <l> or <m> is not a list.

     Examples:

          (%i1) L1: [a, sin(b), c!, d - 1];
          (%o1)                [a, sin(b), c!, d - 1]
          (%i2) join (L1, [1, 2, 3, 4]);
          (%o2)          [a, 1, sin(b), 2, c!, 3, d - 1, 4]
          (%i3) join (L1, [aa, bb, cc, dd, ee, ff]);
          (%o3)        [a, aa, sin(b), bb, c!, cc, d - 1, dd]

Function: last (<expr>)

     Returns the last part (term, row, element, etc.) of the <expr>.

Function: length (<expr>)

     Returns (by default) the number of parts in the external
     (displayed) form of <expr>.  For lists this is the number of
     elements, for matrices it is the number of rows, and for sums it
     is the number of terms (see `dispform' ).

     The `length' command is affected by the `inflag'  switch.  So, e.g.
     `length(a/(b*c));' gives 2 if `inflag' is `false' (Assuming
     `exptdispflag'  is `true'), but 3 if `inflag' is `true' (the
     internal representation is essentially `a*b^-1*c^-1').

Option variable: listarith

     Default value: `true'

     If `false' causes any arithmetic operations with lists to be
     suppressed; when `true', list-matrix operations are contagious
     causing lists to be converted to matrices yielding a result which
     is always a matrix.  However, list-list operations should return
     lists.

Function: listp (<expr>)

     Returns `true' if <expr> is a list else `false'.

Function: makelist ()
Function: makelist (<expr>, <n>)
Function: makelist (<expr>, <i>, <i_max>)
Function: makelist (<expr>, <i>, <i_0>, <i_max>)
Function: makelist (<expr>, <i>, <i_0>, <i_max>, <step>)
Function: makelist (<expr>, <x>, <list>)

     The first form, `makelist ()', creates an empty list. The second
     form, `makelist (<expr>)', creates a list with <expr> as its single
     element. `makelist (<expr>, <n>)' creates a list of <n> elements
     generated from <expr>.

     The most general form, `makelist (<expr>, <i>, <i_0>, <i_max>,
     <step>)', returns the list of elements obtained when `ev (<expr>,
     <i>=<j>)' is applied to the elements <j> of the sequence: <i_0>,
     <i_0> + <step>, <i_0> + 2*<step>, ..., with <|j|> less than or
     equal to <|i_max|>.

     The increment <step> can be a number (positive or negative) or an
     expression. If it is omitted, the default value 1 will be used. If
     both <i_0> and <step> are omitted, they will both have a default
     value of 1.

     `makelist (<expr>, <x>, <list>)' returns a list, the `j''th
     element of which is equal to `ev (<expr>, <x>=<list>[j])' for `j'
     equal to 1 through `length (<list>)'.

     Examples:

          (%i1) makelist (concat (x,i), i, 6);
          (%o1)               [x1, x2, x3, x4, x5, x6]
          (%i2) makelist (x=y, y, [a, b, c]);
          (%o2)                 [x = a, x = b, x = c]
          (%i3) makelist (x^2, x, 3, 2*%pi, 2);
          (%o3)                        [9, 25]
          (%i4) makelist (random(6), 4);
          (%o4)                     [2, 0, 2, 5]
          (%i5) flatten (makelist (makelist (i^2, 3), i, 4));
          (%o5)        [1, 1, 1, 4, 4, 4, 9, 9, 9, 16, 16, 16]
          (%i6) flatten (makelist (makelist (i^2, i, 3), 4));
          (%o6)         [1, 4, 9, 1, 4, 9, 1, 4, 9, 1, 4, 9]

Function: member (<expr_1>, <expr_2>)

     Returns `true' if `is(<expr_1> = <a>)' for some element <a> in
     `args(<expr_2>)', otherwise returns `false'.

     `expr_2' is typically a list, in which case `args(<expr_2>) =
     <expr_2>' and `is(<expr_1> = <a>)' for some element <a> in
     `expr_2' is the test.

     `member' does not inspect parts of the arguments of `expr_2', so it
     may return `false' even if `expr_1' is a part of some argument of
     `expr_2'.

     See also `elementp'.

     Examples:

          (%i1) member (8, [8, 8.0, 8b0]);
          (%o1)                         true
          (%i2) member (8, [8.0, 8b0]);
          (%o2)                         false
          (%i3) member (b, [a, b, c]);
          (%o3)                         true
          (%i4) member (b, [[a, b], [b, c]]);
          (%o4)                         false
          (%i5) member ([b, c], [[a, b], [b, c]]);
          (%o5)                         true
          (%i6) F (1, 1/2, 1/4, 1/8);
                                         1  1  1
          (%o6)                     F(1, -, -, -)
                                         2  4  8
          (%i7) member (1/8, %);
          (%o7)                         true
          (%i8) member ("ab", ["aa", "ab", sin(1), a + b]);
          (%o8)                         true

Function: ninth (<expr>)

     Returns the 9'th item of expression or list <expr>.  See `first'
     for more details.

Function: pop (<list>)

     `pop' removes the first element from the list <list> and returns
     this element.  <list> must be a symbol, which is bound to a list
     and not the list itself.

     If the argument <list> is not bound to a list or the list is
     empty, Maxima generates an error message.

     See also `push'  for examples.

     To use this function, the additional package "basic" must be loaded
     first: `load("basic")'.

Function: push (<item>, <list>)

     `push' prepends the item <item> to the list <list> and returns a
     copy of the new list.  <list> must be a symbol, which is bound to
     a list and not the list itself.  <item> can be any Maxima symbol
     or expression.

     If the argument <list> is not bound to a list, Maxima generates an
     error message.

     See also `pop'  to remove the first item from a list.

     `load("basic")' loads this function.

     Examples:

          (%i1) load ("basic")$
          (%i2) ll: [];
          (%o2)                          []
          (%i3) push (x, ll);
          (%o3)                          [x]
          (%i4) push (x^2+y, ll);
                                           2
          (%o4)                      [y + x , x]
          (%i5) a: push ("string", ll);
                                               2
          (%o5)                  [string, y + x , x]
          (%i6) pop (ll);
          (%o6)                        string
          (%i7) pop (ll);
                                            2
          (%o7)                        y + x
          (%i8) pop (ll);
          (%o8)                           x
          (%i9) ll;
          (%o9)                          []
          (%i10) a;
                                               2
          (%o10)                 [string, y + x , x]

Function: rest (<expr>, <n>)
Function: rest (<expr>)

     Returns <expr> with its first <n> elements removed if <n> is
     positive and its last `- <n>' elements removed if <n> is negative.
     If <n> is 1 it may be omitted.  <expr> may be a list, matrix, or
     other expression.

Function: reverse (<list>)

     Reverses the order of the members of the <list> (not the members
     themselves).  `reverse' also works on general expressions, e.g.
     `reverse(a=b);' gives `b=a'.

Function: second (<expr>)

     Returns the 2'nd item of expression or list <expr>.  See `first'
     for more details.

Function: seventh (<expr>)

     Returns the 7'th item of expression or list <expr>.  See `first'
     for more details.

Function: sixth (<expr>)

     Returns the 6'th item of expression or list <expr>.  See `first'
     for more details.

Function: sort (<L>, <P>)
Function: sort (<L>)

     `sort(<L>, <P>)' sorts a list <L> according to a predicate `P' of
     two arguments which defines a strict weak order on the elements of
     <L>.  If `<P>(a, b)' is `true', then `a' appears before `b' in the
     result.  If neither `<P>(a, b)' nor `<P>(b, a)' are `true', then
     `a' and `b' are equivalent, and appear in the result in the same
     order as in the input.  That is, `sort' is a stable sort.

     If `<P>(a, b)' and `<P>(b, a)' are both `true' for some elements
     of <L>, then <P> is not a valid sort predicate, and the result is
     undefined.  If `<P>(a, b)' is something other than `true' or
     `false', `sort' signals an error.

     The predicate may be specified as the name of a function or binary
     infix operator, or as a `lambda' expression.  If specified as the
     name of an operator, the name must be enclosed in double quotes.

     The sorted list is returned as a new object; the argument <L> is
     not modified.

     `sort(<L>)' is equivalent to `sort(<L>, orderlessp)'.

     The default sorting order is ascending, as determined by
     `orderlessp'.   The predicate `ordergreatp' sorts a list in
     descending order.

     All Maxima atoms and expressions are comparable under `orderlessp'
     and `ordergreatp'.

     Operators `<' and `>' order numbers, constants, and constant
     expressions by magnitude.  Note that `orderlessp' and
     `ordergreatp' do not order numbers, constants, and constant
     expressions by magnitude.

     `ordermagnitudep' orders numbers, constants, and constant
     expressions the same as `<', and all other elements the same as
     `orderlessp'.

     Examples:

     `sort' sorts a list according to a predicate of two arguments
     which defines a strict weak order on the elements of the list.

          (%i1) sort ([1, a, b, 2, 3, c], 'orderlessp);
          (%o1)                  [1, 2, 3, a, b, c]
          (%i2) sort ([1, a, b, 2, 3, c], 'ordergreatp);
          (%o2)                  [c, b, a, 3, 2, 1]

     The predicate may be specified as the name of a function or binary
     infix operator, or as a `lambda' expression.  If specified as the
     name of an operator, the name must be enclosed in double quotes.

          (%i1) L : [[1, x], [3, y], [4, w], [2, z]];
          (%o1)           [[1, x], [3, y], [4, w], [2, z]]
          (%i2) foo (a, b) := a[1] > b[1];
          (%o2)                 foo(a, b) := a  > b
                                              1    1
          (%i3) sort (L, 'foo);
          (%o3)           [[4, w], [3, y], [2, z], [1, x]]
          (%i4) infix (">>");
          (%o4)                          >>
          (%i5) a >> b := a[1] > b[1];
          (%o5)                   a >> b := a  > b
                                             1    1
          (%i6) sort (L, ">>");
          (%o6)           [[4, w], [3, y], [2, z], [1, x]]
          (%i7) sort (L, lambda ([a, b], a[1] > b[1]));
          (%o7)           [[4, w], [3, y], [2, z], [1, x]]

     `sort(<L>)' is equivalent to `sort(<L>, orderlessp)'.

          (%i1) L : [a, 2*b, -5, 7, 1 + %e, %pi];
          (%o1)             [a, 2 b, - 5, 7, %e + 1, %pi]
          (%i2) sort (L);
          (%o2)             [- 5, 7, %e + 1, %pi, a, 2 b]
          (%i3) sort (L, 'orderlessp);
          (%o3)             [- 5, 7, %e + 1, %pi, a, 2 b]

     The default sorting order is ascending, as determined by
     `orderlessp'.   The predicate `ordergreatp' sorts a list in
     descending order.

          (%i1) L : [a, 2*b, -5, 7, 1 + %e, %pi];
          (%o1)                    [a, 2 b, - 5, 7, %e + 1, %pi]
          (%i2) sort (L);
          (%o2)                    [- 5, 7, %e + 1, %pi, a, 2 b]
          (%i3) sort (L, 'ordergreatp);
          (%o3)                    [2 b, a, %pi, %e + 1, 7, - 5]

     All Maxima atoms and expressions are comparable under `orderlessp'
     and `ordergreatp'.

          (%i1) L : [11, -17, 29b0, 9*c, 7.55, foo(x, y), -5/2, b + a];
                                                           5
          (%o1)  [11, - 17, 2.9b1, 9 c, 7.55, foo(x, y), - -, b + a]
                                                           2
          (%i2) sort (L, orderlessp);
                          5
          (%o2)  [- 17, - -, 7.55, 11, 2.9b1, b + a, 9 c, foo(x, y)]
                          2
          (%i3) sort (L, ordergreatp);
                                                            5
          (%o3)  [foo(x, y), 9 c, b + a, 2.9b1, 11, 7.55, - -, - 17]
                                                            2

     Operators `<' and `>' order numbers, constants, and constant
     expressions by magnitude.  Note that `orderlessp' and
     `ordergreatp' do not order numbers, constants, and constant
     expressions by magnitude.

          (%i1) L : [%pi, 3, 4, %e, %gamma];
          (%o1)                [%pi, 3, 4, %e, %gamma]
          (%i2) sort (L, ">");
          (%o2)                [4, %pi, 3, %e, %gamma]
          (%i3) sort (L, ordergreatp);
          (%o3)                [%pi, %gamma, %e, 4, 3]

     `ordermagnitudep' orders numbers, constants, and constant
     expressions the same as `<', and all other elements the same as
     `orderlessp'.

          (%i1) L : [%i, 1+%i, 2*x, minf, inf, %e, sin(1), 0, 1, 2, 3, 1.0, 1.0b0];
          (%o1) [%i, %i + 1, 2 x, minf, inf, %e, sin(1), 0, 1, 2, 3, 1.0,
                                                                     1.0b0]
          (%i2) sort (L, ordermagnitudep);
          (%o2) [minf, 0, sin(1), 1, 1.0, 1.0b0, 2, %e, 3, inf, %i,
                                                               %i + 1, 2 x]
          (%i3) sort (L, orderlessp);
          (%o3) [0, 1, 1.0, 2, 3, %e, %i, %i + 1, inf, minf, sin(1),
                                                                1.0b0, 2 x]

Function: sublist (<list>, <p>)

     Returns the list of elements of <list> for which the predicate `p'
     returns `true'.

     Example:

          (%i1) L: [1, 2, 3, 4, 5, 6];
          (%o1)                  [1, 2, 3, 4, 5, 6]
          (%i2) sublist (L, evenp);
          (%o2)                       [2, 4, 6]

Function: sublist_indices (<L>, <P>)

     Returns the indices of the elements `x' of the list <L> for which
     the predicate `maybe(<P>(x))' returns `true'; this excludes
     `unknown' as well as `false'.  <P> may be the name of a function
     or a lambda expression.  <L> must be a literal list.

     Examples:

          (%i1) sublist_indices ('[a, b, b, c, 1, 2, b, 3, b],
                                 lambda ([x], x='b));
          (%o1)                     [2, 3, 7, 9]
          (%i2) sublist_indices ('[a, b, b, c, 1, 2, b, 3, b], symbolp);
          (%o2)                  [1, 2, 3, 4, 7, 9]
          (%i3) sublist_indices ([1 > 0, 1 < 0, 2 < 1, 2 > 1, 2 > 0],
                                 identity);
          (%o3)                       [1, 4, 5]
          (%i4) assume (x < -1);
          (%o4)                       [x < - 1]
          (%i5) map (maybe, [x > 0, x < 0, x < -2]);
          (%o5)                [false, true, unknown]
          (%i6) sublist_indices ([x > 0, x < 0, x < -2], identity);
          (%o6)                          [2]

Function: unique (<L>)

     Returns the unique elements of the list <L>.

     When all the elements of <L> are unique, `unique' returns a
     shallow copy of <L>, not <L> itself.

     If <L> is not a list, `unique' returns <L>.

     Example:

          (%i1) unique ([1, %pi, a + b, 2, 1, %e, %pi, a + b, [1]]);
          (%o1)              [1, 2, %e, %pi, [1], b + a]

Function: tenth (<expr>)

     Returns the 10'th item of expression or list <expr>.  See `first'
     for more details.

Function: third (<expr>)

     Returns the 3'rd item of expression or list <expr>.  See `first'
     for more details.

5.5 Arrays


* Functions and Variables for Arrays::
Function: array (<name>, <dim_1>, ..., <dim_n>)
Function: array (<name>, <type>, <dim_1>, ..., <dim_n>)
Function: array ([<name_1>, ..., <name_m>], <dim_1>, ..., <dim_n>)

     Creates an n-dimensional array.  n may be less than or equal to 5.
     The subscripts for the i'th dimension are the integers running
     from 0 to <dim_i>.

     `array (<name>, <dim_1>, ..., <dim_n>)' creates a general array.

     `array (<name>, <type>, <dim_1>, ..., <dim_n>)' creates an array,
     with elements of a specified type.  <type> can be `fixnum' for
     integers of limited size or `flonum' for floating-point numbers.

     `array ([<name_1>, ..., <name_m>], <dim_1>, ..., <dim_n>)' creates
     m arrays, all of the same dimensions.

     If the user assigns to a subscripted variable before declaring the
     corresponding array, an undeclared array is created.  Undeclared
     arrays, otherwise known as hashed arrays (because hash coding is
     done on the subscripts), are more general than declared arrays.
     The user does not declare their maximum size, and they grow
     dynamically by hashing as more elements are assigned values.  The
     subscripts of undeclared arrays need not even be numbers.  However,
     unless an array is rather sparse, it is probably more efficient to
     declare it when possible than to leave it undeclared.  The `array'
     function can be used to transform an undeclared array into a
     declared array.

Function: arrayapply (<A>, [<i_1>, ..., <i_n>])

     Evaluates `<A> [<i_1>, ..., <i_n>]', where <A> is an array and
     <i_1>, ..., <i_n> are integers.

     This is reminiscent of `apply',  except the first argument is an
     array instead of a function.

Function: arrayinfo (<A>)

     Returns information about the array <A>.  The argument <A> may be
     a declared array, an undeclared (hashed) array, an array function,
     or a subscripted function.

     For declared arrays, `arrayinfo' returns a list comprising the atom
     `declared', the number of dimensions, and the size of each
     dimension.  The elements of the array, both bound and unbound, are
     returned by `listarray'.

     For undeclared arrays (hashed arrays), `arrayinfo' returns a list
     comprising the atom `hashed', the number of subscripts, and the
     subscripts of every element which has a value.  The values are
     returned by `listarray'.

     For array functions, `arrayinfo' returns a list comprising the atom
     `hashed', the number of subscripts, and any subscript values for
     which there are stored function values.  The stored function
     values are returned by `listarray'.

     For subscripted functions, `arrayinfo' returns a list comprising
     the atom `hashed', the number of subscripts, and any subscript
     values for which there are lambda expressions.  The lambda
     expressions are returned by `listarray'.

     See also `listarray'.

     Examples:

     `arrayinfo' and `listarray' applied to a declared array.

          (%i1) array (aa, 2, 3);
          (%o1)                          aa
          (%i2) aa [2, 3] : %pi;
          (%o2)                          %pi
          (%i3) aa [1, 2] : %e;
          (%o3)                          %e
          (%i4) arrayinfo (aa);
          (%o4)                 [declared, 2, [2, 3]]
          (%i5) listarray (aa);
          (%o5) [#####, #####, #####, #####, #####, #####, %e, #####,
                                                  #####, #####, #####, %pi]

     `arrayinfo' and `listarray' applied to an undeclared (hashed)
     array.

          (%i1) bb [FOO] : (a + b)^2;
                                             2
          (%o1)                       (b + a)
          (%i2) bb [BAR] : (c - d)^3;
                                             3
          (%o2)                       (c - d)
          (%i3) arrayinfo (bb);
          (%o3)               [hashed, 1, [BAR], [FOO]]
          (%i4) listarray (bb);
                                        3         2
          (%o4)                 [(c - d) , (b + a) ]

     `arrayinfo' and `listarray' applied to an array function.

          (%i1) cc [x, y] := y / x;
                                               y
          (%o1)                      cc     := -
                                       x, y    x
          (%i2) cc [u, v];
                                          v
          (%o2)                           -
                                          u
          (%i3) cc [4, z];
                                          z
          (%o3)                           -
                                          4
          (%i4) arrayinfo (cc);
          (%o4)              [hashed, 2, [4, z], [u, v]]
          (%i5) listarray (cc);
                                        z  v
          (%o5)                        [-, -]
                                        4  u

     `arrayinfo' and `listarray' applied to a subscripted function.

          (%i1) dd [x] (y) := y ^ x;
                                               x
          (%o1)                     dd (y) := y
                                      x
          (%i2) dd [a + b];
                                              b + a
          (%o2)                  lambda([y], y     )
          (%i3) dd [v - u];
                                              v - u
          (%o3)                  lambda([y], y     )
          (%i4) arrayinfo (dd);
          (%o4)             [hashed, 1, [b + a], [v - u]]
          (%i5) listarray (dd);
                                   b + a                v - u
          (%o5)      [lambda([y], y     ), lambda([y], y     )]

Function: arraymake (<A>, [<i_1>, ..., <i_n>])

     Returns the expression `<A>[<i_1>, ..., <i_n>]'.  The result is an
     unevaluated array reference.

     `arraymake' is reminiscent of `funmake',  except the return value
     is an unevaluated array reference instead of an unevaluated
     function call.

     Examples:
          (%i1) arraymake (A, [1]);
          (%o1)                          A
                                          1
          (%i2) arraymake (A, [k]);
          (%o2)                          A
                                          k
          (%i3) arraymake (A, [i, j, 3]);
          (%o3)                       A
                                       i, j, 3
          (%i4) array (A, fixnum, 10);
          (%o4)                           A
          (%i5) fillarray (A, makelist (i^2, i, 1, 11));
          (%o5)                           A
          (%i6) arraymake (A, [5]);
          (%o6)                          A
                                          5
          (%i7) ''%;
          (%o7)                          36
          (%i8) L : [a, b, c, d, e];
          (%o8)                    [a, b, c, d, e]
          (%i9) arraymake ('L, [n]);
          (%o9)                          L
                                          n
          (%i10) ''%, n = 3;
          (%o10)                          c
          (%i11) A2 : make_array (fixnum, 10);
          (%o11)          {Array:  #(0 0 0 0 0 0 0 0 0 0)}
          (%i12) fillarray (A2, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
          (%o12)          {Array:  #(1 2 3 4 5 6 7 8 9 10)}
          (%i13) arraymake ('A2, [8]);
          (%o13)                         A2
                                           8
          (%i14) ''%;
          (%o14)                          9

System variable: arrays

     Default value: `[]'

     `arrays' is a list of arrays that have been allocated.  These
     comprise arrays declared by `array', hashed arrays constructed by
     implicit definition (assigning something to an array element), and
     array functions defined by `:=' and `define'.  Arrays defined by
     `make_array'  are not included.

     See also `array',  `arrayapply',  `arrayinfo',

     `arraymake',  `fillarray',  `listarray',  and `rearray'.

     Examples:

          (%i1) array (aa, 5, 7);
          (%o1)                          aa
          (%i2) bb [FOO] : (a + b)^2;
                                             2
          (%o2)                       (b + a)
          (%i3) cc [x] := x/100;
                                             x
          (%o3)                      cc  := ---
                                       x    100
          (%i4) dd : make_array ('any, 7);
          (%o4)       {Array:  #(NIL NIL NIL NIL NIL NIL NIL)}
          (%i5) arrays;
          (%o5)                     [aa, bb, cc]

Function: fillarray (<A>, <B>)

     Fills array <A> from <B>, which is a list or an array.

     If a specific type was declared for <A> when it was created, it
     can only be filled with elements of that same type; it is an error
     if an attempt is made to copy an element of a different type.

     If the dimensions of the arrays <A> and <B> are different, <A> is
     filled in row-major order.  If there are not enough elements in
     <B> the last element is used to fill out the rest of <A>.  If
     there are too many, the remaining ones are ignored.

     `fillarray' returns its first argument.

     Examples:

     Create an array of 9 elements and fill it from a list.
          (%i1) array (a1, fixnum, 8);
          (%o1)                          a1
          (%i2) listarray (a1);
          (%o2)              [0, 0, 0, 0, 0, 0, 0, 0, 0]
          (%i3) fillarray (a1, [1, 2, 3, 4, 5, 6, 7, 8, 9]);
          (%o3)                          a1
          (%i4) listarray (a1);
          (%o4)              [1, 2, 3, 4, 5, 6, 7, 8, 9]

     When there are too few elements to fill the array, the last
     element is repeated.  When there are too many elements, the extra
     elements are ignored.
          (%i1) a2 : make_array (fixnum, 8);
          (%o1)             {Array:  #(0 0 0 0 0 0 0 0)}
          (%i2) fillarray (a2, [1, 2, 3, 4, 5]);
          (%o2)             {Array:  #(1 2 3 4 5 5 5 5)}
          (%i3) fillarray (a2, [4]);
          (%o3)             {Array:  #(4 4 4 4 4 4 4 4)}
          (%i4) fillarray (a2, makelist (i, i, 1, 100));
          (%o4)             {Array:  #(1 2 3 4 5 6 7 8)}

     Multple-dimension arrays are filled in row-major order.

          (%i1) a3 : make_array (fixnum, 2, 5);
          (%o1)        {Array:  #2A((0 0 0 0 0) (0 0 0 0 0))}
          (%i2) fillarray (a3, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
          (%o2)        {Array:  #2A((1 2 3 4 5) (6 7 8 9 10))}
          (%i3) a4 : make_array (fixnum, 5, 2);
          (%o3)     {Array:  #2A((0 0) (0 0) (0 0) (0 0) (0 0))}
          (%i4) fillarray (a4, a3);
          (%o4)     {Array:  #2A((1 2) (3 4) (5 6) (7 8) (9 10))}

Function: listarray (<A>)

     Returns a list of the elements of the array <A>.  The argument <A>
     may be a declared array, an undeclared (hashed) array, an array
     function, or a subscripted function.

     Elements are listed in row-major order.  That is, elements are
     sorted according to the first index, then according to the second
     index, and so on.  The sorting order of index values is the same as
     the order established by `orderless'.

     For undeclared arrays, array functions, and subscripted functions,
     the elements correspond to the index values returned by
     `arrayinfo'.

     Unbound elements of declared general arrays (that is, not `fixnum'
     and not `flonum') are returned as `#####'.  Unbound elements of
     declared `fixnum' or `flonum' arrays are returned as 0 or 0.0,
     respectively.  Unbound elements of undeclared arrays, array
     functions, and subscripted functions are not returned.

     Examples:

     `listarray' and `arrayinfo' applied to a declared array.

          (%i1) array (aa, 2, 3);
          (%o1)                          aa
          (%i2) aa [2, 3] : %pi;
          (%o2)                          %pi
          (%i3) aa [1, 2] : %e;
          (%o3)                          %e
          (%i4) listarray (aa);
          (%o4) [#####, #####, #####, #####, #####, #####, %e, #####,
                                                  #####, #####, #####, %pi]
          (%i5) arrayinfo (aa);
          (%o5)                 [declared, 2, [2, 3]]

     `listarray' and `arrayinfo' applied to an undeclared (hashed)
     array.

          (%i1) bb [FOO] : (a + b)^2;
                                             2
          (%o1)                       (b + a)
          (%i2) bb [BAR] : (c - d)^3;
                                             3
          (%o2)                       (c - d)
          (%i3) listarray (bb);
                                        3         2
          (%o3)                 [(c - d) , (b + a) ]
          (%i4) arrayinfo (bb);
          (%o4)               [hashed, 1, [BAR], [FOO]]

     `listarray' and `arrayinfo' applied to an array function.

          (%i1) cc [x, y] := y / x;
                                               y
          (%o1)                      cc     := -
                                       x, y    x
          (%i2) cc [u, v];
                                          v
          (%o2)                           -
                                          u
          (%i3) cc [4, z];
                                          z
          (%o3)                           -
                                          4
          (%i4) listarray (cc);
                                        z  v
          (%o4)                        [-, -]
                                        4  u
          (%i5) arrayinfo (cc);
          (%o5)              [hashed, 2, [4, z], [u, v]]

     `listarray' and `arrayinfo' applied to a subscripted function.

          (%i1) dd [x] (y) := y ^ x;
                                               x
          (%o1)                     dd (y) := y
                                      x
          (%i2) dd [a + b];
                                              b + a
          (%o2)                  lambda([y], y     )
          (%i3) dd [v - u];
                                              v - u
          (%o3)                  lambda([y], y     )
          (%i4) listarray (dd);
                                   b + a                v - u
          (%o4)      [lambda([y], y     ), lambda([y], y     )]
          (%i5) arrayinfo (dd);
          (%o5)             [hashed, 1, [b + a], [v - u]]

Function: make_array (<type>, <dim_1>, ..., <dim_n>)

     Creates and returns a Lisp array.  <type> may be `any', `flonum',
     `fixnum', `hashed' or `functional'.  There are n indices, and the
     i'th index runs from 0 to <dim_i> - 1.

     The advantage of `make_array' over `array'  is that the return
     value doesn't have a name, and once a pointer to it goes away, it
     will also go away.  For example, if `y: make_array (...)' then `y'
     points to an object which takes up space, but after `y: false',
     `y' no longer points to that object, so the object can be garbage
     collected.

     Examples:

          (%i1) A1 : make_array (fixnum, 10);
          (%o1)           {Array:  #(0 0 0 0 0 0 0 0 0 0)}
          (%i2) A1 [8] : 1729;
          (%o2)                         1729
          (%i3) A1;
          (%o3)          {Array:  #(0 0 0 0 0 0 0 0 1729 0)}
          (%i4) A2 : make_array (flonum, 10);
          (%o4) {Array:  #(0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0)}
          (%i5) A2 [2] : 2.718281828;
          (%o5)                      2.718281828
          (%i6) A2;
          (%o6)
               {Array:  #(0.0 0.0 2.718281828 0.0 0.0 0.0 0.0 0.0 0.0 0.0)}
          (%i7) A3 : make_array (any, 10);
          (%o7) {Array:  #(NIL NIL NIL NIL NIL NIL NIL NIL NIL NIL)}
          (%i8) A3 [4] : x - y - z;
          (%o8)                      - z - y + x
          (%i9) A3;
          (%o9) {Array:  #(NIL NIL NIL NIL ((MPLUS SIMP) $X ((MTIMES SIMP)\
           -1 $Y) ((MTIMES SIMP) -1 $Z))
            NIL NIL NIL NIL NIL)}
          (%i10) A4 : make_array (fixnum, 2, 3, 5);
          (%o10) {Array:  #3A(((0 0 0 0 0) (0 0 0 0 0) (0 0 0 0 0)) ((0 0 \
          0 0 0) (0 0 0 0 0) (0 0 0 0 0)))}
          (%i11) fillarray (A4, makelist (i, i, 1, 2*3*5));
          (%o11) {Array:  #3A(((1 2 3 4 5) (6 7 8 9 10) (11 12 13 14 15))
              ((16 17 18 19 20) (21 22 23 24 25) (26 27 28 29 30)))}
          (%i12) A4 [0, 2, 1];
          (%o12)                         12

Function: rearray (<A>, <dim_1>, ..., <dim_n>)

     Changes the dimensions of an array.  The new array will be filled
     with the elements of the old one in row-major order.  If the old
     array was too small, the remaining elements are filled with
     `false', `0.0' or `0', depending on the type of the array.  The
     type of the array cannot be changed.

Function: remarray (<A_1>, ..., <A_n>)
Function: remarray (all)

     Removes arrays and array associated functions and frees the
     storage occupied.  The arguments may be declared arrays,
     undeclared (hashed) arrays, array functions, and subscripted
     functions.

     `remarray (all)' removes all items in the global list `arrays'.

     It may be necessary to use this function if it is desired to
     redefine the values in a hashed array.

     `remarray' returns the list of arrays removed.

     `remarray' quotes its arguments.

Function: subvar (<x>, <i>)

     Evaluates the subscripted expression `<x>[<i>]'.

     `subvar' evaluates its arguments.

     `arraymake (<x>, [<i>]' constructs the expression `<x>[<i>]', but
     does not evaluate it.

     Examples:

          (%i1) x : foo $
          (%i2) i : 3 $
          (%i3) subvar (x, i);
          (%o3)                         foo
                                           3
          (%i4) foo : [aa, bb, cc, dd, ee]$
          (%i5) subvar (x, i);
          (%o5)                          cc
          (%i6) arraymake (x, [i]);
          (%o6)                         foo
                                           3
          (%i7) ''%;
          (%o7)                          cc

Function: subvarp (<expr>)

     Returns `true' if <expr> is a subscripted variable, for example
     `a[i]'.

Option variable: use_fast_arrays

     If `true' then only two types of arrays are recognized:

       1. The art-q array (t in Common Lisp) which may have several
          dimensions indexed by integers, and may hold any Lisp or
          Maxima object as an entry.  To construct such an array, enter
          `a:make_array(any,3,4);' then `a' will have as value, an
          array with twelve slots, and the indexing is zero based.

       2. The Hash_table array which is the default type of array
          created if one does `b[x+1]:y^2' (and `b' is not already an
          array, a list, or a matrix - if it were one of these an error
          would be caused since `x+1' would not be a valid subscript
          for an art-q array, a list or a matrix).  Its indices (also
          known as keys) may be any object.  It only takes one key at a
          time (`b[x+1,u]:y' would ignore the `u').  Referencing is
          done by `b[x+1] ==> y^2'.  Of course the key may be a list,
          e.g. `b[[x+1,u]]:y' would be valid.  This is incompatible
          with the old Maxima hash arrays, but saves consing.

     An advantage of storing the arrays as values of the symbol is that
     the usual conventions about local variables of a function apply to
     arrays as well.  The Hash_table type also uses less consing and is
     more efficient than the old type of Maxima hashar.  To obtain
     consistent behaviour in translated and compiled code set
     `translate_fast_arrays' to be `true'.

5.6 Structures


* Introduction to Structures::
* Functions and Variables for Structures::
Global variable: structures

     `structures' is the list of user-defined structures defined by
     `defstruct'.

Function: defstruct (<S>(<a_1>, ..., <a_n>))
Function: defstruct (<S>(<a_1> = <v_1>, ..., <a_n> = <v_n>))

     Define a structure, which is a list of named fields <a_1>, ...,
     <a_n> associated with a symbol <S>.  An instance of a structure is
     just an expression which has operator <S> and exactly `n'
     arguments.  `new(<S>)' creates a new instance of structure <S>.

     An argument which is just a symbol <a> specifies the name of a
     field.  An argument which is an equation `<a> = <v>' specifies the
     field name <a> and its default value <v>.  The default value can
     be any expression.

     `defstruct' puts <S> on the list of user-defined structures,
     `structures'.

     `kill(<S>)' removes <S> from the list of user-defined structures,
     and removes the structure definition.

     Examples:

          (%i1) defstruct (foo (a, b, c));
          (%o1)                    [foo(a, b, c)]
          (%i2) structures;
          (%o2)                    [foo(a, b, c)]
          (%i3) new (foo);
          (%o3)                     foo(a, b, c)
          (%i4) defstruct (bar (v, w, x = 123, y = %pi));
          (%o4)             [bar(v, w, x = 123, y = %pi)]
          (%i5) structures;
          (%o5)      [foo(a, b, c), bar(v, w, x = 123, y = %pi)]
          (%i6) new (bar);
          (%o6)              bar(v, w, x = 123, y = %pi)
          (%i7) kill (foo);
          (%o7)                         done
          (%i8) structures;
          (%o8)             [bar(v, w, x = 123, y = %pi)]

Function: new (<S>)
Function: new (<S> (<v_1>, ..., <v_n>))

     `new' creates new instances of structures.

     `new(<S>)' creates a new instance of structure <S> in which each
     field is assigned its default value, if any, or no value at all if
     no default was specified in the structure definition.

     `new(<S>(<v_1>, ..., <v_n>))' creates a new instance of <S> in
     which fields are assigned the values <v_1>, ..., <v_n>.

     Examples:

          (%i1) defstruct (foo (w, x = %e, y = 42, z));
          (%o1)              [foo(w, x = %e, y = 42, z)]
          (%i2) new (foo);
          (%o2)               foo(w, x = %e, y = 42, z)
          (%i3) new (foo (1, 2, 4, 8));
          (%o3)            foo(w = 1, x = 2, y = 4, z = 8)

Operator: @

     `@' is the structure field access operator.  The expression `<x>@
     <a>' refers to the value of field <a> of the structure instance
     <x>.  The field name is not evaluated.

     If the field <a> in <x> has not been assigned a value, `<x>@ <a>'
     evaluates to itself.

     `kill(<x>@ <a>)' removes the value of field <a> in <x>.

     Examples:

          (%i1) defstruct (foo (x, y, z));
          (%o1)                    [foo(x, y, z)]
          (%i2) u : new (foo (123, a - b, %pi));
          (%o2)           foo(x = 123, y = a - b, z = %pi)
          (%i3) u@z;
          (%o3)                          %pi
          (%i4) u@z : %e;
          (%o4)                          %e
          (%i5) u;
          (%o5)            foo(x = 123, y = a - b, z = %e)
          (%i6) kill (u@z);
          (%o6)                         done
          (%i7) u;
          (%o7)              foo(x = 123, y = a - b, z)
          (%i8) u@z;
          (%o8)                          u@z

     The field name is not evaluated.

          (%i1) defstruct (bar (g, h));
          (%o1)                      [bar(g, h)]
          (%i2) x : new (bar);
          (%o2)                       bar(g, h)
          (%i3) x@h : 42;
          (%o3)                          42
          (%i4) h : 123;
          (%o4)                          123
          (%i5) x@h;
          (%o5)                          42
          (%i6) x@h : 19;
          (%o6)                          19
          (%i7) x;
          (%o7)                    bar(g, h = 19)
          (%i8) h;
          (%o8)                          123

6 Expressions


Introduction to Expressions
Nouns and Verbs
Identifiers
Inequality
Functions and Variables for Expressions

6.1 Introduction to Expressions

There are a number of reserved words which should not be used as
variable names.  Their use would cause a possibly cryptic syntax error.

     integrate            next           from                 diff
     in                   at             limit                sum
     for                  and            elseif               then
     else                 do             or                   if
     unless               product        while                thru
     step

   Most things in Maxima are expressions.  A sequence of expressions
can be made into an expression by separating them by commas and putting
parentheses around them.  This is similar to the C comma expression.

     (%i1) x: 3$
     (%i2) (x: x+1, x: x^2);
     (%o2)                          16
     (%i3) (if (x > 17) then 2 else 4);
     (%o3)                           4
     (%i4) (if (x > 17) then x: 2 else y: 4, y+x);
     (%o4)                          20

   Even loops in Maxima are expressions, although the value they return
is the not too useful `done'.

     (%i1) y: (x: 1, for i from 1 thru 10 do (x: x*i))$
     (%i2) y;
     (%o2)                         done

   Whereas what you really want is probably to include a third term in
the comma expression which actually gives back the value.

     (%i3) y: (x: 1, for i from 1 thru 10 do (x: x*i), x)$
     (%i4) y;
     (%o4)                        3628800

6.2 Nouns and Verbs

Maxima distinguishes between operators which are "nouns" and operators
which are "verbs".  A verb is an operator which can be executed.  A
noun is an operator which appears as a symbol in an expression, without
being executed.  By default, function names are verbs.  A verb can be
changed into a noun by quoting the function name or applying the
`nounify'  function.  A noun can be changed into a verb by applying the
`verbify'  function.  The evaluation flag `nouns'  causes `ev'  to
evaluate nouns in an expression.

   The verb form is distinguished by a leading dollar sign `$' on the
corresponding Lisp symbol.  In contrast, the noun form is distinguished
by a leading percent sign `%' on the corresponding Lisp symbol.  Some
nouns have special display properties, such as `'integrate' and
`'derivative' (returned by `diff' ), but most do not.  By default, the
noun and verb forms of a function are identical when displayed.  The
global flag `noundisp'

   causes Maxima to display nouns with a leading quote mark `''.

   See also `noun',  `nouns',  `nounify',  and `verbify'.

   Examples:

     (%i1) foo (x) := x^2;
                                          2
     (%o1)                     foo(x) := x
     (%i2) foo (42);
     (%o2)                         1764
     (%i3) 'foo (42);
     (%o3)                        foo(42)
     (%i4) 'foo (42), nouns;
     (%o4)                         1764
     (%i5) declare (bar, noun);
     (%o5)                         done
     (%i6) bar (x) := x/17;
                                          x
     (%o6)                    ''bar(x) := --
                                          17
     (%i7) bar (52);
     (%o7)                        bar(52)
     (%i8) bar (52), nouns;
                                    52
     (%o8)                          --
                                    17
     (%i9) integrate (1/x, x, 1, 42);
     (%o9)                        log(42)
     (%i10) 'integrate (1/x, x, 1, 42);
                                  42
                                 /
                                 [   1
     (%o10)                      I   - dx
                                 ]   x
                                 /
                                  1
     (%i11) ev (%, nouns);
     (%o11)                       log(42)

6.3 Identifiers

Maxima identifiers may comprise alphabetic characters, plus the
numerals 0 through 9, plus any special character preceded by the
backslash `\' character.

   A numeral may be the first character of an identifier if it is
preceded by a backslash.  Numerals which are the second or later
characters need not be preceded by a backslash.

   Characters may be declared alphabetic by the `declare' function.  If
so declared, they need not be preceded by a backslash in an identifier.
The alphabetic characters are initially `A' through `Z', `a' through
`z', `%', and `_'.

   Maxima is case-sensitive.  The identifiers `foo', `FOO', and `Foo'
are distinct.  See *note Lisp and Maxima:: for more on this point.

   A Maxima identifier is a Lisp symbol which begins with a dollar sign
`$'.  Any other Lisp symbol is preceded by a question mark `?' when it
appears in Maxima.  See *note Lisp and Maxima:: for more on this point.

   Examples:

     (%i1) %an_ordinary_identifier42;
     (%o1)               %an_ordinary_identifier42
     (%i2) embedded\ spaces\ in\ an\ identifier;
     (%o2)           embedded spaces in an identifier
     (%i3) symbolp (%);
     (%o3)                         true
     (%i4) [foo+bar, foo\+bar];
     (%o4)                 [foo + bar, foo+bar]
     (%i5) [1729, \1729];
     (%o5)                     [1729, 1729]
     (%i6) [symbolp (foo\+bar), symbolp (\1729)];
     (%o6)                     [true, true]
     (%i7) [is (foo\+bar = foo+bar), is (\1729 = 1729)];
     (%o7)                    [false, false]
     (%i8) baz\~quux;
     (%o8)                       baz~quux
     (%i9) declare ("~", alphabetic);
     (%o9)                         done
     (%i10) baz~quux;
     (%o10)                      baz~quux
     (%i11) [is (foo = FOO), is (FOO = Foo), is (Foo = foo)];
     (%o11)                [false, false, false]
     (%i12) :lisp (defvar *my-lisp-variable* '$foo)
     *MY-LISP-VARIABLE*
     (%i12) ?\*my\-lisp\-variable\*;
     (%o12)                         foo

6.4 Inequality

Maxima has the inequality operators `<', `<=', `>=', `>', `#', and
`notequal'.  See `if' for a description of conditional expressions.

6.5 Functions and Variables for Expressions

Function: alias (<new_name_1>, <old_name_1>, ..., <new_name_n>, <old_name_n>)

     provides an alternate name for a (user or system) function,
     variable, array, etc.  Any even number of arguments may be used.

System variable: aliases

     Default value: `[]'

     `aliases' is the list of atoms which have a user defined alias
     (set up by the `alias',  `ordergreat',  `orderless'  functions or
     by declaring the atom a `noun'  with `declare'.  )

Keyword: allbut

     works with the `part' commands (i.e.  `part',

     `inpart',  `substpart',  `substinpart',

     `dpart',  and `lpart' ).  For example,

          (%i1) expr : e + d + c + b + a;
          (%o1)                   e + d + c + b + a
          (%i2) part (expr, [2, 5]);
          (%o2)                         d + a

     while

          (%i1) expr : e + d + c + b + a;
          (%o1)                   e + d + c + b + a
          (%i2) part (expr, allbut (2, 5));
          (%o2)                       e + c + b

     `allbut' is also recognized by `kill'.

          (%i1) [aa : 11, bb : 22, cc : 33, dd : 44, ee : 55];
          (%o1)                 [11, 22, 33, 44, 55]
          (%i2) kill (allbut (cc, dd));
          (%o0)                         done
          (%i1) [aa, bb, cc, dd];
          (%o1)                   [aa, bb, 33, 44]

     `kill(allbut(<a_1>, <a_2>, ...))' has the effect of `kill(all)'
     except that it does not kill the symbols <a_1>, <a_2>, ...

Function: args (<expr>)

     Returns the list of arguments of `expr', which may be any kind of
     expression other than an atom.  Only the arguments of the
     top-level operator are extracted; subexpressions of `expr' appear
     as elements or subexpressions of elements of the list of arguments.

     The order of the items in the list may depend on the global flag
     `inflag'.

     `args (<expr>)' is equivalent to `substpart ("[", <expr>, 0)'.
     See also `substpart',  and `op'.

Function: atom (<expr>)

     Returns `true' if <expr> is atomic (i.e. a number, name or string)
     else `false'.  Thus `atom(5)' is `true' while `atom(a[1])' and
     `atom(sin(x))' are `false' (assuming `a[1]' and `x' are unbound).

Function: box (<expr>)
Function: box (<expr>, <a>)

     Returns <expr> enclosed in a box.  The return value is an
     expression with `box' as the operator and <expr> as the argument.
     A box is drawn on the display when `display2d' is `true'.

     `box (<expr>, <a>)' encloses <expr> in a box labelled by the
     symbol <a>.  The label is truncated if it is longer than the width
     of the box.

     `box' evaluates its argument.  However, a boxed expression does not
     evaluate to its content, so boxed expressions are effectively
     excluded from computations.

     `boxchar'  is the character used to draw the box in `box' and in
     the `dpart'  and `lpart'  functions.

     Examples:

          (%i1) box (a^2 + b^2);
                                      """""""""
                                      " 2    2"
          (%o1)                       "b  + a "
                                      """""""""
          (%i2) a : 1234;
          (%o2)                         1234
          (%i3) b : c - d;
          (%o3)                         c - d
          (%i4) box (a^2 + b^2);
                                """"""""""""""""""""
                                "       2          "
          (%o4)                 "(c - d)  + 1522756"
                                """"""""""""""""""""
          (%i5) box (a^2 + b^2, term_1);
                                term_1""""""""""""""
                                "       2          "
          (%o5)                 "(c - d)  + 1522756"
                                """"""""""""""""""""
          (%i6) 1729 - box (1729);
                                           """"""
          (%o6)                     1729 - "1729"
                                           """"""
          (%i7) boxchar: "-";
          (%o7)                           -
          (%i8) box (sin(x) + cos(y));
                                  -----------------
          (%o8)                   -cos(y) + sin(x)-
                                  -----------------

Option variable: boxchar

     Default value: `"'

     `boxchar' is the character used to draw the box in the `box'

     and in the `dpart'  and `lpart'  functions.

     All boxes in an expression are drawn with the current value of
     `boxchar'; the drawing character is not stored with the box
     expression.

Function: collapse (<expr>)

     Collapses <expr> by causing all of its common (i.e., equal)
     subexpressions to share (i.e., use the same cells), thereby saving
     space.  (`collapse' is a subroutine used by the `optimize'
     command.)  Thus, calling `collapse' may be useful after loading in
     a `save'  file.  You can collapse several expressions together by
     using `collapse ([<expr_1>, ..., <expr_n>])'.  Similarly, you can
     collapse the elements of the array `A' by doing `collapse
     (listarray ('A))'.

Function: disolate (<expr>, <x_1>, ..., <x_n>)

     is similar to `isolate (<expr>, <x>)' except that it enables the
     user to isolate more than one variable simultaneously.  This might
     be useful, for example, if one were attempting to change variables
     in a multiple integration, and that variable change involved two
     or more of the integration variables.  This function is autoloaded
     from `simplification/disol.mac'.  A demo is available by
     `demo("disol")$'.

Function: dispform (<expr>)
Function: dispform (<expr>, all)

     Returns the external representation of <expr>.

     `dispform(<expr>)' returns the external representation with
     respect to the main (top-level) operator.  `dispform(<expr>, all)'
     returns the external representation with respect to all operators
     in <expr>.

     See also `part',  `inpart',  and `inflag'.

     Examples:

     The internal representation of `- x' is "negative one times `x'"
     while the external representation is "minus `x'".

          (%i1) - x;
          (%o1)                          - x
          (%i2) ?format (true, "~S~%", %);
          ((MTIMES SIMP) -1 $X)
          (%o2)                         false
          (%i3) dispform (- x);
          (%o3)                          - x
          (%i4) ?format (true, "~S~%", %);
          ((MMINUS SIMP) $X)
          (%o4)                         false

     The internal representation of `sqrt(x)' is "`x' to the power 1/2"
     while the external representation is "square root of `x'".

          (%i1) sqrt (x);
          (%o1)                        sqrt(x)
          (%i2) ?format (true, "~S~%", %);
          ((MEXPT SIMP) $X ((RAT SIMP) 1 2))
          (%o2)                         false
          (%i3) dispform (sqrt (x));
          (%o3)                        sqrt(x)
          (%i4) ?format (true, "~S~%", %);
          ((%SQRT SIMP) $X)
          (%o4)                         false

     Use of the optional argument `all'.

          (%i1) expr : sin (sqrt (x));
          (%o1)                     sin(sqrt(x))
          (%i2) freeof (sqrt, expr);
          (%o2)                         true
          (%i3) freeof (sqrt, dispform (expr));
          (%o3)                         true
          (%i4) freeof (sqrt, dispform (expr, all));
          (%o4)                         false

Function: dpart (<expr>, <n_1>, ..., <n_k>)

     Selects the same subexpression as `part',  but instead of just
     returning that subexpression as its value, it returns the whole
     expression with the selected subexpression displayed inside a box.
     The box is actually part of the expression.

          (%i1) dpart (x+y/z^2, 1, 2, 1);
                                       y
          (%o1)                       ---- + x
                                         2
                                      """
                                      "z"
                                      """

Option variable: exptisolate

     Default value: `false'

     `exptisolate', when `true', causes `isolate (expr, var)' to
     examine exponents of atoms (such as `%e') which contain `var'.

Option variable: exptsubst

     Default value: `false'

     `exptsubst', when `true', permits substitutions such as `y' for
     `%e^x' in `%e^(a x)'.

Function: freeof (<x_1>, ..., <x_n>, <expr>)

     `freeof (<x_1>, <expr>)' returns `true' if no subexpression of
     <expr> is equal to <x_1> or if <x_1> occurs only as a dummy
     variable in <expr>, or if <x_1> is neither the noun nor verb form
     of any operator in <expr>, and returns `false' otherwise.

     `freeof (<x_1>, ..., <x_n>, <expr>)' is equivalent to `freeof
     (<x_1>, <expr>) and ... and freeof (<x_n>, <expr>)'.

     The arguments <x_1>, ..., <x_n> may be names of functions and
     variables, subscripted names, operators (enclosed in double
     quotes), or general expressions.  `freeof' evaluates its arguments.

     `freeof' operates only on <expr> as it stands (after
     simplification and evaluation) and does not attempt to determine
     if some equivalent expression would give a different result.  In
     particular, simplification may yield an equivalent but different
     expression which comprises some different elements than the
     original form of <expr>.

     A variable is a dummy variable in an expression if it has no
     binding outside of the expression.  Dummy variables recognized by
     `freeof' are the index of a sum or product, the limit variable in
     `limit',  the integration variable in the definite integral form
     of `integrate' , the original variable in `laplace',  formal
     variables in `at'  expressions, and arguments in `lambda'
     expressions.

     The indefinite form of `integrate' is not free of its variable of
     integration.

     Examples:

     Arguments are names of functions, variables, subscripted names,
     operators, and expressions.  `freeof (a, b, expr)' is equivalent to
     `freeof (a, expr) and freeof (b, expr)'.

          (%i1) expr: z^3 * cos (a[1]) * b^(c+d);
                                           d + c  3
          (%o1)                   cos(a ) b      z
                                       1
          (%i2) freeof (z, expr);
          (%o2)                         false
          (%i3) freeof (cos, expr);
          (%o3)                         false
          (%i4) freeof (a[1], expr);
          (%o4)                         false
          (%i5) freeof (cos (a[1]), expr);
          (%o5)                         false
          (%i6) freeof (b^(c+d), expr);
          (%o6)                         false
          (%i7) freeof ("^", expr);
          (%o7)                         false
          (%i8) freeof (w, sin, a[2], sin (a[2]), b*(c+d), expr);
          (%o8)                         true

     `freeof' evaluates its arguments.

          (%i1) expr: (a+b)^5$
          (%i2) c: a$
          (%i3) freeof (c, expr);
          (%o3)                         false

     `freeof' does not consider equivalent expressions.  Simplification
     may yield an equivalent but different expression.

          (%i1) expr: (a+b)^5$
          (%i2) expand (expr);
                    5        4       2  3       3  2      4      5
          (%o2)    b  + 5 a b  + 10 a  b  + 10 a  b  + 5 a  b + a
          (%i3) freeof (a+b, %);
          (%o3)                         true
          (%i4) freeof (a+b, expr);
          (%o4)                         false
          (%i5) exp (x);
                                           x
          (%o5)                          %e
          (%i6) freeof (exp, exp (x));
          (%o6)                         true

     A summation or definite integral is free of its dummy variable.
     An indefinite integral is not free of its variable of integration.

          (%i1) freeof (i, 'sum (f(i), i, 0, n));
          (%o1)                         true
          (%i2) freeof (x, 'integrate (x^2, x, 0, 1));
          (%o2)                         true
          (%i3) freeof (x, 'integrate (x^2, x));
          (%o3)                         false

Option variable: inflag

     Default value: `false'

     When `inflag' is `true', functions for part extraction inspect the
     internal form of `expr'.

     Note that the simplifier re-orders expressions.  Thus `first (x +
     y)' returns `x' if `inflag' is `true' and `y' if `inflag' is
     `false'.  (`first (y + x)' gives the same results.)

     Also, setting `inflag' to `true' and calling `part'  or `substpart'
     is the same as calling `inpart'  or `substinpart'.

     Functions affected by the setting of `inflag' are: `part',

     `substpart',  `first',  `rest',  `last',

     `length',  the `for'  ... `in' construct, `map',  `fullmap',
     `maplist',  `reveal'  and `pickapart'.

Function: inpart (<expr>, <n_1>, ..., <n_k>)

     is similar to `part'  but works on the internal representation of
     the expression rather than the displayed form and thus may be
     faster since no formatting is done.  Care should be taken with
     respect to the order of subexpressions in sums and products (since
     the order of variables in the internal form is often different
     from that in the displayed form) and in dealing with unary minus,
     subtraction, and division (since these operators are removed from
     the expression).  `part (x+y, 0)' or `inpart (x+y, 0)' yield `+',
     though in order to refer to the operator it must be enclosed in "s.
     For example `... if inpart (%o9,0) = "+" then ...'.

     Examples:

          (%i1) x + y + w*z;
          (%o1)                      w z + y + x
          (%i2) inpart (%, 3, 2);
          (%o2)                           z
          (%i3) part (%th (2), 1, 2);
          (%o3)                           z
          (%i4) 'limit (f(x)^g(x+1), x, 0, minus);
                                            g(x + 1)
          (%o4)                 limit   f(x)
                                x -> 0-
          (%i5) inpart (%, 1, 2);
          (%o5)                       g(x + 1)

Function: isolate (<expr>, <x>)

     Returns <expr> with subexpressions which are sums and which do not
     contain <var> replaced by intermediate expression labels (these
     being atomic symbols like `%t1', `%t2', ...).  This is often
     useful to avoid unnecessary expansion of subexpressions which
     don't contain the variable of interest.  Since the intermediate
     labels are bound to the subexpressions they can all be substituted
     back by evaluating the expression in which they occur.

     `exptisolate'  (default value: `false') if `true' will cause
     `isolate' to examine exponents of atoms (like `%e') which contain
     <var>.

     `isolate_wrt_times' if `true', then `isolate' will also isolate
     with respect to products.  See `isolate_wrt_times'.

     Do `example (isolate)' for examples.

Option variable: isolate_wrt_times

     Default value: `false'

     When `isolate_wrt_times' is `true', `isolate' will also isolate
     with respect to products.  E.g. compare both settings of the
     switch on

          (%i1) isolate_wrt_times: true$
          (%i2) isolate (expand ((a+b+c)^2), c);

          (%t2)                          2 a

Option variable: listconstvars

     Default value: `false'

     When `listconstvars' is `true', it will cause `listofvars'  to
     include `%e', `%pi', `%i', and any variables declared constant in
     the list it returns if they appear in the expression `listofvars'
     is called on.  The default is to omit these.

Option variable: listdummyvars

     Default value: `true'

     When `listdummyvars' is `false', "dummy variables" in the
     expression will not be included in the list returned by
     `listofvars'.    (The meaning of "dummy variables" is as given in
     `freeof'.    "Dummy variables" are mathematical things like the
     index of a sum or product, the limit variable, and the definite
     integration variable.)

     Example:

          (%i1) listdummyvars: true$
          (%i2) listofvars ('sum(f(i), i, 0, n));
          (%o2)                        [i, n]
          (%i3) listdummyvars: false$
          (%i4) listofvars ('sum(f(i), i, 0, n));
          (%o4)                          [n]

Function: listofvars (<expr>)

     Returns a list of the variables in <expr>.

     `listconstvars'  if `true' causes `listofvars' to include `%e',
     `%pi', `%i', and any variables declared constant in the list it
     returns if they appear in <expr>.  The default is to omit these.

     See also the option variable `listdummyvars'  to exclude or include
     "dummy variables" in the list of variables.

          (%i1) listofvars (f (x[1]+y) / g^(2+a));
          (%o1)                     [g, a, x , y]
                                            1

Function: lfreeof (<list>, <expr>)

     For each member <m> of <list>, calls `freeof (<m>, <expr>)'.  It
     returns `false' if any call to `freeof'  does and `true' otherwise.

Function: lpart (<label>, <expr>, <n_1>, ..., <n_k>)

     is similar to `dpart'  but uses a labelled box.  A labelled box is
     similar to the one produced by `dpart' but it has a name in the
     top line.

Property: mainvar

     You may declare variables to be `mainvar'.  The ordering scale for
     atoms is essentially: numbers < constants (e.g., `%e', `%pi') <
     scalars < other variables < mainvars.  E.g., compare `expand
     ((X+Y)^4)' with `(declare (x, mainvar), expand ((x+y)^4))'.
     (Note: Care should be taken if you elect to use the above feature.
     E.g., if you subtract an expression in which `x' is a `mainvar'
     from one in which `x' isn't a `mainvar', resimplification e.g.
     with `ev (expr, simp)' may be necessary if cancellation is to
     occur.  Also, if you save an expression in which `x' is a
     `mainvar', you probably should also save `x'.)

Property: noun

     `noun' is one of the options of the `declare'  command.  It makes a
     function so declared a "noun", meaning that it won't be evaluated
     automatically.

     Example:

          (%i1) factor (12345678);
                                       2
          (%o1)                     2 3  47 14593
          (%i2) declare (factor, noun);
          (%o2)                         done
          (%i3) factor (12345678);
          (%o3)                   factor(12345678)
          (%i4) ''%, nouns;
                                       2
          (%o4)                     2 3  47 14593

Option variable: noundisp

     Default value: `false'

     When `noundisp' is `true', nouns display with a single quote.
     This switch is always `true' when displaying function definitions.

Function: nounify (<f>)

     Returns the noun form of the function name <f>.  This is needed if
     one wishes to refer to the name of a verb function as if it were a
     noun.  Note that some verb functions will return their noun forms
     if they can't be evaluated for certain arguments.  This is also
     the form returned if a function call is preceded by a quote.

     See also `verbify'.

Function: nterms (<expr>)

     Returns the number of terms that <expr> would have if it were fully
     expanded out and no cancellations or combination of terms occurred.
     Note that expressions like `sin (<expr>)', `sqrt (<expr>)', `exp
     (<expr>)', etc. count as just one term regardless of how many
     terms <expr> has (if it is a sum).

Function: op (<expr>)

     Returns the main operator of the expression <expr>.  `op (<expr>)'
     is equivalent to `part (<expr>, 0)'.

     `op' returns a string if the main operator is a built-in or
     user-defined prefix, binary or n-ary infix, postfix, matchfix, or
     nofix operator.  Otherwise, if <expr> is a subscripted function
     expression, `op' returns the subscripted function; in this case
     the return value is not an atom.  Otherwise, <expr> is an array
     function or ordinary function expression, and `op' returns a
     symbol.

     `op' observes the value of the global flag `inflag'.

     `op' evaluates it argument.

     See also `args'.

     Examples:

          (%i1) stringdisp: true$
          (%i2) op (a * b * c);
          (%o2)                          "*"
          (%i3) op (a * b + c);
          (%o3)                          "+"
          (%i4) op ('sin (a + b));
          (%o4)                          sin
          (%i5) op (a!);
          (%o5)                          "!"
          (%i6) op (-a);
          (%o6)                          "-"
          (%i7) op ([a, b, c]);
          (%o7)                          "["
          (%i8) op ('(if a > b then c else d));
          (%o8)                         "if"
          (%i9) op ('foo (a));
          (%o9)                          foo
          (%i10) prefix (foo);
          (%o10)                        "foo"
          (%i11) op (foo a);
          (%o11)                        "foo"
          (%i12) op (F [x, y] (a, b, c));
          (%o12)                        F
                                         x, y
          (%i13) op (G [u, v, w]);
          (%o13)                          G

Function: operatorp (<expr>, <op>)
Function: operatorp (<expr>, [<op_1>, ..., <op_n>])

     `operatorp (<expr>, <op>)' returns `true' if <op> is equal to the
     operator of <expr>.

     `operatorp (<expr>, [<op_1>, ..., <op_n>])' returns `true' if some
     element <op_1>, ..., <op_n> is equal to the operator of <expr>.

Option variable: opsubst

     Default value: `true'

     When `opsubst' is `false', `subst'  does not attempt to substitute
     into the operator of an expression.  E.g., `(opsubst: false, subst
     (x^2, r, r+r[0]))' will work.

Function: optimize (<expr>)

     Returns an expression that produces the same value and side
     effects as <expr> but does so more efficiently by avoiding the
     recomputation of common subexpressions.  `optimize' also has the
     side effect of "collapsing" its argument so that all common
     subexpressions are shared.  Do `example (optimize)' for examples.

Option variable: optimprefix

     Default value: `%'

     `optimprefix' is the prefix used for generated symbols by the
     `optimize'  command.

Function: ordergreat (<v_1>, ..., <v_n>)
Function: orderless (<v_1>, ..., <v_n>)

     `ordergreat' changes the canonical ordering of Maxima expressions
     such that <v_1> succeeds <v_2> succeeds ...  succeeds <v_n>, and
     <v_n> succeeds any other symbol not mentioned as an argument.

     `orderless' changes the canonical ordering of Maxima expressions
     such that <v_1> precedes <v_2> precedes ... precedes <v_n>, and
     <v_n> precedes any other variable not mentioned as an argument.

     The order established by `ordergreat' and `orderless' is dissolved
     by `unorder'.    `ordergreat' and `orderless' can be called only
     once each, unless `unorder' is called; only the last call to
     `ordergreat' and `orderless' has any effect.

     See also `ordergreatp'.

Function: ordergreatp (<expr_1>, <expr_2>)
Function: orderlessp (<expr_1>, <expr_2>)

     `ordergreatp' returns `true' if <expr_1> succeeds <expr_2> in the
     canonical ordering of Maxima expressions, and `false' otherwise.

     `orderlessp' returns `true' if <expr_1> precedes <expr_2> in the
     canonical ordering of Maxima expressions, and `false' otherwise.

     All Maxima atoms and expressions are comparable under
     `ordergreatp' and `orderlessp', although there are isolated
     examples of expressions for which these predicates are not
     transitive; that is a bug.

     The canonical ordering of atoms (symbols, literal numbers, and
     strings) is the following.

     (integers and floats) precede (bigfloats) precede (declared
     constants) precede (strings) precede (declared scalars) precede
     (first argument to `orderless' ) precedes ... precedes (last
     argument to `orderless') precedes (other symbols) precede (last
     argument to `ordergreat' ) precedes ... precedes (first argument
     to `ordergreat') precedes (declared main variables)

     For non-atomic expressions, the canonical ordering is derived from
     the ordering for atoms.  For the built-in `+' `*' and `^'
     operators, the ordering is not easily summarized.  For other
     built-in operators and all other functions and operators,
     expressions are ordered by their arguments (beginning with the
     first argument), then by the name of the operator or function.  In
     the case of subscripted expressions, the subscripted symbol is
     considered the operator and the subscript is considered an
     argument.

     The canonical ordering of expressions is modified by the functions
     `ordergreat'  and `orderless',  and the `mainvar',

     `constant',  and `scalar' declarations.

     See also `sort'.

     Examples:

     Ordering ordinary symbols and constants.  Note that `%pi' is not
     ordered according to its numerical value.

          (%i1) stringdisp : true;
          (%o1)                         true
          (%i2) sort([%pi, 3b0, 3.0, x, X, "foo", 3, a, 4, "bar", 4.0, 4b0]);
          (%o2) [3, 3.0, 4, 4.0, 3.0b0, 4.0b0, %pi, "bar", "foo", a, x, X]

     Effect of `ordergreat' and `orderless' functions.

          (%i1) sort ([M, H, K, T, E, W, G, A, P, J, S]);
          (%o1)           [A, E, G, H, J, K, M, P, S, T, W]
          (%i2) ordergreat (S, J);
          (%o2)                         done
          (%i3) orderless (M, H);
          (%o3)                         done
          (%i4) sort ([M, H, K, T, E, W, G, A, P, J, S]);
          (%o4)           [M, H, A, E, G, K, P, T, W, J, S]

     Effect of `mainvar', `constant', and `scalar' declarations.

          (%i1) sort ([aa, foo, bar, bb, baz, quux, cc, dd, A1, B1, C1]);
          (%o1)   [aa, bar, baz, bb, cc, dd, foo, quux, A1, B1, C1]
          (%i2) declare (aa, mainvar);
          (%o2)                         done
          (%i3) declare ([baz, quux], constant);
          (%o3)                         done
          (%i4) declare ([A1, B1], scalar);
          (%o4)                         done
          (%i5) sort ([aa, foo, bar, bb, baz, quux, cc, dd, A1, B1, C1]);
          (%o5)   [baz, quux, A1, B1, bar, bb, cc, dd, foo, C1, aa]

     Ordering non-atomic expressions.

          (%i1) sort([1, 2, n, f(1), f(2), f(2, 1), g(1), g(1, 2), g(n),
                      f(n, 1)]);
          (%o1) [1, 2, f(1), g(1), g(1, 2), f(2), f(2, 1), n, g(n),
                                                                   f(n, 1)]
          (%i2) sort ([foo(1), X[1], X[k], foo(k), 1, k]);
          (%o2)            [1, foo(1), X , k, foo(k), X ]
                                        1              k

Function: part (<expr>, <n_1>, ..., <n_k>)

     Returns parts of the displayed form of `expr'.  It obtains the
     part of `expr' as specified by the indices <n_1>, ..., <n_k>.
     First part <n_1> of `expr' is obtained, then part <n_2> of that,
     etc.  The result is part <n_k> of ... part <n_2> of part <n_1> of
     `expr'.  If no indices are specified `expr' is returned.

     `part' can be used to obtain an element of a list, a row of a
     matrix, etc.

     If the last argument to a `part' function is a list of indices then
     several subexpressions are picked out, each one corresponding to an
     index of the list.  Thus `part (x + y + z, [1, 3])' is `z+x'.

     `piece'  holds the last expression selected when using the `part'
     functions.  It is set during the execution of the function and thus
     may be referred to in the function itself as shown below.

     If `partswitch'  is set to `true' then `end' is returned when a
     selected part of an expression doesn't exist, otherwise an error
     message is given.

     See also `inpart',  `substpart',  `substinpart',

     `dpart',  and `lpart'.

     Examples:

          (%i1) part(z+2*y+a,2);
          (%o1)                                 2 y
          (%i2) part(z+2*y+a,[1,3]);
          (%o2)                                z + a
          (%i3) part(z+2*y+a,2,1);
          (%o3)                                  2

     `example (part)' displays additional examples.

Function: partition (<expr>, <x>)

     Returns a list of two expressions.  They are (1) the factors of
     <expr> (if it is a product), the terms of <expr> (if it is a sum),
     or the list (if it is a list) which don't contain <x> and, (2) the
     factors, terms, or list which do.

     Examples:

          (%i1) partition (2*a*x*f(x), x);
          (%o1)                     [2 a, x f(x)]
          (%i2) partition (a+b, x);
          (%o2)                      [b + a, 0]
          (%i3) partition ([a, b, f(a), c], a);
          (%o3)                  [[b, c], [a, f(a)]]

Option variable: partswitch

     Default value: `false'

     When `partswitch' is `true', `end' is returned when a selected
     part of an expression doesn't exist, otherwise an error message is
     given.

Function: pickapart (<expr>, <n>)

     Assigns intermediate expression labels to subexpressions of <expr>
     at depth <n>, an integer.  Subexpressions at greater or lesser
     depths are not assigned labels.  `pickapart' returns an expression
     in terms of intermediate expressions equivalent to the original
     expression <expr>.

     See also `part',  `dpart',  `lpart',

     `inpart',  and `reveal'.

     Examples:

          (%i1) expr: (a+b)/2 + sin (x^2)/3 - log (1 + sqrt(x+1));
                                                    2
                                               sin(x )   b + a
          (%o1)       - log(sqrt(x + 1) + 1) + ------- + -----
                                                  3        2
          (%i2) pickapart (expr, 0);
                                                    2
                                               sin(x )   b + a
          (%t2)       - log(sqrt(x + 1) + 1) + ------- + -----
                                                  3        2
          (%o2)                          %t2
          (%i3) pickapart (expr, 1);

          (%t3)                - log(sqrt(x + 1) + 1)

System variable: piece

     Holds the last expression selected when using the `part'
     functions.  It is set during the execution of the function and
     thus may be referred to in the function itself.

Function: psubst (<list>, <expr>)
Function: psubst (<a>, <b>, <expr>)

     `psubst(<a>, <b>, <expr>)' is simliar to `subst'.  See `subst'.

     In distinction from `subst' the function `psubst' makes parallel
     substitutions, if the first argument <list> is a list of equations.

     See also `sublis'  for making parallel substitutions.

     Example:

     The first example shows parallel substitution with `psubst'.  The
     second example shows the result for the function `subst', which
     does a serial substitution.

          (%i4) psubst ([a^2=b, b=a], sin(a^2) + sin(b));
          (%o4)                           sin(b) + sin(a)
          (%i5) subst ([a^2=b, b=a], sin(a^2) + sin(b));
          (%o5)                              2 sin(a)

Function: rembox (<expr>, unlabelled)
Function: rembox (<expr>, <label>)
Function: rembox (<expr>)

     Removes boxes from <expr>.

     `rembox (<expr>, unlabelled)' removes all unlabelled boxes from
     <expr>.

     `rembox (<expr>, <label>)' removes only boxes bearing <label>.

     `rembox (<expr>)' removes all boxes, labelled and unlabelled.

     Boxes are drawn by the `box',  `dpart',  and `lpart'

     functions.

     Examples:

          (%i1) expr: (a*d - b*c)/h^2 + sin(%pi*x);
                                            a d - b c
          (%o1)                sin(%pi x) + ---------
                                                2
                                               h
          (%i2) dpart (dpart (expr, 1, 1), 2, 2);
                                  """""""    a d - b c
          (%o2)               sin("%pi x") + ---------
                                  """""""      """"
                                               " 2"
                                               "h "
                                               """"
          (%i3) expr2: lpart (BAR, lpart (FOO, %, 1), 2);
                            FOO"""""""""""   BAR""""""""
                            "    """"""" "   "a d - b c"
          (%o3)             "sin("%pi x")" + "---------"
                            "    """"""" "   "  """"   "
                            """"""""""""""   "  " 2"   "
                                             "  "h "   "
                                             "  """"   "
                                             """""""""""
          (%i4) rembox (expr2, unlabelled);
                                            BAR""""""""
                             FOO"""""""""   "a d - b c"
          (%o4)              "sin(%pi x)" + "---------"
                             """"""""""""   "    2    "
                                            "   h     "
                                            """""""""""
          (%i5) rembox (expr2, FOO);
                                            BAR""""""""
                                 """""""    "a d - b c"
          (%o5)              sin("%pi x") + "---------"
                                 """""""    "  """"   "
                                            "  " 2"   "
                                            "  "h "   "
                                            "  """"   "
                                            """""""""""
          (%i6) rembox (expr2, BAR);
                             FOO"""""""""""
                             "    """"""" "   a d - b c
          (%o6)              "sin("%pi x")" + ---------
                             "    """"""" "     """"
                             """"""""""""""     " 2"
                                                "h "
                                                """"
          (%i7) rembox (expr2);
                                            a d - b c
          (%o7)                sin(%pi x) + ---------
                                                2
                                               h

Function: reveal (<expr>, <depth>)

     Replaces parts of <expr> at the specified integer <depth> with
     descriptive summaries.

        * Sums and differences are replaced by `Sum(<n>)' where <n> is
          the number of operands of the sum.

        * Products are replaced by `Product(<n>)' where <n> is the
          number of operands of the product.

        * Exponentials are replaced by `Expt'.

        * Quotients are replaced by `Quotient'.

        * Unary negation is replaced by `Negterm'.

        * Lists are replaced by `List(<n>)' where <n> ist the number of
          elements of the list.

     When <depth> is greater than or equal to the maximum depth of
     <expr>, `reveal (<expr>, <depth>)' returns <expr> unmodified.

     `reveal' evaluates its arguments.  `reveal' returns the summarized
     expression.

     Example:

          (%i1) e: expand ((a - b)^2)/expand ((exp(a) + exp(b))^2);
                                    2            2
                                   b  - 2 a b + a
          (%o1)               -------------------------
                                  b + a     2 b     2 a
                              2 %e      + %e    + %e
          (%i2) reveal (e, 1);
          (%o2)                       Quotient
          (%i3) reveal (e, 2);
                                       Sum(3)
          (%o3)                        ------
                                       Sum(3)
          (%i4) reveal (e, 3);
                               Expt + Negterm + Expt
          (%o4)               ------------------------
                              Product(2) + Expt + Expt
          (%i5) reveal (e, 4);
                                 2                 2
                                b  - Product(3) + a
          (%o5)         ------------------------------------
                                   Product(2)     Product(2)
                        2 Expt + %e           + %e
          (%i6) reveal (e, 5);
                                   2            2
                                  b  - 2 a b + a
          (%o6)              --------------------------
                                 Sum(2)     2 b     2 a
                             2 %e       + %e    + %e
          (%i7) reveal (e, 6);
                                    2            2
                                   b  - 2 a b + a
          (%o7)               -------------------------
                                  b + a     2 b     2 a
                              2 %e      + %e    + %e

Function: sublis (<list>, <expr>)

     Makes multiple parallel substitutions into an expression.  <list>
     is a list of equations.  The left hand side of the equations must
     be an atom.

     The variable `sublis_apply_lambda'  controls simplification after
     `sublis'.

     See also `psubst'  for making parallel substitutions.

     Example:

          (%i1) sublis ([a=b, b=a], sin(a) + cos(b));
          (%o1)                    sin(b) + cos(a)

Option variable: sublis_apply_lambda

     Default value: `true'

     Controls whether `lambda''s substituted are applied in
     simplification after `sublis' is used or whether you have to do an
     `ev'  to get things to apply.  `true' means do the application.

Option variable: subnumsimp

     Default value: `false'

     If `true' then the functions `subst'  and `psubst'  can substitute
     a subscripted variable `f[x]' with a number, when only the symbol
     `f' is given.

     See also `subst'.

          (%i1) subst(100,g,g[x]+2);

          subst: cannot substitute 100 for operator g in expression g
                                                                     x
           -- an error. To debug this try: debugmode(true);

          (%i2) subst(100,g,g[x]+2),subnumsimp:true;
          (%o2)                          102

Function: subst (<a>, <b>, <c>)

     Substitutes <a> for <b> in <c>.  <b> must be an atom or a complete
     subexpression of <c>.  For example, `x+y+z' is a complete
     subexpression of `2*(x+y+z)/w' while `x+y' is not.  When <b> does
     not have these characteristics, one may sometimes use `substpart'
     or `ratsubst'  (see below).  Alternatively, if <b> is of the form
     `e/f' then one could use `subst (a*f, e, c)' while if <b> is of
     the form `e^(1/f)' then one could use `subst (a^f, e, c)'.  The
     `subst' command also discerns the `x^y' in `x^-y' so that `subst
     (a, sqrt(x), 1/sqrt(x))' yields `1/a'.  <a> and <b> may also be
     operators of an expression enclosed in double-quotes `"' or they
     may be function names.  If one wishes to substitute for the
     independent variable in derivative forms then the `at' function
     (see below) should be used.

     `subst' is an alias for `substitute'.

     The commands `subst (<eq_1>, <expr>)' or `subst ([<eq_1>, ...,
     <eq_k>], <expr>)' are other permissible forms.  The <eq_i> are
     equations indicating substitutions to be made.  For each equation,
     the right side will be substituted for the left in the expression
     <expr>.  The equations are substituted in serial from left to
     right in <expr>.  See the functions `sublis' and `psubst' for
     making parallel substitutions.

     `exptsubst'  if `true' permits substitutions like `y' for `%e^x'
     in `%e^(a*x)' to take place.

     When `opsubst' is `false', `subst' will not attempt to substitute
     into the operator of an expression.  E.g. `(opsubst: false, subst
     (x^2, r, r+r[0]))' will work.

     Examples:

          (%i1) subst (a, x+y, x + (x+y)^2 + y);
                                              2
          (%o1)                      y + x + a
          (%i2) subst (-%i, %i, a + b*%i);
          (%o2)                       a - %i b

     The substitution is done in serial for a list of equations.
     Compare this with a parallel substitution:

          (%i3) subst([a=b, b=c], a+b);
          (%o3)                                 2 c
          (%i4) sublis([a=b, b=c], a+b);
          (%o4)                                c + b

     For further examples, do `example (subst)'.

Function: substinpart (<x>, <expr>, <n_1>, ..., <n_k>)

     Similar to `substpart',  but `substinpart' works on the internal
     representation of <expr>.

     Examples:

          (%i1) x . 'diff (f(x), x, 2);
                                        2
                                       d
          (%o1)                   x . (--- (f(x)))
                                         2
                                       dx
          (%i2) substinpart (d^2, %, 2);
                                            2
          (%o2)                        x . d
          (%i3) substinpart (f1, f[1](x + 1), 0);
          (%o3)                       f1(x + 1)

     If the last argument to a `part' function is a list of indices then
     several subexpressions are picked out, each one corresponding to an
     index of the list.  Thus

          (%i1) part (x + y + z, [1, 3]);
          (%o1)                         z + x

     `piece'  holds the value of the last expression selected when
     using the `part' functions.  It is set during the execution of the
     function and thus may be referred to in the function itself as
     shown below.  If `partswitch'  is set to `true' then `end' is
     returned when a selected part of an expression doesn't exist,
     otherwise an error message is given.

          (%i1) expr: 27*y^3 + 54*x*y^2 + 36*x^2*y + y + 8*x^3 + x + 1;
                        3         2       2            3
          (%o1)     27 y  + 54 x y  + 36 x  y + y + 8 x  + x + 1
          (%i2) part (expr, 2, [1, 3]);
                                            2
          (%o2)                         54 y
          (%i3) sqrt (piece/54);
          (%o3)                        abs(y)
          (%i4) substpart (factor (piece), expr, [1, 2, 3, 5]);
                                         3
          (%o4)               (3 y + 2 x)  + y + x + 1
          (%i5) expr: 1/x + y/x - 1/z;
                                       1   y   1
          (%o5)                      - - + - + -
                                       z   x   x
          (%i6) substpart (xthru (piece), expr, [2, 3]);
                                      y + 1   1
          (%o6)                       ----- - -
                                        x     z

     Also, setting the option `inflag'  to `true' and calling `part'

     or `substpart'  is the same as calling `inpart'  or `substinpart'.

Function: substpart (<x>, <expr>, <n_1>, ..., <n_k>)

     Substitutes <x> for the subexpression picked out by the rest of the
     arguments as in `part'.    It returns the new value of <expr>.  <x>
     may be some operator to be substituted for an operator of <expr>.
     In some cases <x> needs to be enclosed in double-quotes `"' (e.g.
     `substpart ("+", a*b, 0)' yields `b + a').

     Example:

          (%i1) 1/(x^2 + 2);
                                         1
          (%o1)                        ------
                                        2
                                       x  + 2
          (%i2) substpart (3/2, %, 2, 1, 2);
                                         1
          (%o2)                       --------
                                       3/2
                                      x    + 2
          (%i3) a*x + f(b, y);
          (%o3)                     a x + f(b, y)
          (%i4) substpart ("+", %, 1, 0);
          (%o4)                    x + f(b, y) + a

     Also, setting the option `inflag'  to `true' and calling `part'

     or `substpart' is the same as calling `inpart' or `substinpart'.

Function: symbolp (<expr>)

     Returns `true' if <expr> is a symbol, else `false'.  In effect,
     `symbolp(x)' is equivalent to the predicate `atom(x) and not
     numberp(x)'.

     See also *note Identifiers::.

Function: unorder ()

     Disables the aliasing created by the last use of the ordering
     commands `ordergreat' and `orderless'.  `ordergreat' and
     `orderless' may not be used more than one time each without
     calling `unorder'.  `unorder' does not substitute back in
     expressions the original symbols for the aliases introduced by
     `ordergreat' and `orderless'.  Therefore, after execution of
     `unorder' the aliases appear in previous expressions.

     See also `ordergreat' and `orderless'.

     Examples:

     `ordergreat(a)' introduces an alias for the symbol `a'.  Therefore,
     the difference of `%o2' and `%o4' does not vanish.  `unorder' does
     not substitute back the symbol `a' and the alias appears in the
     output `%o7'.

          (%i1) unorder();
          (%o1)                          []
          (%i2) b*x+a^2;
                                             2
          (%o2)                       b x + a
          (%i3) ordergreat(a);
          (%o3)                         done
          (%i4) b*x+a^2;
                                       2
          (%o4)                       a  + b x
          (%i5) %th(1)-%th(3);
                                        2    2
          (%o5)                        a  - a
          (%i6) unorder();
          (%o6)                          [a]
          (%i7) %th(2);
                                          2    2
          (%o7)                      _101a  - a

Function: verbify (<f>)

     Returns the verb form of the function name <f>.  See also `verb',
     `noun', and `nounify'.

     Examples:

          (%i1) verbify ('foo);
          (%o1)                          foo
          (%i2) :lisp $%
          $FOO
          (%i2) nounify (foo);
          (%o2)                          foo
          (%i3) :lisp $%
          %FOO

7 Operators


Introduction to operators
Arithmetic operators
Relational operators
Logical operators
Operators for Equations
Assignment operators
User defined operators

7.1 Introduction to operators

It is possible to define new operators with specified precedence, to
undefine existing operators, or to redefine the precedence of existing
operators.  An operator may be unary prefix or unary postfix, binary
infix, n-ary infix, matchfix, or nofix.  "Matchfix" means a pair of
symbols which enclose their argument or arguments, and "nofix" means an
operator which takes no arguments.  As examples of the different types
of operators, there are the following.

unary prefix
     negation `- a'

unary postfix
     factorial `a!'

binary infix
     exponentiation `a^b'

n-ary infix
     addition `a + b'

matchfix
     list construction `[a, b]'

   (There are no built-in nofix operators; for an example of such an
operator, see `nofix'.)

   The mechanism to define a new operator is straightforward.  It is
only necessary to declare a function as an operator; the operator
function might or might not be defined.

   An example of user-defined operators is the following.  Note that
the explicit function call `"dd" (a)' is equivalent to `dd a', likewise
`"<-" (a, b)' is equivalent to `a <- b'.  Note also that the functions
`"dd"' and `"<-"' are undefined in this example.

     (%i1) prefix ("dd");
     (%o1)                          dd
     (%i2) dd a;
     (%o2)                         dd a
     (%i3) "dd" (a);
     (%o3)                         dd a
     (%i4) infix ("<-");
     (%o4)                          <-
     (%i5) a <- dd b;
     (%o5)                      a <- dd b
     (%i6) "<-" (a, "dd" (b));
     (%o6)                      a <- dd b

   The Maxima functions which define new operators are summarized in
this table, stating the default left and right binding powers (lbp and
rbp, respectively).  (Binding power determines operator precedence.
However, since left and right binding powers can differ, binding power
is somewhat more complicated than precedence.) Some of the operation
definition functions take additional arguments; see the function
descriptions for details.

`prefix'
     rbp=180

`postfix'
     lbp=180

`infix'
     lbp=180, rbp=180

`nary'
     lbp=180, rbp=180

`matchfix'
     (binding power not applicable)

`nofix'
     (binding power not applicable)

   For comparison, here are some built-in operators and their left and
right binding powers.

     Operator   lbp     rbp

       :        180     20
       ::       180     20
       :=       180     20
       ::=      180     20
       !        160
       !!       160
       ^        140     139
       .        130     129
       *        120
       /        120     120
       +        100     100
       -        100     134
       =        80      80
       #        80      80
       >        80      80
       >=       80      80
       <        80      80
       <=       80      80
       not              70
       and      65
       or       60
       ,        10
       $        -1
       ;        -1

   `remove'  and `kill'  remove operator properties from an atom.
`remove ("<a>", op)' removes only the operator properties of <a>.
`kill ("<a>")' removes all properties of <a>, including the operator
properties.  Note that the name of the operator must be enclosed in
quotation marks.

     (%i1) infix ("##");
     (%o1)                          ##
     (%i2) "##" (a, b) := a^b;
                                          b
     (%o2)                     a ## b := a
     (%i3) 5 ## 3;
     (%o3)                          125
     (%i4) remove ("##", op);
     (%o4)                         done
     (%i5) 5 ## 3;
     Incorrect syntax: # is not a prefix operator
     5 ##
       ^
     (%i5) "##" (5, 3);
     (%o5)                          125
     (%i6) infix ("##");
     (%o6)                          ##
     (%i7) 5 ## 3;
     (%o7)                          125
     (%i8) kill ("##");
     (%o8)                         done
     (%i9) 5 ## 3;
     Incorrect syntax: # is not a prefix operator
     5 ##
       ^
     (%i9) "##" (5, 3);
     (%o9)                       ##(5, 3)

7.2 Arithmetic operators

Operator: +
Operator: -
Operator: *
Operator: /
Operator: ^

     The symbols `+' `*' `/' and `^' represent addition,
     multiplication, division, and exponentiation, respectively.  The
     names of these operators are `"+"' `"*"' `"/"' and `"^"', which
     may appear where the name of a function or operator is required.

     The symbols `+' and `-' represent unary addition and negation,
     respectively, and the names of these operators are `"+"' and `"-"',
     respectively.

     Subtraction `a - b' is represented within Maxima as addition, `a +
     (- b)'.  Expressions such as `a + (- b)' are displayed as
     subtraction.  Maxima recognizes `"-"' only as the name of the unary
     negation operator, and not as the name of the binary subtraction
     operator.

     Division `a / b' is represented within Maxima as multiplication,
     `a * b^(- 1)'.  Expressions such as `a * b^(- 1)' are displayed as
     division.  Maxima recognizes `"/"' as the name of the division
     operator.

     Addition and multiplication are n-ary, commutative operators.
     Division and exponentiation are binary, noncommutative operators.

     Maxima sorts the operands of commutative operators to construct a
     canonical representation.  For internal storage, the ordering is
     determined by `orderlessp'.    For display, the ordering for
     addition is determined by `ordergreatp',  and for multiplication,
     it is the same as the internal ordering.

     Arithmetic computations are carried out on literal numbers
     (integers, rationals, ordinary floats, and bigfloats).  Except for
     exponentiation, all arithmetic operations on numbers are
     simplified to numbers.  Exponentiation is simplified to a number
     if either operand is an ordinary float or bigfloat or if the result
     is an exact integer or rational; otherwise an exponentiation may
     be simplified to `sqrt'  or another exponentiation or left
     unchanged.

     Floating-point contagion applies to arithmetic computations: if
     any operand is a bigfloat, the result is a bigfloat; otherwise, if
     any operand is an ordinary float, the result is an ordinary float;
     otherwise, the operands are rationals or integers and the result
     is a rational or integer.

     Arithmetic computations are a simplification, not an evaluation.
     Thus arithmetic is carried out in quoted (but simplified)
     expressions.

     Arithmetic operations are applied element-by-element to lists when
     the global flag `listarith'  is `true', and always applied
     element-by-element to matrices.  When one operand is a list or
     matrix and another is an operand of some other type, the other
     operand is combined with each of the elements of the list or
     matrix.

     Examples:

     Addition and multiplication are n-ary, commutative operators.
     Maxima sorts the operands to construct a canonical representation.
     The names of these operators are `"+"' and `"*"'.

          (%i1) c + g + d + a + b + e + f;
          (%o1)               g + f + e + d + c + b + a
          (%i2) [op (%), args (%)];
          (%o2)              [+, [g, f, e, d, c, b, a]]
          (%i3) c * g * d * a * b * e * f;
          (%o3)                     a b c d e f g
          (%i4) [op (%), args (%)];
          (%o4)              [*, [a, b, c, d, e, f, g]]
          (%i5) apply ("+", [a, 8, x, 2, 9, x, x, a]);
          (%o5)                    3 x + 2 a + 19
          (%i6) apply ("*", [a, 8, x, 2, 9, x, x, a]);
                                           2  3
          (%o6)                       144 a  x

     Division and exponentiation are binary, noncommutative operators.
     The names of these operators are `"/"' and `"^"'.

          (%i1) [a / b, a ^ b];
                                        a   b
          (%o1)                        [-, a ]
                                        b
          (%i2) [map (op, %), map (args, %)];
          (%o2)              [[/, ^], [[a, b], [a, b]]]
          (%i3) [apply ("/", [a, b]), apply ("^", [a, b])];
                                        a   b
          (%o3)                        [-, a ]
                                        b

     Subtraction and division are represented internally in terms of
     addition and multiplication, respectively.

          (%i1) [inpart (a - b, 0), inpart (a - b, 1), inpart (a - b, 2)];
          (%o1)                      [+, a, - b]
          (%i2) [inpart (a / b, 0), inpart (a / b, 1), inpart (a / b, 2)];
                                             1
          (%o2)                       [*, a, -]
                                             b

     Computations are carried out on literal numbers.  Floating-point
     contagion applies.

          (%i1) 17 + b - (1/2)*29 + 11^(2/4);
                                                 5
          (%o1)                   b + sqrt(11) + -
                                                 2
          (%i2) [17 + 29, 17 + 29.0, 17 + 29b0];
          (%o2)                   [46, 46.0, 4.6b1]

     Arithmetic computations are a simplification, not an evaluation.

          (%i1) simp : false;
          (%o1)                         false
          (%i2) '(17 + 29*11/7 - 5^3);
                                        29 11    3
          (%o2)                    17 + ----- - 5
                                          7
          (%i3) simp : true;
          (%o3)                         true
          (%i4) '(17 + 29*11/7 - 5^3);
                                          437
          (%o4)                         - ---
                                           7

     Arithmetic is carried out element-by-element for lists (depending
     on `listarith') and matrices.

          (%i1) matrix ([a, x], [h, u]) - matrix ([1, 2], [3, 4]);
                                  [ a - 1  x - 2 ]
          (%o1)                   [              ]
                                  [ h - 3  u - 4 ]
          (%i2) 5 * matrix ([a, x], [h, u]);
                                    [ 5 a  5 x ]
          (%o2)                     [          ]
                                    [ 5 h  5 u ]
          (%i3) listarith : false;
          (%o3)                         false
          (%i4) [a, c, m, t] / [1, 7, 2, 9];
                                    [a, c, m, t]
          (%o4)                     ------------
                                    [1, 7, 2, 9]
          (%i5) [a, c, m, t] ^ x;
                                                x
          (%o5)                     [a, c, m, t]
          (%i6) listarith : true;
          (%o6)                         true
          (%i7) [a, c, m, t] / [1, 7, 2, 9];
                                        c  m  t
          (%o7)                     [a, -, -, -]
                                        7  2  9
          (%i8) [a, c, m, t] ^ x;
                                    x   x   x   x
          (%o8)                   [a , c , m , t ]

Operator: **

     Exponentiation operator.  Maxima recognizes `**' as the same
     operator as `^'  in input, and it is displayed as `^' in
     1-dimensional output, or by placing the exponent as a superscript
     in 2-dimensional output.

     The `fortran'  function displays the exponentiation operator as
     `**', whether it was input as `**' or `^'.

     Examples:

          (%i1) is (a**b = a^b);
          (%o1)                         true
          (%i2) x**y + x^z;
                                        z    y
          (%o2)                        x  + x
          (%i3) string (x**y + x^z);
          (%o3)                        x^z+x^y
          (%i4) fortran (x**y + x^z);
                x**z+x**y
          (%o4)                         done

Operator: ^^

     Noncommutative exponentiation operator.  `^^' is the
     exponentiation operator corresponding to noncommutative
     multiplication `.', just as the ordinary exponentiation operator
     `^' corresponds to commutative multiplication `*'.

     Noncommutative exponentiation is displayed by `^^' in
     1-dimensional output, and by placing the exponent as a superscript
     within angle brackets `< >' in 2-dimensional output.

     Examples:

          (%i1) a . a . b . b . b + a * a * a * b * b;
                                  3  2    <2>    <3>
          (%o1)                  a  b  + a    . b
          (%i2) string (a . a . b . b . b + a * a * a * b * b);
          (%o2)                  a^3*b^2+a^^2 . b^^3

Operator: .

     The dot operator, for matrix (non-commutative) multiplication.
     When `"."' is used in this way, spaces should be left on both
     sides of it, e.g.  `A . B'.  This distinguishes it plainly from a
     decimal point in a floating point number.

     See also `dot', `dot0nscsimp', `dot0simp', `dot1simp', `dotassoc',
     `dotconstrules', `dotdistrib', `dotexptsimp', `dotident', and
     `dotscrules'.

7.3 Relational operators

Operator: <
Operator: <=
Operator: >=
Operator: >

     The symbols `<' `<=' `>=' and `>' represent less than, less than
     or equal, greater than or equal, and greater than, respectively.
     The names of these operators are `"<"' `"<="' `">="' and `">"',
     which may appear where the name of a function or operator is
     required.

     These relational operators are all binary operators; constructs
     such as `a < b < c' are not recognized by Maxima.

     Relational expressions are evaluated to Boolean values by the
     functions `is'  and `maybe',  and the programming constructs `if',
     `while',  and `unless'.    Relational expressions are not
     otherwise evaluated or simplified to Boolean values, although the
     arguments of relational expressions are evaluated (when evaluation
     is not otherwise prevented by quotation).

     When a relational expression cannot be evaluated to `true' or
     `false', the behavior of `is' and `if' are governed by the global
     flag `prederror'.    When `prederror' is `true', `is' and `if'
     trigger an error.  When `prederror' is `false', `is' returns
     `unknown', and `if' returns a partially-evaluated conditional
     expression.

     `maybe' always behaves as if `prederror' were `false', and `while'
     and `unless' always behave as if `prederror' were `true'.

     Relational operators do not distribute over lists or other
     aggregates.

     See also `=',  `#',  `equal',  and `notequal'.

     Examples:

     Relational expressions are evaluated to Boolean values by some
     functions and programming constructs.

          (%i1) [x, y, z] : [123, 456, 789];
          (%o1)                    [123, 456, 789]
          (%i2) is (x < y);
          (%o2)                         true
          (%i3) maybe (y > z);
          (%o3)                         false
          (%i4) if x >= z then 1 else 0;
          (%o4)                           0
          (%i5) block ([S], S : 0, for i:1 while i <= 100 do S : S + i,
                       return (S));
          (%o5)                         5050

     Relational expressions are not otherwise evaluated or simplified
     to Boolean values, although the arguments of relational
     expressions are evaluated.

          (%o1)                    [123, 456, 789]
          (%i2) [x < y, y <= z, z >= y, y > z];
          (%o2)    [123 < 456, 456 <= 789, 789 >= 456, 456 > 789]
          (%i3) map (is, %);
          (%o3)               [true, true, true, false]

7.4 Logical operators

Operator: and

     The logical conjunction operator.  `and' is an n-ary infix
     operator; its operands are Boolean expressions, and its result is
     a Boolean value.

     `and' forces evaluation (like `is' ) of one or more operands, and
     may force evaluation of all operands.

     Operands are evaluated in the order in which they appear.  `and'
     evaluates only as many of its operands as necessary to determine
     the result.  If any operand is `false', the result is `false' and
     no further operands are evaluated.

     The global flag `prederror'  governs the behavior of `and' when an
     evaluated operand cannot be determined to be `true' or `false'.
     `and' prints an error message when `prederror' is `true'.
     Otherwise, operands which do not evaluate to `true' or `false' are
     accepted, and the result is a Boolean expression.

     `and' is not commutative: `a and b' might not be equal to `b and
     a' due to the treatment of indeterminate operands.

Operator: not

     The logical negation operator.  `not' is a prefix operator; its
     operand is a Boolean expression, and its result is a Boolean value.

     `not' forces evaluation (like `is') of its operand.

     The global flag `prederror'  governs the behavior of `not' when its
     operand cannot be determined to be `true' or `false'.  `not'
     prints an error message when `prederror' is `true'.  Otherwise,
     operands which do not evaluate to `true' or `false' are accepted,
     and the result is a Boolean expression.

Operator: or

     The logical disjunction operator.  `or' is an n-ary infix operator;
     its operands are Boolean expressions, and its result is a Boolean
     value.

     `or' forces evaluation (like `is' ) of one or more operands, and
     may force evaluation of all operands.

     Operands are evaluated in the order in which they appear.  `or'
     evaluates only as many of its operands as necessary to determine
     the result.  If any operand is `true', the result is `true' and no
     further operands are evaluated.

     The global flag `prederror'  governs the behavior of `or' when an
     evaluated operand cannot be determined to be `true' or `false'.
     `or' prints an error message when `prederror' is `true'.
     Otherwise, operands which do not evaluate to `true' or `false' are
     accepted, and the result is a Boolean expression.

     `or' is not commutative: `a or b' might not be equal to `b or a'
     due to the treatment of indeterminate operands.

7.5 Operators for Equations

Operator: #

     Represents the negation of syntactic equality `='.

     Note that because of the rules for evaluation of predicate
     expressions (in particular because `not <expr>' causes evaluation
     of <expr>), `not <a> = <b>' is equivalent to `is(<a> # <b>)',
     instead of `<a> # <b>'.

     Examples:

          (%i1) a = b;
          (%o1)                         a = b
          (%i2) is (a = b);
          (%o2)                         false
          (%i3) a # b;
          (%o3)                         a # b
          (%i4) not a = b;
          (%o4)                         true
          (%i5) is (a # b);
          (%o5)                         true
          (%i6) is (not a = b);
          (%o6)                         true

Operator: =

     The equation operator.

     An expression `<a> = <b>', by itself, represents an unevaluated
     equation, which might or might not hold.  Unevaluated equations
     may appear as arguments to `solve'  and `algsys'  or some other
     functions.

     The function `is'  evaluates `=' to a Boolean value.  `is(<a> =
     <b>)' evaluates `<a> = <b>' to `true' when <a> and <b> are
     identical.  That is, <a> and <b> are atoms which are identical, or
     they are not atoms and their operators are identical and their
     arguments are identical.  Otherwise, `is(<a> = <b>)' evaluates to
     `false'; it never evaluates to `unknown'.  When `is(<a> = <b>)' is
     `true', <a> and <b> are said to be syntactically equal, in
     contrast to equivalent expressions, for which `is(equal(<a>,
     <b>))' is `true'.  Expressions can be equivalent and not
     syntactically equal.

     The negation of `=' is represented by `#'.

     As with `=', an expression `<a> # <b>', by itself, is not
     evaluated.  `is(<a> # <b>)' evaluates `<a> # <b>' to `true' or
     `false'.

     In addition to `is', some other operators evaluate `=' and `#' to
     `true' or `false', namely `if',  `and',

     `or',  and `not'.

     Note that because of the rules for evaluation of predicate
     expressions (in particular because `not <expr>' causes evaluation
     of <expr>), `not <a> = <b>' is equivalent to `is(<a> # <b>)',
     instead of `<a> # <b>'.

     `rhs'  and `lhs'  return the right-hand and left-hand sides,
     respectively, of an equation or inequation.

     See also `equal'  and `notequal'.

     Examples:

     An expression `<a> = <b>', by itself, represents an unevaluated
     equation, which might or might not hold.

          (%i1) eq_1 : a * x - 5 * y = 17;
          (%o1)                    a x - 5 y = 17
          (%i2) eq_2 : b * x + 3 * y = 29;
          (%o2)                    3 y + b x = 29
          (%i3) solve ([eq_1, eq_2], [x, y]);
                                  196         29 a - 17 b
          (%o3)          [[x = ---------, y = -----------]]
                               5 b + 3 a       5 b + 3 a
          (%i4) subst (%, [eq_1, eq_2]);
                   196 a     5 (29 a - 17 b)
          (%o4) [--------- - --------------- = 17,
                 5 b + 3 a      5 b + 3 a
                                            196 b     3 (29 a - 17 b)
                                          --------- + --------------- = 29]
                                          5 b + 3 a      5 b + 3 a
          (%i5) ratsimp (%);
          (%o5)                  [17 = 17, 29 = 29]

     `is(<a> = <b>)' evaluates `<a> = <b>' to `true' when <a> and <b>
     are syntactically equal (that is, identical).  Expressions can be
     equivalent and not syntactically equal.

          (%i1) a : (x + 1) * (x - 1);
          (%o1)                    (x - 1) (x + 1)
          (%i2) b : x^2 - 1;
                                        2
          (%o2)                        x  - 1
          (%i3) [is (a = b), is (a # b)];
          (%o3)                     [false, true]
          (%i4) [is (equal (a, b)), is (notequal (a, b))];
          (%o4)                     [true, false]

     Some operators evaluate `=' and `#' to `true' or `false'.

          (%i1) if expand ((x + y)^2) = x^2 + 2 * x * y + y^2 then FOO else
                BAR;
          (%o1)                          FOO
          (%i2) eq_3 : 2 * x = 3 * x;
          (%o2)                       2 x = 3 x
          (%i3) eq_4 : exp (2) = %e^2;
                                        2     2
          (%o3)                       %e  = %e
          (%i4) [eq_3 and eq_4, eq_3 or eq_4, not eq_3];
          (%o4)                  [false, true, true]

     Because `not <expr>' causes evaluation of <expr>, `not <a> = <b>'
     is equivalent to `is(<a> # <b>)'.

          (%i1) [2 * x # 3 * x, not (2 * x = 3 * x)];
          (%o1)                   [2 x # 3 x, true]
          (%i2) is (2 * x # 3 * x);
          (%o2)                         true

7.6 Assignment operators

Operator: :

     Assignment operator.

     When the left-hand side is a simple variable (not subscripted), `:'
     evaluates its right-hand side and associates that value with the
     left-hand side.

     When the left-hand side is a subscripted element of a list,
     matrix, declared Maxima array, or Lisp array, the right-hand side
     is assigned to that element.  The subscript must name an existing
     element; such objects cannot be extended by naming nonexistent
     elements.

     When the left-hand side is a subscripted element of an undeclared
     Maxima array, the right-hand side is assigned to that element, if
     it already exists, or a new element is allocated, if it does not
     already exist.

     When the left-hand side is a list of simple and/or subscripted
     variables, the right-hand side must evaluate to a list, and the
     elements of the right-hand side are assigned to the elements of
     the left-hand side, in parallel.

     See also `kill'  and `remvalue',  which undo the association
     between the left-hand side and its value.

     Examples:

     Assignment to a simple variable.

          (%i1) a;
          (%o1)                           a
          (%i2) a : 123;
          (%o2)                          123
          (%i3) a;
          (%o3)                          123

     Assignment to an element of a list.

          (%i1) b : [1, 2, 3];
          (%o1)                       [1, 2, 3]
          (%i2) b[3] : 456;
          (%o2)                          456
          (%i3) b;
          (%o3)                      [1, 2, 456]

     Assignment creates an undeclared array.

          (%i1) c[99] : 789;
          (%o1)                          789
          (%i2) c[99];
          (%o2)                          789
          (%i3) c;
          (%o3)                           c
          (%i4) arrayinfo (c);
          (%o4)                   [hashed, 1, [99]]
          (%i5) listarray (c);
          (%o5)                         [789]

     Multiple assignment.

          (%i1) [a, b, c] : [45, 67, 89];
          (%o1)                     [45, 67, 89]
          (%i2) a;
          (%o2)                          45
          (%i3) b;
          (%o3)                          67
          (%i4) c;
          (%o4)                          89

     Multiple assignment is carried out in parallel.  The values of `a'
     and `b' are exchanged in this example.

          (%i1) [a, b] : [33, 55];
          (%o1)                       [33, 55]
          (%i2) [a, b] : [b, a];
          (%o2)                       [55, 33]
          (%i3) a;
          (%o3)                          55
          (%i4) b;
          (%o4)                          33

Operator: ::

     Assignment operator.

     `::' is the same as `:'  (which see) except that `::' evaluates
     its left-hand side as well as its right-hand side.

     Examples:

          (%i1) x : 'foo;
          (%o1)                          foo
          (%i2) x :: 123;
          (%o2)                          123
          (%i3) foo;
          (%o3)                          123
          (%i4) x : '[a, b, c];
          (%o4)                       [a, b, c]
          (%i5) x :: [11, 22, 33];
          (%o5)                     [11, 22, 33]
          (%i6) a;
          (%o6)                          11
          (%i7) b;
          (%o7)                          22
          (%i8) c;
          (%o8)                          33

Operator: ::=

     Macro function definition operator.  `::=' defines a function
     (called a "macro" for historical reasons) which quotes its
     arguments, and the expression which it returns (called the "macro
     expansion") is evaluated in the context from which the macro was
     called.  A macro function is otherwise the same as an ordinary
     function.

     `macroexpand'  returns a macro expansion (without evaluating it).
     `macroexpand (foo (x))' followed by `''%' is equivalent to `foo
     (x)' when `foo' is a macro function.

     `::=' puts the name of the new macro function onto the global list
     `macros'.    `kill',  `remove',  and `remfunction'

     unbind macro function definitions and remove names from `macros'.

     `fundef'  or `dispfun'  return a macro function definition or
     assign it to a label, respectively.

     Macro functions commonly contain `buildq'  and `splice'
     expressions to construct an expression, which is then evaluated.

     Examples

     A macro function quotes its arguments, so message (1) shows `y -
     z', not the value of `y - z'.  The macro expansion (the quoted
     expression `'(print ("(2) x is equal to", x))' is evaluated in the
     context from which the macro was called, printing message (2).

          (%i1) x: %pi$
          (%i2) y: 1234$
          (%i3) z: 1729 * w$
          (%i4) printq1 (x) ::= block (print ("(1) x is equal to", x),
                '(print ("(2) x is equal to", x)))$
          (%i5) printq1 (y - z);
          (1) x is equal to y - z
          (2) x is equal to %pi
          (%o5)                                 %pi

     An ordinary function evaluates its arguments, so message (1) shows
     the value of `y - z'.  The return value is not evaluated, so
     message (2) is not printed until the explicit evaluation `''%'.

          (%i1) x: %pi$
          (%i2) y: 1234$
          (%i3) z: 1729 * w$
          (%i4) printe1 (x) := block (print ("(1) x is equal to", x),
                '(print ("(2) x is equal to", x)))$
          (%i5) printe1 (y - z);
          (1) x is equal to 1234 - 1729 w
          (%o5)                     print((2) x is equal to, x)
          (%i6) ''%;
          (2) x is equal to %pi
          (%o6)                                 %pi

     `macroexpand' returns a macro expansion.  `macroexpand (foo (x))'
     followed by `''%' is equivalent to `foo (x)' when `foo' is a macro
     function.

          (%i1) x: %pi$
          (%i2) y: 1234$
          (%i3) z: 1729 * w$
          (%i4) g (x) ::= buildq ([x], print ("x is equal to", x))$
          (%i5) macroexpand (g (y - z));
          (%o5)                     print(x is equal to, y - z)
          (%i6) ''%;
          x is equal to 1234 - 1729 w
          (%o6)                            1234 - 1729 w
          (%i7) g (y - z);
          x is equal to 1234 - 1729 w
          (%o7)                            1234 - 1729 w

Operator: :=

     The function definition operator.  `<f>(<x_1>, ..., <x_n>) :=
     <expr>' defines a function named <f> with arguments <x_1>, ...,
     <x_n> and function body <expr>.  `:=' never evaluates the function
     body (unless explicitly evaluated by quote-quote `''').  The
     function so defined may be an ordinary Maxima function (with
     arguments enclosed in parentheses) or an array function (with
     arguments enclosed in square brackets).

     When the last or only function argument <x_n> is a list of one
     element, the function defined by `:=' accepts a variable number of
     arguments.  Actual arguments are assigned one-to-one to formal
     arguments <x_1>, ..., <x_(n - 1)>, and any further actual
     arguments, if present, are assigned to <x_n> as a list.

     All function definitions appear in the same namespace; defining a
     function `f' within another function `g' does not automatically
     limit the scope of `f' to `g'.  However, `local(f)' makes the
     definition of function `f' effective only within the block or
     other compound expression in which `local'  appears.

     If some formal argument <x_k> is a quoted symbol, the function
     defined by `:=' does not evaluate the corresponding actual
     argument.  Otherwise all actual arguments are evaluated.

     See also `define'  and `::='.

     Examples:

     `:=' never evaluates the function body (unless explicitly
     evaluated by quote-quote).

          (%i1) expr : cos(y) - sin(x);
          (%o1)                    cos(y) - sin(x)
          (%i2) F1 (x, y) := expr;
          (%o2)                   F1(x, y) := expr
          (%i3) F1 (a, b);
          (%o3)                    cos(y) - sin(x)
          (%i4) F2 (x, y) := ''expr;
          (%o4)              F2(x, y) := cos(y) - sin(x)
          (%i5) F2 (a, b);
          (%o5)                    cos(b) - sin(a)

     The function defined by `:=' may be an ordinary Maxima function or
     an array function.

          (%i1) G1 (x, y) := x.y - y.x;
          (%o1)               G1(x, y) := x . y - y . x
          (%i2) G2 [x, y] := x.y - y.x;
          (%o2)                G2     := x . y - y . x
                                 x, y

     When the last or only function argument <x_n> is a list of one
     element, the function defined by `:=' accepts a variable number of
     arguments.

          (%i1) H ([L]) := apply ("+", L);
          (%o1)                H([L]) := apply("+", L)
          (%i2) H (a, b, c);
          (%o2)                       c + b + a

     `local' makes a local function definition.

          (%i1) foo (x) := 1 - x;
          (%o1)                    foo(x) := 1 - x
          (%i2) foo (100);
          (%o2)                         - 99
          (%i3) block (local (foo), foo (x) := 2 * x, foo (100));
          (%o3)                          200
          (%i4) foo (100);
          (%o4)                         - 99

7.7 User defined operators

Function: infix (<op>)
Function: infix (<op>, <lbp>, <rbp>)
Function: infix (<op>, <lbp>, <rbp>, <lpos>, <rpos>, <pos>)

     Declares <op> to be an infix operator.  An infix operator is a
     function of two arguments, with the name of the function written
     between the arguments.  For example, the subtraction operator `-'
     is an infix operator.

     `infix (<op>)' declares <op> to be an infix operator with default
     binding powers (left and right both equal to 180) and parts of
     speech (left and right both equal to `any').

     `infix (<op>, <lbp>, <rbp>)' declares <op> to be an infix operator
     with stated left and right binding powers and default parts of
     speech (left and right both equal to `any').

     `infix (<op>, <lbp>, <rbp>, <lpos>, <rpos>, <pos>)' declares <op>
     to be an infix operator with stated left and right binding powers
     and parts of speech <lpos>, <rpos>, and <pos> for the left
     operand, the right operand, and the operator result, respectively.

     "Part of speech", in reference to operator declarations, means
     expression type.  Three types are recognized: `expr', `clause',
     and `any', indicating an algebraic expression, a Boolean
     expression, or any kind of expression, respectively.  Maxima can
     detect some syntax errors by comparing the declared part of speech
     to an actual expression.

     The precedence of <op> with respect to other operators derives
     from the left and right binding powers of the operators in
     question.  If the left and right binding powers of <op> are both
     greater the left and right binding powers of some other operator,
     then <op> takes precedence over the other operator.  If the
     binding powers are not both greater or less, some more complicated
     relation holds.

     The associativity of <op> depends on its binding powers.  Greater
     left binding power (<lbp>) implies an instance of <op> is
     evaluated before other operators to its left in an expression,
     while greater right binding power (<rbp>) implies  an instance of
     <op> is evaluated before other operators to its right in an
     expression.  Thus greater <lbp> makes <op> right-associative,
     while greater <rbp> makes <op> left-associative.  If <lbp> is
     equal to <rbp>, <op> is left-associative.

     See also *note Introduction to operators::.

     Examples:

     If the left and right binding powers of <op> are both greater the
     left and right binding powers of some other operator, then <op>
     takes precedence over the other operator.

          (%i1) :lisp (get '$+ 'lbp)
          100
          (%i1) :lisp (get '$+ 'rbp)
          100
          (%i1) infix ("##", 101, 101);
          (%o1)                          ##
          (%i2) "##"(a, b) := sconcat("(", a, ",", b, ")");
          (%o2)       (a ## b) := sconcat("(", a, ",", b, ")")
          (%i3) 1 + a ## b + 2;
          (%o3)                       (a,b) + 3
          (%i4) infix ("##", 99, 99);
          (%o4)                          ##
          (%i5) 1 + a ## b + 2;
          (%o5)                       (a+1,b+2)

     Greater <lbp> makes <op> right-associative, while greater <rbp>
     makes <op> left-associative.

          (%i1) infix ("##", 100, 99);
          (%o1)                          ##
          (%i2) "##"(a, b) := sconcat("(", a, ",", b, ")")$
          (%i3) foo ## bar ## baz;
          (%o3)                    (foo,(bar,baz))
          (%i4) infix ("##", 100, 101);
          (%o4)                          ##
          (%i5) foo ## bar ## baz;
          (%o5)                    ((foo,bar),baz)

     Maxima can detect some syntax errors by comparing the declared
     part of speech to an actual expression.

          (%i1) infix ("##", 100, 99, expr, expr, expr);
          (%o1)                          ##
          (%i2) if x ## y then 1 else 0;
          Incorrect syntax: Found algebraic expression where logical
          expression expected
          if x ## y then
                       ^
          (%i2) infix ("##", 100, 99, expr, expr, clause);
          (%o2)                          ##
          (%i3) if x ## y then 1 else 0;
          (%o3)                if x ## y then 1 else 0

Function: matchfix (<ldelimiter>, <rdelimiter>)
Function: matchfix (<ldelimiter>, <rdelimiter>, <arg_pos>, <pos>)

     Declares a matchfix operator with left and right delimiters
     <ldelimiter> and <rdelimiter>.  The delimiters are specified as
     strings.

     A "matchfix" operator is a function of any number of arguments,
     such that the arguments occur between matching left and right
     delimiters.  The delimiters may be any strings, so long as the
     parser can distinguish the delimiters from the operands and other
     expressions and operators.  In practice this rules out unparseable
     delimiters such as `%', `,', `$' and `;', and may require
     isolating the delimiters with white space.  The right delimiter
     can be the same or different from the left delimiter.

     A left delimiter can be associated with only one right delimiter;
     two different matchfix operators cannot have the same left
     delimiter.

     An existing operator may be redeclared as a matchfix operator
     without changing its other properties.  In particular, built-in
     operators such as addition `+' can be declared matchfix, but
     operator functions cannot be defined for built-in operators.

     The command `matchfix (<ldelimiter>, <rdelimiter>, <arg_pos>,
     <pos>)' declares the argument part-of-speech <arg_pos> and result
     part-of-speech <pos>, and the delimiters <ldelimiter> and
     <rdelimiter>.

     "Part of speech", in reference to operator declarations, means
     expression type.  Three types are recognized: `expr', `clause',
     and `any', indicating an algebraic expression, a Boolean
     expression, or any kind of expression, respectively.  Maxima can
     detect some syntax errors by comparing the declared part of speech
     to an actual expression.

     The function to carry out a matchfix operation is an ordinary
     user-defined function.  The operator function is defined in the
     usual way with the function definition operator `:=' or `define'.
     The arguments may be written between the delimiters, or with the
     left delimiter as a quoted string and the arguments following in
     parentheses.  `dispfun (<ldelimiter>)' displays the function
     definition.

     The only built-in matchfix operator is the list constructor `[ ]'.
     Parentheses `( )' and double-quotes `" "' act like matchfix
     operators, but are not treated as such by the Maxima parser.

     `matchfix' evaluates its arguments.  `matchfix' returns its first
     argument, <ldelimiter>.

     Examples:

     Delimiters may be almost any strings.

          (%i1) matchfix ("@@", "~");
          (%o1)                          @@
          (%i2) @@ a, b, c ~;
          (%o2)                      @@a, b, c~
          (%i3) matchfix (">>", "<<");
          (%o3)                          >>
          (%i4) >> a, b, c <<;
          (%o4)                      >>a, b, c<<
          (%i5) matchfix ("foo", "oof");
          (%o5)                          foo
          (%i6) foo a, b, c oof;
          (%o6)                     fooa, b, coof
          (%i7) >> w + foo x, y oof + z << / @@ p, q ~;
                               >>z + foox, yoof + w<<
          (%o7)                ----------------------
                                      @@p, q~

     Matchfix operators are ordinary user-defined functions.

          (%i1) matchfix ("!-", "-!");
          (%o1)                         "!-"
          (%i2) !- x, y -! := x/y - y/x;
                                              x   y
          (%o2)                   !-x, y-! := - - -
                                              y   x
          (%i3) define (!-x, y-!, x/y - y/x);
                                              x   y
          (%o3)                   !-x, y-! := - - -
                                              y   x
          (%i4) define ("!-" (x, y), x/y - y/x);
                                              x   y
          (%o4)                   !-x, y-! := - - -
                                              y   x
          (%i5) dispfun ("!-");
                                              x   y
          (%t5)                   !-x, y-! := - - -
                                              y   x

          (%o5)                         done
          (%i6) !-3, 5-!;
                                          16
          (%o6)                         - --
                                          15
          (%i7) "!-" (3, 5);
                                          16
          (%o7)                         - --
                                          15

Function: nary (<op>)
Function: nary (<op>, <bp>, <arg_pos>, <pos>)

     An `nary' operator is used to denote a function of any number of
     arguments, each of which is separated by an occurrence of the
     operator, e.g.  A+B or A+B+C.  The `nary("x")' function is a
     syntax extension function to declare `x' to be an `nary' operator.
     Functions may be declared to be `nary'.  If `declare(j,nary);' is
     done, this tells the simplifier to simplify, e.g.
     `j(j(a,b),j(c,d))' to `j(a, b, c, d)'.

     See also *note Introduction to operators::.

Function: nofix (<op>)
Function: nofix (<op>, <pos>)

     `nofix' operators are used to denote functions of no arguments.
     The mere presence of such an operator in a command will cause the
     corresponding function to be evaluated.  For example, when one
     types "exit;" to exit from a Maxima break, "exit" is behaving
     similar to a `nofix' operator.  The function `nofix("x")' is a
     syntax extension function which declares `x' to be a `nofix'
     operator.

     See also *note Introduction to operators::.

Function: postfix (<op>)
Function: postfix (<op>, <lbp>, <lpos>, <pos>)

     `postfix' operators like the `prefix' variety denote functions of a
     single argument, but in this case the argument immediately
     precedes an occurrence of the operator in the input string, e.g.
     3!.  The `postfix("x")' function is a syntax extension function to
     declare `x' to be a `postfix' operator.

     See also *note Introduction to operators::.

Function: prefix (<op>)
Function: prefix (<op>, <rbp>, <rpos>, <pos>)

     A `prefix' operator is one which signifies a function of one
     argument, which argument immediately follows an occurrence of the
     operator.  `prefix("x")' is a syntax extension function to declare
     `x' to be a `prefix' operator.

     See also *note Introduction to operators::.

8 Evaluation


Functions and Variables for Evaluation

8.1 Functions and Variables for Evaluation

Operator: '

     The single quote operator `'' prevents evaluation.

     Applied to a symbol, the single quote prevents evaluation of the
     symbol.

     Applied to a function call, the single quote prevents evaluation
     of the function call, although the arguments of the function are
     still evaluated (if evaluation is not otherwise prevented).  The
     result is the noun form of the function call.

     Applied to a parenthesized expression, the single quote prevents
     evaluation of all symbols and function calls in the expression.
     E.g., `'(f(x))' means do not evaluate the expression `f(x)'.
     `'f(x)' (with the single quote applied to `f' instead of `f(x)')
     means return the noun form of `f' applied to `[x]'.

     The single quote does not prevent simplification.

     When the global flag `noundisp'  is `true', nouns display with a
     single quote.  This switch is always `true' when displaying
     function definitions.

     See also the quote-quote operator `'''  and `nouns'.

     Examples:

     Applied to a symbol, the single quote prevents evaluation of the
     symbol.

          (%i1) aa: 1024;
          (%o1)                         1024
          (%i2) aa^2;
          (%o2)                        1048576
          (%i3) 'aa^2;
                                           2
          (%o3)                          aa
          (%i4) ''%;
          (%o4)                        1048576

     Applied to a function call, the single quote prevents evaluation
     of the function call.  The result is the noun form of the function
     call.

          (%i1) x0: 5;
          (%o1)                           5
          (%i2) x1: 7;
          (%o2)                           7
          (%i3) integrate (x^2, x, x0, x1);
                                         218
          (%o3)                          ---
                                          3
          (%i4) 'integrate (x^2, x, x0, x1);
                                       7
                                      /
                                      [   2
          (%o4)                       I  x  dx
                                      ]
                                      /
                                       5
          (%i5) %, nouns;
                                         218
          (%o5)                          ---
                                          3

     Applied to a parenthesized expression, the single quote prevents
     evaluation of all symbols and function calls in the expression.

          (%i1) aa: 1024;
          (%o1)                         1024
          (%i2) bb: 19;
          (%o2)                          19
          (%i3) sqrt(aa) + bb;
          (%o3)                          51
          (%i4) '(sqrt(aa) + bb);
          (%o4)                     bb + sqrt(aa)
          (%i5) ''%;
          (%o5)                          51

     The single quote does not prevent simplification.

          (%i1) sin (17 * %pi) + cos (17 * %pi);
          (%o1)                          - 1
          (%i2) '(sin (17 * %pi) + cos (17 * %pi));
          (%o2)                          - 1

     Maxima considers floating point operations by its in-built
     mathematical functions to be a simplification.

          (%i1) sin(1.0);
          (%o1)                          .8414709848078965
          (%i2) '(sin(1.0));
          (%o2)                          .8414709848078965

Operator: "

     The quote-quote operator `''' (two single quote marks) modifies
     evaluation in input expressions.

     Applied to a general expression <expr>, quote-quote causes the
     value of <expr> to be substituted for <expr> in the input
     expression.

     Applied to the operator of an expression, quote-quote changes the
     operator from a noun to a verb (if it is not already a verb).

     The quote-quote operator is applied by the input parser; it is not
     stored as part of a parsed input expression.  The quote-quote
     operator is always applied as soon as it is parsed, and cannot be
     quoted.  Thus quote-quote causes evaluation when evaluation is
     otherwise suppressed, such as in function definitions, lambda
     expressions, and expressions quoted by single quote `''.

     Quote-quote is recognized by `batch'  and `load'.

     See also the single-quote operator `''  and `nouns'.

     Examples:

     Applied to a general expression <expr>, quote-quote causes the
     value of <expr> to be substituted for <expr> in the input
     expression.

          (%i1) expand ((a + b)^3);
                               3        2      2      3
          (%o1)               b  + 3 a b  + 3 a  b + a
          (%i2) [_, ''_];
                                   3    3        2      2      3
          (%o2)     [expand((b + a) ), b  + 3 a b  + 3 a  b + a ]
          (%i3) [%i1, ''%i1];
                                   3    3        2      2      3
          (%o3)     [expand((b + a) ), b  + 3 a b  + 3 a  b + a ]
          (%i4) [aa : cc, bb : dd, cc : 17, dd : 29];
          (%o4)                   [cc, dd, 17, 29]
          (%i5) foo_1 (x) := aa - bb * x;
          (%o5)                 foo_1(x) := aa - bb x
          (%i6) foo_1 (10);
          (%o6)                      cc - 10 dd
          (%i7) ''%;
          (%o7)                         - 273
          (%i8) ''(foo_1 (10));
          (%o8)                         - 273
          (%i9) foo_2 (x) := ''aa - ''bb * x;
          (%o9)                 foo_2(x) := cc - dd x
          (%i10) foo_2 (10);
          (%o10)                        - 273
          (%i11) [x0 : x1, x1 : x2, x2 : x3];
          (%o11)                    [x1, x2, x3]
          (%i12) x0;
          (%o12)                         x1
          (%i13) ''x0;
          (%o13)                         x2
          (%i14) '' ''x0;
          (%o14)                         x3

     Applied to the operator of an expression, quote-quote changes the
     operator from a noun to a verb (if it is not already a verb).

          (%i1) declare (foo, noun);
          (%o1)                         done
          (%i2) foo (x) := x - 1729;
          (%o2)                 ''foo(x) := x - 1729
          (%i3) foo (100);
          (%o3)                       foo(100)
          (%i4) ''foo (100);
          (%o4)                        - 1629

     The quote-quote operator is applied by the input parser; it is not
     stored as part of a parsed input expression.

          (%i1) [aa : bb, cc : dd, bb : 1234, dd : 5678];
          (%o1)                 [bb, dd, 1234, 5678]
          (%i2) aa + cc;
          (%o2)                        dd + bb
          (%i3) display (_, op (_), args (_));
                                     _ = cc + aa

                                   op(cc + aa) = +

                              args(cc + aa) = [cc, aa]

          (%o3)                         done
          (%i4) ''(aa + cc);
          (%o4)                         6912
          (%i5) display (_, op (_), args (_));
                                     _ = dd + bb

                                   op(dd + bb) = +

                              args(dd + bb) = [dd, bb]

          (%o5)                         done

     Quote-quote causes evaluation when evaluation is otherwise
     suppressed, such as in function definitions, lambda expressions,
     and expressions quoted by single quote `''.

          (%i1) foo_1a (x) := ''(integrate (log (x), x));
          (%o1)               foo_1a(x) := x log(x) - x
          (%i2) foo_1b (x) := integrate (log (x), x);
          (%o2)           foo_1b(x) := integrate(log(x), x)
          (%i3) dispfun (foo_1a, foo_1b);
          (%t3)               foo_1a(x) := x log(x) - x

          (%t4)           foo_1b(x) := integrate(log(x), x)

          (%o4)                      [%t3, %t4]
          (%i5) integrate (log (x), x);
          (%o5)                     x log(x) - x
          (%i6) foo_2a (x) := ''%;
          (%o6)               foo_2a(x) := x log(x) - x
          (%i7) foo_2b (x) := %;
          (%o7)                    foo_2b(x) := %
          (%i8) dispfun (foo_2a, foo_2b);
          (%t8)               foo_2a(x) := x log(x) - x

          (%t9)                    foo_2b(x) := %

          (%o9)                      [%t7, %t8]
          (%i10) F : lambda ([u], diff (sin (u), u));
          (%o10)             lambda([u], diff(sin(u), u))
          (%i11) G : lambda ([u], ''(diff (sin (u), u)));
          (%o11)                  lambda([u], cos(u))
          (%i12) '(sum (a[k], k, 1, 3) + sum (b[k], k, 1, 3));
          (%o12)         sum(b , k, 1, 3) + sum(a , k, 1, 3)
                              k                  k
          (%i13) '(''(sum (a[k], k, 1, 3)) + ''(sum (b[k], k, 1, 3)));
          (%o13)             b  + a  + b  + a  + b  + a
                              3    3    2    2    1    1

Function: ev (<expr>, <arg_1>, ..., <arg_n>)

     Evaluates the expression <expr> in the environment specified by the
     arguments <arg_1>, ..., <arg_n>.  The arguments are switches
     (Boolean flags), assignments, equations, and functions.  `ev'
     returns the result (another expression) of the evaluation.

     The evaluation is carried out in steps, as follows.

       1. First the environment is set up by scanning the arguments
          which may be any or all of the following.

             * `simp'  causes <expr> to be simplified regardless of the
               setting of the switch `simp' which inhibits
               simplification if `false'.

             * `noeval'  suppresses the evaluation phase of `ev' (see
               step (4) below).  This is useful in conjunction with the
               other switches and in causing <expr> to be resimplified
               without being reevaluated.

             * `nouns'  causes the evaluation of noun forms (typically
               unevaluated functions such as `'integrate' or `'diff')
               in <expr>.

             * `expand'  causes expansion.

             * `expand (<m>, <n>)' causes expansion, setting the values
               of `maxposex'  and `maxnegex'  to <m> and <n>
               respectively.

             * `detout'  causes any matrix inverses computed in <expr>
               to have their determinant kept outside of the inverse
               rather than dividing through each element.

             * `diff'  causes all differentiations indicated in <expr>
               to be performed.

             * `derivlist (<x>, <y>, <z>, ...)' causes only
               differentiations with respect to the indicated
               variables.  See also `derivlist'.

             * `risch' causes integrals in <expr> to be evaluated using
               the Risch algorithm.  See `risch'.    The standard
               integration routine is invoked when using the special
               symbol `nouns'.

             * `float'  causes non-integral rational numbers to be
               converted to floating point.

             * `numer'  causes some mathematical functions (including
               exponentiation) with numerical arguments to be evaluated
               in floating point.  It causes variables in <expr> which
               have been given numervals to be replaced by their
               values.  It also sets the `float'  switch on.

             * `pred'  causes predicates (expressions which evaluate to
               `true' or `false') to be evaluated.

             * `eval'  causes an extra post-evaluation of <expr> to
               occur.  (See step (5) below.)  `eval' may occur multiple
               times.  For each instance of `eval', the expression is
               evaluated again.

             * `A' where `A' is an atom declared to be an evaluation
               flag `evflag'  causes `A' to be bound to `true' during
               the evaluation of <expr>.

             * `V: expression' (or alternately `V=expression') causes
               `V' to be bound to the value of `expression' during the
               evaluation of <expr>.  Note that if `V' is a Maxima
               option, then `expression' is used for its value during
               the evaluation of <expr>.  If more than one argument to
               `ev' is of this type then the binding is done in
               parallel.  If `V' is a non-atomic expression then a
               substitution rather than a binding is performed.

             * `F' where `F', a function name, has been declared to be
               an evaluation function `evfun'  causes `F' to be applied
               to <expr>.

             * Any other function names, e.g. `sum',  cause evaluation
               of occurrences of those names in <expr> as though they
               were verbs.

             * In addition a function occurring in <expr> (say `F(x)')
               may be defined locally for the purpose of this
               evaluation of <expr> by giving `F(x) := expression' as
               an argument to `ev'.

             * If an atom not mentioned above or a subscripted variable
               or subscripted expression was given as an argument, it
               is evaluated and if the result is an equation or
               assignment then the indicated binding or substitution is
               performed.  If the result is a list then the members of
               the list are treated as if they were additional
               arguments given to `ev'.  This permits a list of
               equations to be given (e.g. `[X=1, Y=A**2]') or a list
               of names of equations (e.g., `[%t1, %t2]' where `%t1'
               and `%t2' are equations) such as that returned by
               `solve'.

Special symbol: eval

     As an argument in a call to `ev (<expr>)', `eval' causes an extra
     evaluation of <expr>.  See `ev'.

     Example:

          (%i1) [a:b,b:c,c:d,d:e];
          (%o1)                            [b, c, d, e]
          (%i2) a;
          (%o2)                                  b
          (%i3) ev(a);
          (%o3)                                  c
          (%i4) ev(a),eval;
          (%o4)                                  e
          (%i5) a,eval,eval;
          (%o5)                                  e

Property: evflag

     When a symbol <x> has the `evflag' property, the expressions
     `ev(<expr>, <x>)' and `<expr>, <x>' (at the interactive prompt)
     are equivalent to `ev(<expr>, <x> = true)'.  That is, <x> is bound
     to `true' while <expr> is evaluated.

     The expression `declare(<x>, evflag)' gives the `evflag' property
     to the variable <x>.

     The flags which have the `evflag' property by default are the
     following:

        algebraic          cauchysum       demoivre
        dotscrules         %emode          %enumer
        exponentialize     exptisolate     factorflag
        float              halfangles      infeval
        isolate_wrt_times  keepfloat       letrat
        listarith          logabs          logarc
        logexpand          lognegint
        m1pbranch          numer_pbranch   programmode
        radexpand          ratalgdenom     ratfac
        ratmx              ratsimpexpons   simp
        simpproduct        simpsum         sumexpand
        trigexpand

     Examples:

          (%i1) sin (1/2);
                                           1
          (%o1)                        sin(-)
                                           2
          (%i2) sin (1/2), float;
          (%o2)                   0.479425538604203
          (%i3) sin (1/2), float=true;
          (%o3)                   0.479425538604203
          (%i4) simp : false;
          (%o4)                         false
          (%i5) 1 + 1;
          (%o5)                         1 + 1
          (%i6) 1 + 1, simp;
          (%o6)                           2
          (%i7) simp : true;
          (%o7)                         true
          (%i8) sum (1/k^2, k, 1, inf);
                                      inf
                                      ====
                                      \     1
          (%o8)                        >    --
                                      /      2
                                      ====  k
                                      k = 1
          (%i9) sum (1/k^2, k, 1, inf), simpsum;
                                           2
                                        %pi
          (%o9)                         ----
                                         6
          (%i10) declare (aa, evflag);
          (%o10)                        done
          (%i11) if aa = true then YES else NO;
          (%o11)                         NO
          (%i12) if aa = true then YES else NO, aa;
          (%o12)                         YES

Property: evfun

     When a function <F> has the `evfun' property, the expressions
     `ev(<expr>, <F>)' and `<expr>, <F>' (at the interactive prompt)
     are equivalent to `<F>(ev(<expr>))'.

     If two or more `evfun' functions <F>, <G>, etc., are specified,
     the functions are applied in the order that they are specified.

     The expression `declare(<F>, evfun)' gives the `evfun' property to
     the function <F>.  The functions which have the `evfun' property by
     default are the following:

        bfloat          factor       fullratsimp
        logcontract     polarform    radcan
        ratexpand       ratsimp      rectform
        rootscontract   trigexpand   trigreduce

     Examples:

          (%i1) x^3 - 1;
                                        3
          (%o1)                        x  - 1
          (%i2) x^3 - 1, factor;
                                          2
          (%o2)                 (x - 1) (x  + x + 1)
          (%i3) factor (x^3 - 1);
                                          2
          (%o3)                 (x - 1) (x  + x + 1)
          (%i4) cos(4 * x) / sin(x)^4;
                                      cos(4 x)
          (%o4)                       --------
                                         4
                                      sin (x)
          (%i5) cos(4 * x) / sin(x)^4, trigexpand;
                           4           2       2         4
                        sin (x) - 6 cos (x) sin (x) + cos (x)
          (%o5)         -------------------------------------
                                          4
                                       sin (x)
          (%i6) cos(4 * x) / sin(x)^4, trigexpand, ratexpand;
                                     2         4
                                6 cos (x)   cos (x)
          (%o6)               - --------- + ------- + 1
                                    2          4
                                 sin (x)    sin (x)
          (%i7) ratexpand (trigexpand (cos(4 * x) / sin(x)^4));
                                     2         4
                                6 cos (x)   cos (x)
          (%o7)               - --------- + ------- + 1
                                    2          4
                                 sin (x)    sin (x)
          (%i8) declare ([F, G], evfun);
          (%o8)                         done
          (%i9) (aa : bb, bb : cc, cc : dd);
          (%o9)                          dd
          (%i10) aa;
          (%o10)                         bb
          (%i11) aa, F;
          (%o11)                        F(cc)
          (%i12) F (aa);
          (%o12)                        F(bb)
          (%i13) F (ev (aa));
          (%o13)                        F(cc)
          (%i14) aa, F, G;
          (%o14)                      G(F(cc))
          (%i15) G (F (ev (aa)));
          (%o15)                      G(F(cc))

Option variable: infeval

     Enables "infinite evaluation" mode.  `ev'  repeatedly evaluates an
     expression until it stops changing.  To prevent a variable, say
     `X', from being evaluated away in this mode, simply include `X='X'
     as an argument to `ev'.  Of course expressions such as `ev (X,
     X=X+1, infeval)' will generate an infinite loop.

Special symbol: noeval

     `noeval' suppresses the evaluation phase of `ev'.    This is
     useful in conjunction with other switches and in causing
     expressions to be resimplified without being reevaluated.

Special symbol: nouns

     `nouns' is an `evflag'.    When used as an option to the `ev'

     command, `nouns' converts all "noun" forms occurring in the
     expression being `ev''d to "verbs", i.e., evaluates them.  See also
     `noun',  `nounify',  `verb', and `verbify'.

Special symbol: pred

     As an argument in a call to `ev (<expr>)', `pred' causes
     predicates (expressions which evaluate to `true' or `false') to be
     evaluated.  See `ev'.

     Example:

          (%i1) 1<2;
          (%o1)                                1 < 2
          (%i2) 1<2,pred;
          (%o2)                                true

9 Simplification


Functions and Variables for Simplification

9.1 Functions and Variables for Simplification

Property: additive

     If `declare(f,additive)' has been executed, then:

     (1) If `f' is univariate, whenever the simplifier encounters `f'
     applied to a sum, `f' will be distributed over that sum.  I.e.
     `f(y+x)' will simplify to `f(y)+f(x)'.

     (2) If `f' is a function of 2 or more arguments, additivity is
     defined as additivity in the first argument to `f', as in the case
     of `sum' or `integrate', i.e.  `f(h(x)+g(x),x)' will simplify to
     `f(h(x),x)+f(g(x),x)'.  This simplification does not occur when
     `f' is applied to expressions of the form
     `sum(x[i],i,lower-limit,upper-limit)'.

     Example:

          (%i1) F3 (a + b + c);
          (%o1)                     F3(c + b + a)
          (%i2) declare (F3, additive);
          (%o2)                         done
          (%i3) F3 (a + b + c);
          (%o3)                 F3(c) + F3(b) + F3(a)

Property: antisymmetric

     If `declare(h,antisymmetric)' is done, this tells the simplifier
     that `h' is antisymmetric.  E.g.  `h(x,z,y)' will simplify to `-
     h(x, y, z)'.  That is, it will give (-1)^n times the result given
     by `symmetric' or `commutative', where n is the number of
     interchanges of two arguments necessary to convert it to that form.

     Examples:

          (%i1) S (b, a);
          (%o1)                        S(b, a)
          (%i2) declare (S, symmetric);
          (%o2)                         done
          (%i3) S (b, a);
          (%o3)                        S(a, b)
          (%i4) S (a, c, e, d, b);
          (%o4)                   S(a, b, c, d, e)
          (%i5) T (b, a);
          (%o5)                        T(b, a)
          (%i6) declare (T, antisymmetric);
          (%o6)                         done
          (%i7) T (b, a);
          (%o7)                       - T(a, b)
          (%i8) T (a, c, e, d, b);
          (%o8)                   T(a, b, c, d, e)

Function: combine (<expr>)

     Simplifies the sum <expr> by combining terms with the same
     denominator into a single term.

Property: commutative

     If `declare(h, commutative)' is done, this tells the simplifier
     that `h' is a commutative function.  E.g.  `h(x, z, y)' will
     simplify to `h(x, y, z)'.  This is the same as `symmetric'.

Function: demoivre (<expr>)
Option variable: demoivre

     The function `demoivre (expr)' converts one expression without
     setting the global variable `demoivre'.

     When the variable `demoivre' is `true', complex exponentials are
     converted into equivalent expressions in terms of circular
     functions: `exp (a + b*%i)' simplifies to `%e^a * (cos(b) +
     %i*sin(b))' if `b' is free of `%i'.  `a' and `b' are not expanded.

     The default value of `demoivre' is `false'.

     `exponentialize' converts circular and hyperbolic functions to
     exponential form.  `demoivre' and `exponentialize' cannot both be
     true at the same time.

Function: distrib (<expr>)

     Distributes sums over products.  It differs from `expand' in that
     it works at only the top level of an expression, i.e., it doesn't
     recurse and it is faster than `expand'.  It differs from
     `multthru' in that it expands all sums at that level.

     Examples:

          (%i1) distrib ((a+b) * (c+d));
          (%o1)                 b d + a d + b c + a c
          (%i2) multthru ((a+b) * (c+d));
          (%o2)                 (b + a) d + (b + a) c
          (%i3) distrib (1/((a+b) * (c+d)));
                                          1
          (%o3)                    ---------------
                                   (b + a) (d + c)
          (%i4) expand (1/((a+b) * (c+d)), 1, 0);
                                          1
          (%o4)                 ---------------------
                                b d + a d + b c + a c

Option variable: distribute_over

     Default value: `true'

     `distribute_over' controls the mapping of functions over bags like
     lists, matrices, and equations.  At this time not all Maxima
     functions have this property.  It is possible to look up this
     property with the command `properties'.

     The mapping of functions is switched off, when setting
     `distribute_over' to the value `false'.

     Examples:

     The `sin' function maps over a list:

          (%i1) sin([x,1,1.0]);
          (%o1)                 [sin(x), sin(1), .8414709848078965]

     `mod' is a function with two arguments which maps over lists.
     Mapping over nested lists is possible too:

          (%i2) mod([x,11,2*a],10);
          (%o2)                    [mod(x, 10), 1, 2 mod(a, 5)]
          (%i3) mod([[x,y,z],11,2*a],10);
          (%o3)       [[mod(x, 10), mod(y, 10), mod(z, 10)], 1, 2 mod(a, 5)]

     Mapping of the `floor' function over a matrix and an equation:

          (%i4) floor(matrix([a,b],[c,d]));
                                      [ floor(a)  floor(b) ]
          (%o4)                       [                    ]
                                      [ floor(c)  floor(d) ]
          (%i5) floor(a=b);
          (%o5)                         floor(a) = floor(b)

     Functions with more than one argument map over any of the
     arguments or all arguments:

          (%i6) expintegral_e([1,2],[x,y]);
          (%o6) [[expintegral_e(1, x), expintegral_e(1, y)],
                 [expintegral_e(2, x), expintegral_e(2, y)]]

     Check if a function has the property distribute_over:

          (%i7) properties(abs);
          (%o7) [integral, distributes over bags, noun, rule, gradef]

Option variable: domain

     Default value: `real'

     When `domain' is set to `complex', `sqrt (x^2)' will remain `sqrt
     (x^2)' instead of returning `abs(x)'.

Property: evenfun
Property: oddfun

     `declare(f, evenfun' or `declare(f, oddfun' tells Maxima to
     recognize the function `f' as an even or odd function.

     Examples:

          (%i1) o (- x) + o (x);
          (%o1)                     o(x) + o(- x)
          (%i2) declare (o, oddfun);
          (%o2)                         done
          (%i3) o (- x) + o (x);
          (%o3)                           0
          (%i4) e (- x) - e (x);
          (%o4)                     e(- x) - e(x)
          (%i5) declare (e, evenfun);
          (%o5)                         done
          (%i6) e (- x) - e (x);
          (%o6)                           0

Function: expand (<expr>)
Function: expand (<expr>, <p>, <n>)

     Expand expression <expr>.  Products of sums and exponentiated sums
     are multiplied out, numerators of rational expressions which are
     sums are split into their respective terms, and multiplication
     (commutative and non-commutative) are distributed over addition at
     all levels of <expr>.

     For polynomials one should usually use `ratexpand' which uses a
     more efficient algorithm.

     `maxnegex' and `maxposex' control the maximum negative and
     positive exponents, respectively, which will expand.

     `expand (<expr>, <p>, <n>)' expands <expr>, using <p> for
     `maxposex' and <n> for `maxnegex'.  This is useful in order to
     expand part but not all of an expression.

     `expon' - the exponent of the largest negative power which is
     automatically expanded (independent of calls to `expand').  For
     example if `expon' is 4 then `(x+1)^(-5)' will not be
     automatically expanded.

     `expop' - the highest positive exponent which is automatically
     expanded.  Thus `(x+1)^3', when typed, will be automatically
     expanded only if `expop' is greater than or equal to 3.  If it is
     desired to have `(x+1)^n' expanded where `n' is greater than
     `expop' then executing `expand ((x+1)^n)' will work only if
     `maxposex' is not less than `n'.

     `expand(expr, 0, 0)' causes a resimplification of `expr'.  `expr'
     is not reevaluated.  In distinction from `ev(expr, noeval)' a
     special representation (e. g. a CRE form) is removed.  See also
     `ev'.

     The `expand' flag used with `ev' causes expansion.

     The file `share/simplification/facexp.mac' contains several
     related functions (in particular `facsum', `factorfacsum' and
     `collectterms', which are autoloaded) and variables
     (`nextlayerfactor' and `facsum_combine') that provide the user with
     the ability to structure expressions by controlled expansion.
     Brief function descriptions are available in
     `simplification/facexp.usg'.  A demo is available by doing
     `demo("facexp")'.

     Examples:

          (%i1) expr:(x+1)^2*(y+1)^3;
                                         2        3
          (%o1)                   (x + 1)  (y + 1)
          (%i2) expand(expr);
                 2  3        3    3      2  2        2      2      2
          (%o2) x  y  + 2 x y  + y  + 3 x  y  + 6 x y  + 3 y  + 3 x  y
                                                                2
                                               + 6 x y + 3 y + x  + 2 x + 1
          (%i3) expand(expr,2);
                         2        3              3          3
          (%o3)         x  (y + 1)  + 2 x (y + 1)  + (y + 1)
          (%i4) expr:(x+1)^-2*(y+1)^3;
                                             3
                                      (y + 1)
          (%o4)                       --------
                                             2
                                      (x + 1)
          (%i5) expand(expr);
                      3               2
                     y             3 y            3 y             1
          (%o5) ------------ + ------------ + ------------ + ------------
                 2              2              2              2
                x  + 2 x + 1   x  + 2 x + 1   x  + 2 x + 1   x  + 2 x + 1
          (%i6) expand(expr,2,2);
                                             3
                                      (y + 1)
          (%o6)                     ------------
                                     2
                                    x  + 2 x + 1

     Resimplify an expression without expansion:

          (%i7) expr:(1+x)^2*sin(x);
                                                 2
          (%o7)                           (x + 1)  sin(x)
          (%i8) exponentialize:true;
          (%o8)                                true
          (%i9) expand(expr,0,0);
                                             2    %i x     - %i x
                                   %i (x + 1)  (%e     - %e      )
          (%o9)                  - -------------------------------
                                                  2

Function: expandwrt (<expr>, <x_1>, ..., <x_n>)

     Expands expression `expr' with respect to the variables <x_1>,
     ..., <x_n>.  All products involving the variables appear
     explicitly.  The form returned will be free of products of sums of
     expressions that are not free of the variables.  <x_1>, ..., <x_n>
     may be variables, operators, or expressions.

     By default, denominators are not expanded, but this can be
     controlled by means of the switch `expandwrt_denom'.

     This function is autoloaded from `simplification/stopex.mac'.

Option variable: expandwrt_denom

     Default value: `false'

     `expandwrt_denom' controls the treatment of rational expressions
     by `expandwrt'.  If `true', then both the numerator and
     denominator of the expression will be expanded according to the
     arguments of `expandwrt', but if `expandwrt_denom' is `false',
     then only the numerator will be expanded in that way.

Function: expandwrt_factored (<expr>, <x_1>, ..., <x_n>)

     is similar to `expandwrt', but treats expressions that are products
     somewhat differently.  `expandwrt_factored' expands only on those
     factors of `expr' that contain the variables <x_1>, ..., <x_n>.

     This function is autoloaded from `simplification/stopex.mac'.

Option variable: expon

     Default value: 0

     `expon' is the exponent of the largest negative power which is
     automatically expanded (independent of calls to `expand').  For
     example, if `expon' is 4 then `(x+1)^(-5)' will not be
     automatically expanded.

Function: exponentialize (<expr>)
Option variable: exponentialize

     The function `exponentialize (expr)' converts circular and
     hyperbolic functions in <expr> to exponentials, without setting
     the global variable `exponentialize'.

     When the variable `exponentialize' is `true', all circular and
     hyperbolic functions are converted to exponential form.  The
     default value is `false'.

     `demoivre' converts complex exponentials into circular functions.
     `exponentialize' and `demoivre' cannot both be true at the same
     time.

Option variable: expop

     Default value: 0

     `expop' is the highest positive exponent which is automatically
     expanded.  Thus `(x + 1)^3', when typed, will be automatically
     expanded only if `expop' is greater than or equal to 3.  If it is
     desired to have `(x + 1)^n' expanded where `n' is greater than
     `expop' then executing `expand ((x + 1)^n)' will work only if
     `maxposex' is not less than n.

Property: lassociative

     `declare (g, lassociative)' tells the Maxima simplifier that `g' is
     left-associative.  E.g., `g (g (a, b), g (c, d))' will simplify to
     `g (g (g (a, b), c), d)'.

Property: linear

     One of Maxima's operator properties.  For univariate `f' so
     declared, "expansion" `f(x + y)' yields `f(x) + f(y)', `f(a*x)'
     yields `a*f(x)' takes place where `a' is a "constant".  For
     functions of two or more arguments, "linearity" is defined to be
     as in the case of `sum' or `integrate', i.e., `f (a*x + b, x)'
     yields `a*f(x,x) + b*f(1,x)' for `a' and `b' free of `x'.

     `linear' is equivalent to `additive' and `outative'.  See also
     `opproperties'.

     Example:

          (%i1) 'sum (F(k) + G(k), k, 1, inf);
                                 inf
                                 ====
                                 \
          (%o1)                   >    (G(k) + F(k))
                                 /
                                 ====
                                 k = 1
          (%i2) declare (nounify (sum), linear);
          (%o2)                         done
          (%i3) 'sum (F(k) + G(k), k, 1, inf);
                               inf          inf
                               ====         ====
                               \            \
          (%o3)                 >    G(k) +  >    F(k)
                               /            /
                               ====         ====
                               k = 1        k = 1

Option variable: maxnegex

     Default value: 1000

     `maxnegex' is the largest negative exponent which will be expanded
     by the `expand' command (see also `maxposex').

Option variable: maxposex

     Default value: 1000

     `maxposex' is the largest exponent which will be expanded with the
     `expand' command (see also `maxnegex').

Property: multiplicative

     `declare(f, multiplicative)' tells the Maxima simplifier that `f'
     is multiplicative.

       1. If `f' is univariate, whenever the simplifier encounters `f'
          applied to a product, `f' distributes over that product.
          E.g., `f(x*y)' simplifies to `f(x)*f(y)'.

       2. If `f' is a function of 2 or more arguments, multiplicativity
          is defined as multiplicativity in the first argument to `f',
          e.g., `f (g(x) * h(x), x)' simplifies to `f (g(x) ,x) * f
          (h(x), x)'.

     This simplification does not occur when `f' is applied to
     expressions of the form `product (x[i], i, m, n)'.

     Example:

          (%i1) F2 (a * b * c);
          (%o1)                       F2(a b c)
          (%i2) declare (F2, multiplicative);
          (%o2)                         done
          (%i3) F2 (a * b * c);
          (%o3)                   F2(a) F2(b) F2(c)

Function: multthru (<expr>)
Function: multthru (<expr_1>, <expr_2>)

     Multiplies a factor (which should be a sum) of <expr> by the other
     factors of <expr>.  That is, <expr> is `<f_1> <f_2> ... <f_n>'
     where at least one factor, say <f_i>, is a sum of terms.  Each
     term in that sum is multiplied by the other factors in the
     product.  (Namely all the factors except <f_i>).  `multthru' does
     not expand exponentiated sums.  This function is the fastest way
     to distribute products (commutative or noncommutative) over sums.
     Since quotients are represented as products `multthru' can be used
     to divide sums by products as well.

     `multthru (<expr_1>, <expr_2>)' multiplies each term in <expr_2>
     (which should be a sum or an equation) by <expr_1>.  If <expr_1>
     is not itself a sum then this form is equivalent to `multthru
     (<expr_1>*<expr_2>)'.

          (%i1) x/(x-y)^2 - 1/(x-y) - f(x)/(x-y)^3;
                                1        x         f(x)
          (%o1)             - ----- + -------- - --------
                              x - y          2          3
                                      (x - y)    (x - y)
          (%i2) multthru ((x-y)^3, %);
                                     2
          (%o2)             - (x - y)  + x (x - y) - f(x)
          (%i3) ratexpand (%);
                                     2
          (%o3)                   - y  + x y - f(x)
          (%i4) ((a+b)^10*s^2 + 2*a*b*s + (a*b)^2)/(a*b*s^2);
                                  10  2              2  2
                           (b + a)   s  + 2 a b s + a  b
          (%o4)            ------------------------------
                                            2
                                       a b s
          (%i5) multthru (%);  /* note that this does not expand (b+a)^10 */
                                                  10
                                 2   a b   (b + a)
          (%o5)                  - + --- + ---------
                                 s    2       a b
                                     s
          (%i6) multthru (a.(b+c.(d+e)+f));
          (%o6)            a . f + a . c . (e + d) + a . b
          (%i7) expand (a.(b+c.(d+e)+f));
          (%o7)         a . f + a . c . e + a . c . d + a . b

Property: nary

     `declare(f, nary)' tells Maxima to recognize the function `f' as an
     n-ary function.

     The `nary' declaration is not the same as calling the `nary'
     function.  The sole effect of `declare(f, nary)' is to instruct
     the Maxima simplifier to flatten nested expressions, for example,
     to simplify `foo(x, foo(y, z))' to `foo(x, y, z)'.  See also
     `declare'.

     Example:

          (%i1) H (H (a, b), H (c, H (d, e)));
          (%o1)               H(H(a, b), H(c, H(d, e)))
          (%i2) declare (H, nary);
          (%o2)                         done
          (%i3) H (H (a, b), H (c, H (d, e)));
          (%o3)                   H(a, b, c, d, e)

Option variable: negdistrib

     Default value: `true'

     When `negdistrib' is `true', -1 distributes over an expression.
     E.g., `-(x + y)' becomes `- y - x'.  Setting it to `false' will
     allow `- (x + y)' to be displayed like that.  This is sometimes
     useful but be very careful: like the `simp' flag, this is one flag
     you do not want to set to `false' as a matter of course or
     necessarily for other than local use in your Maxima.

System variable: opproperties

     `opproperties' is the list of the special operator properties
     recognized by the Maxima simplifier: `linear', `additive',
     `multiplicative', `outative', `evenfun', `oddfun', `commutative',
     `symmetric', `antisymmetric', `nary', `lassociative',
     `rassociative'.

Property: outative

     `declare(f, outative)' tells the Maxima simplifier that constant
     factors in the argument of `f' can be pulled out.

       1. If `f' is univariate, whenever the simplifier encounters `f'
          applied to a product, that product will be partitioned into
          factors that are constant and factors that are not and the
          constant factors will be pulled out.  E.g., `f(a*x)' will
          simplify to `a*f(x)' where `a' is a constant.  Non-atomic
          constant factors will not be pulled out.

       2. If `f' is a function of 2 or more arguments, outativity is
          defined as in the case of `sum' or `integrate', i.e., `f
          (a*g(x), x)' will simplify to `a * f(g(x), x)' for `a' free
          of `x'.

     `sum', `integrate', and `limit' are all `outative'.

     Example:

          (%i1) F1 (100 * x);
          (%o1)                       F1(100 x)
          (%i2) declare (F1, outative);
          (%o2)                         done
          (%i3) F1 (100 * x);
          (%o3)                       100 F1(x)
          (%i4) declare (zz, constant);
          (%o4)                         done
          (%i5) F1 (zz * y);
          (%o5)                       zz F1(y)

Function: radcan (<expr>)

     Simplifies <expr>, which can contain logs, exponentials, and
     radicals, by converting it into a form which is canonical over a
     large class of expressions and a given ordering of variables; that
     is, all functionally equivalent forms are mapped into a unique
     form.  For a somewhat larger class of expressions, `radcan'
     produces a regular form.  Two equivalent expressions in this class
     do not necessarily have the same appearance, but their difference
     can be simplified by `radcan' to zero.

     For some expressions `radcan' is quite time consuming.  This is
     the cost of exploring certain relationships among the components
     of the expression for simplifications based on factoring and
     partial-fraction expansions of exponents.

     Examples:

          (%i1) radcan((log(x+x^2)-log(x))^a/log(1+x)^(a/2));
                                                     a/2
          (%o1)                            log(x + 1)

          (%i2) radcan((log(1+2*a^x+a^(2*x))/log(1+a^x)));
          (%o2)                                  2

          (%i3) radcan((%e^x-1)/(1+%e^(x/2)));
                                               x/2
          (%o3)                              %e    - 1

Option variable: radexpand

     Default value: `true'

     `radexpand' controls some simplifications of radicals.

     When `radexpand' is `all', causes nth roots of factors of a product
     which are powers of n to be pulled outside of the radical.  E.g. if
     `radexpand' is `all', `sqrt (16*x^2)' simplifies to `4*x'.

     More particularly, consider `sqrt (x^2)'.
        * If `radexpand' is `all' or `assume (x > 0)' has been executed,
          `sqrt(x^2)' simplifies to `x'.

        * If `radexpand' is `true' and `domain' is `real' (its
          default), `sqrt(x^2)' simplifies to `abs(x)'.

        * If `radexpand' is `false', or `radexpand' is `true' and
          `domain' is `complex', `sqrt(x^2)' is not simplified.

     Note that `domain' only matters when `radexpand' is `true'.

Property: rassociative

     `declare (g, rassociative)' tells the Maxima simplifier that `g'
     is right-associative.  E.g., `g(g(a, b), g(c, d))' simplifies to
     `g(a, g(b, g(c, d)))'.

Function: scsimp (<expr>, <rule_1>, ..., <rule_n>)

     Sequential Comparative Simplification (method due to Stoute).
     `scsimp' attempts to simplify <expr> according to the rules
     <rule_1>, ..., <rule_n>.  If a smaller expression is obtained, the
     process repeats.  Otherwise after all simplifications are tried,
     it returns the original answer.

     `example (scsimp)' displays some examples.

Option variable: simp

     Default value: `true'

     `simp' enables simplification.  This is the standard.  `simp' is
     also an `evflag', which is recognized by the function `ev'.  See
     `ev'.

     When `simp' is used as an `evflag' with a value `false', the
     simplification is suppressed only during the evaluation phase of
     an expression.  The flag can not suppress the simplification which
     follows the evaluation phase.

     Examples:

     The simplification is switched off globally.  The expression
     `sin(1.0)' is not simplified to its numerical value.  The
     `simp'-flag switches the simplification on.

          (%i1) simp:false;
          (%o1)                                false
          (%i2) sin(1.0);
          (%o2)                              sin(1.0)
          (%i3) sin(1.0),simp;
          (%o3)                          .8414709848078965

     The simplification is switched on again.  The `simp'-flag cannot
     suppress the simplification completely.  The output shows a
     simplified expression, but the variable `x' has an unsimplified
     expression as a value, because the assignment has occurred during
     the evaluation phase of the expression.

          (%i4) simp:true;
          (%o4)                                true
          (%i5) x:sin(1.0),simp:false;
          (%o5)                          .8414709848078965
          (%i6) :lisp $X
          ((%SIN) 1.0)

Property: symmetric

     `declare (h, symmetric)' tells the Maxima simplifier that `h' is a
     symmetric function.  E.g., `h (x, z, y)' simplifies to `h (x, y,
     z)'.

     `commutative' is synonymous with `symmetric'.

Function: xthru (<expr>)

     Combines all terms of <expr> (which should be a sum) over a common
     denominator without expanding products and exponentiated sums as
     `ratsimp' does.  `xthru' cancels common factors in the numerator
     and denominator of rational expressions but only if the factors
     are explicit.

     Sometimes it is better to use `xthru' before `ratsimp'ing an
     expression in order to cause explicit factors of the gcd of the
     numerator and denominator to be canceled thus simplifying the
     expression to be `ratsimp'ed.

     Examples:

          (%i1) ((x+2)^20 - 2*y)/(x+y)^20 + (x+y)^(-19) - x/(x+y)^20;
                                          20
                           1       (x + 2)   - 2 y       x
          (%o1)        --------- + --------------- - ---------
                              19             20             20
                       (y + x)        (y + x)        (y + x)
          (%i2) xthru (%);
                                           20
                                    (x + 2)   - y
          (%o2)                     -------------
                                             20
                                      (y + x)

10 Mathematical Functions


Functions for Numbers
Functions for Complex Numbers
Combinatorial Functions
* Root Exponential and Logarithmic Functions::
Trigonometric Functions
Random Numbers

10.1 Functions for Numbers

Function: abs (<z>)

     The `abs' function represents the mathematical absolute value
     function and works for both numerical and symbolic values. If the
     argument, <z>, is a real or complex number, `abs' returns the
     absolute value of <z>. If possible, symbolic expressions using the
     absolute value function are also simplified.

     Maxima can differentiate, integrate and calculate limits for
     expressions containing `abs'. The `abs_integrate' package further
     extends Maxima's ability to calculate integrals involving the abs
     function. See (%i12) in the examples below.

     When applied to a list or matrix, `abs' automatically distributes
     over the terms. Similarly, it distributes over both sides of an
     equation. To alter this behaviour, see the variable
     `distribute_over'.

     Examples:

     Calculation of `abs' for real and complex numbers, including
     numerical constants and various infinities. The first example
     shows how `abs' distributes over the elements of a list.

          (%i1) abs([-4, 0, 1, 1+%i]);
          (%o1)                  [4, 0, 1, sqrt(2)]

          (%i2) abs((1+%i)*(1-%i));
          (%o2)                           2
          (%i3) abs(%e+%i);
                                          2
          (%o3)                    sqrt(%e  + 1)
          (%i4) abs([inf, infinity, minf]);
          (%o4)                   [inf, inf, inf]

     Simplification of expressions containing `abs':

          (%i5) abs(x^2);
                                          2
          (%o5)                          x
          (%i6) abs(x^3);
                                       2
          (%o6)                       x  abs(x)

          (%i7) abs(abs(x));
          (%o7)                       abs(x)
          (%i8) abs(conjugate(x));
          (%o8)                       abs(x)

     Integrating and differentiating with the `abs' function. Note that
     more integrals involving the `abs' function can be performed, if
     the `abs_integrate' package is loaded. The last example shows the
     Laplace transform of `abs': see `laplace'.

          (%i9) diff(x*abs(x),x),expand;
          (%o9)                       2 abs(x)

          (%i10) integrate(abs(x),x);
                                       x abs(x)
          (%o10)                       --------
                                          2

          (%i11) integrate(x*abs(x),x);
                                     /
                                     [
          (%o11)                     I x abs(x) dx
                                     ]
                                     /

          (%i12) load(abs_integrate)$
          (%i13) integrate(x*abs(x),x);
                                2           3
                               x  abs(x)   x  signum(x)
          (%o13)               --------- - ------------
                                   2            6

          (%i14) integrate(abs(x),x,-2,%pi);
                                         2
                                      %pi
          (%o14)                      ---- + 2
                                       2

          (%i15) laplace(abs(x),x,s);
                                         1
          (%o15)                         --
                                          2
                                         s

Function: ceiling (<x>)

     When <x> is a real number, return the least integer that is
     greater than or equal to <x>.

     If <x> is a constant expression (`10 * %pi', for example),
     `ceiling' evaluates <x> using big floating point numbers, and
     applies `ceiling' to the resulting big float.  Because `ceiling'
     uses floating point evaluation, it's possible, although unlikely,
     that `ceiling' could return an erroneous value for constant
     inputs.  To guard against errors, the floating point evaluation is
     done using three values for `fpprec'.

     For non-constant inputs, `ceiling' tries to return a simplified
     value.  Here are examples of the simplifications that `ceiling'
     knows about:

          (%i1) ceiling (ceiling (x));
          (%o1)                      ceiling(x)
          (%i2) ceiling (floor (x));
          (%o2)                       floor(x)
          (%i3) declare (n, integer)$
          (%i4) [ceiling (n), ceiling (abs (n)), ceiling (max (n, 6))];
          (%o4)                [n, abs(n), max(n, 6)]
          (%i5) assume (x > 0, x < 1)$
          (%i6) ceiling (x);
          (%o6)                           1
          (%i7) tex (ceiling (a));
          $$\left \lceil a \right \rceil$$
          (%o7)                         false

     The function `ceiling' does not automatically map over lists or
     matrices.  Finally, for all inputs that are manifestly complex,
     `ceiling' returns a noun form.

     If the range of a function is a subset of the integers, it can be
     declared to be `integervalued'.  Both the `ceiling' and `floor'
     functions can use this information; for example:

          (%i1) declare (f, integervalued)$
          (%i2) floor (f(x));
          (%o2)                         f(x)
          (%i3) ceiling (f(x) - 1);
          (%o3)                       f(x) - 1

Function: entier (<x>)

     Returns the largest integer less than or equal to <x> where <x> is
     numeric.  `fix'  (as in `fixnum') is a synonym for this, so
     `fix(<x>)' is precisely the same.

Function: floor (<x>)

     When <x> is a real number, return the largest integer that is less
     than or equal to <x>.

     If <x> is a constant expression (`10 * %pi', for example), `floor'
     evaluates <x> using big floating point numbers, and applies
     `floor' to the resulting big float. Because `floor' uses floating
     point evaluation, it's possible, although unlikely, that `floor'
     could return an erroneous value for constant inputs.  To guard
     against errors, the floating point evaluation is done using three
     values for `fpprec'.

     For non-constant inputs, `floor' tries to return a simplified
     value.  Here are examples of the simplifications that `floor'
     knows about:

          (%i1) floor (ceiling (x));
          (%o1)                      ceiling(x)
          (%i2) floor (floor (x));
          (%o2)                       floor(x)
          (%i3) declare (n, integer)$
          (%i4) [floor (n), floor (abs (n)), floor (min (n, 6))];
          (%o4)                [n, abs(n), min(n, 6)]
          (%i5) assume (x > 0, x < 1)$
          (%i6) floor (x);
          (%o6)                           0
          (%i7) tex (floor (a));
          $$\left \lfloor a \right \rfloor$$
          (%o7)                         false

     The function `floor' does not automatically map over lists or
     matrices.  Finally, for all inputs that are manifestly complex,
     `floor' returns a noun form.

     If the range of a function is a subset of the integers, it can be
     declared to be `integervalued'.  Both the `ceiling'  and `floor'
     functions can use this information; for example:

          (%i1) declare (f, integervalued)$
          (%i2) floor (f(x));
          (%o2)                         f(x)
          (%i3) ceiling (f(x) - 1);
          (%o3)                       f(x) - 1

Function: fix (<x>)

     A synonym for `entier (<x>)'.

Function: lmax (<L>)

     When <L> is a list or a set, return `apply ('max, args (<L>))'.
     When <L> is not a list or a set, signal an error.  See also `lmin'
     and `max'.

Function: lmin (<L>)

     When <L> is a list or a set, return `apply ('min, args (<L>))'.
     When <L> is not a list or a set, signal an error.  See also `lmax'
     and `min'.

Function: max (<x_1>, ..., <x_n>)

     Return a simplified value for the maximum of the expressions <x_1>
     through <x_n>.  When `get (trylevel, maxmin)', is 2 or greater,
     `max' uses the simplification `max (e, -e) --> |e|'.  When `get
     (trylevel, maxmin)' is 3 or greater, <max> tries to eliminate
     expressions that are between two other arguments; for example,
     `max (x, 2*x, 3*x) --> max (x, 3*x)'.  To set the value of
     `trylevel' to 2, use `put (trylevel, 2, maxmin)'.

     See also `min'  and `lmax'.

Function: min (<x_1>, ..., <x_n>)

     Return a simplified value for the minimum of the expressions `x_1'
     through `x_n'.  When `get (trylevel, maxmin)', is 2 or greater,
     `min' uses the simplification `min (e, -e) --> -|e|'.  When `get
     (trylevel, maxmin)' is 3 or greater, `min' tries to eliminate
     expressions that are between two other arguments; for example,
     `min (x, 2*x, 3*x) --> min (x, 3*x)'.  To set the value of
     `trylevel' to 2, use `put (trylevel, 2, maxmin)'.

     See also `max'  and `lmin'.

Function: round (<x>)

     When <x> is a real number, returns the closest integer to <x>.
     Multiples of 1/2 are rounded to the nearest even integer.
     Evaluation of <x> is similar to `floor'  and `ceiling'.

Function: signum (<x>)

     For either real or complex numbers <x>, the signum function returns
     0 if <x> is zero; for a nonzero numeric input <x>, the signum
     function returns `abs(x)/x'.

     For non-numeric inputs, Maxima attempts to determine the sign of
     the input.  When the sign is negative, zero, or positive, `signum'
     returns -1,0, 1, respectively.  For all other values for the sign,
     `signum' a simplified but equivalent form.  The simplifications
     include reflection (`signum(-x)' gives `-signum(x)') and
     multiplicative identity (`signum(x*y)' gives `signum(x) *
     signum(y)').

     The `signum' function distributes over a list, a matrix, or an
     equation.  See `distribute_over'.

10.2 Functions for Complex Numbers

Function: cabs (<expr>)

     Calculates the absolute value of an expression representing a
     complex number.  Unlike the function `abs',  the `cabs' function
     always decomposes its argument into a real and an imaginary part.
     If `x' and `y' represent real variables or expressions, the `cabs'
     function calculates the absolute value of `x + %i*y' as
                                     2    2
                               sqrt(y  + x )

     The `cabs' function can use known properties like symmetry
     properties of complex functions to help it calculate the absolute
     value of an expression.  If such identities exist, they can be
     advertised to `cabs' using function properties.  The symmetries
     that `cabs' understands are: mirror symmetry, conjugate function
     and complex characteristic.

     `cabs' is a verb function and is not suitable for symbolic
     calculations.  For such calculations (including integration,
     differentiation and taking limits of expressions containing
     absolute values), use `abs'.

     The result of `cabs' can include the absolute value function,
     `abs',  and the arc tangent, `atan2'.

     When applied to a list or matrix, `cabs' automatically distributes
     over the terms.  Similarly, it distributes over both sides of an
     equation.

     For further ways to compute with complex numbers, see the functions
     `rectform',  `realpart',  `imagpart',

     `carg',  `conjugate'  and `polarform'.

     Examples:

     Examples with `sqrt'  and `sin'.

          (%i1) cabs(sqrt(1+%i*x));
                                       2     1/4
          (%o1)                      (x  + 1)
          (%i2) cabs(sin(x+%i*y));
                              2        2         2        2
          (%o2)       sqrt(cos (x) sinh (y) + sin (x) cosh (y))

     The error function, `erf',  has mirror symmetry, which is used
     here in the calculation of the absolute value with a complex
     argument:

          (%i3) cabs(erf(x+%i*y));
                                                    2
                     (erf(%i y + x) - erf(%i y - x))
          (%o3) sqrt(--------------------------------
                                    4
                                                                         2
                                          (erf(%i y + x) + erf(%i y - x))
                                        - --------------------------------)
                                                         4

     Maxima knows complex identities for the Bessel functions, which
     allow it to compute the absolute value for complex arguments.
     Here is an example for `bessel_j'.

          (%i4) cabs(bessel_j(1,%i));
          (%o4)                 abs(bessel_j(1, %i))

Function: carg (<z>)

     Returns the complex argument of <z>.  The complex argument is an
     angle `theta' in `(-%pi, %pi]' such that `r exp (theta %i) = <z>'
     where `r' is the magnitude of <z>.

     `carg' is a computational function, not a simplifying function.

     See also `abs'  (complex magnitude), `polarform',

     `rectform',  `realpart',  and `imagpart'.

     Examples:

          (%i1) carg (1);
          (%o1)                           0
          (%i2) carg (1 + %i);
                                         %pi
          (%o2)                          ---
                                          4
          (%i3) carg (exp (%i));
          (%o3)                           1
          (%i4) carg (exp (%pi * %i));
          (%o4)                          %pi
          (%i5) carg (exp (3/2 * %pi * %i));
                                          %pi
          (%o5)                         - ---
                                           2
          (%i6) carg (17 * exp (2 * %i));
          (%o6)                           2

Function: conjugate (<x>)

     Returns the complex conjugate of <x>.

          (%i1) declare ([aa, bb], real, cc, complex, ii, imaginary);

          (%o1)                         done
          (%i2) conjugate (aa + bb*%i);

          (%o2)                      aa - %i bb
          (%i3) conjugate (cc);

          (%o3)                     conjugate(cc)
          (%i4) conjugate (ii);

          (%o4)                         - ii
          (%i5) conjugate (xx + yy);

          (%o5)             conjugate(yy) + conjugate(xx)

Function: imagpart (<expr>)

     Returns the imaginary part of the expression <expr>.

     `imagpart' is a computational function, not a simplifying function.

     See also `abs',  `carg',  `polarform',

     `rectform',  and `realpart'.

Function: polarform (<expr>)

     Returns an expression `r %e^(%i theta)' equivalent to <expr>, such
     that `r' and `theta' are purely real.

Function: realpart (<expr>)

     Returns the real part of <expr>.  `realpart' and `imagpart'  will
     work on expressions involving trigonometric and hyperbolic
     functions, as well as square root, logarithm, and exponentiation.

Function: rectform (<expr>)

     Returns an expression `a + b %i' equivalent to <expr>, such that
     <a> and <b> are purely real.

10.3 Combinatorial Functions

Operator: !!

     The double factorial operator.

     For an integer, float, or rational number `n', `n!!' evaluates to
     the product `n (n-2) (n-4) (n-6) ... (n - 2 (k-1))' where `k' is
     equal to `entier (n/2)', that is, the largest integer less than or
     equal to `n/2'.  Note that this definition does not coincide with
     other published definitions for arguments which are not integers.

     For an even (or odd) integer `n', `n!!' evaluates to the product of
     all the consecutive even (or odd) integers from 2 (or 1) through
     `n' inclusive.

     For an argument `n' which is not an integer, float, or rational,
     `n!!' yields a noun form `genfact (n, n/2, 2)'.

Function: binomial (<x>, <y>)

     The binomial coefficient `<x>!/(<y>! (<x> - <y>)!)'.  If <x> and
     <y> are integers, then the numerical value of the binomial
     coefficient is computed.  If <y>, or <x - y>, is an integer, the
     binomial coefficient is expressed as a polynomial.

     Examples:

          (%i1) binomial (11, 7);
          (%o1)                          330
          (%i2) 11! / 7! / (11 - 7)!;
          (%o2)                          330
          (%i3) binomial (x, 7);
                  (x - 6) (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) x
          (%o3)   -------------------------------------------------
                                        5040
          (%i4) binomial (x + 7, x);
                (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x + 6) (x + 7)
          (%o4) -------------------------------------------------------
                                         5040
          (%i5) binomial (11, y);
          (%o5)                    binomial(11, y)

Function: factcomb (<expr>)

     Tries to combine the coefficients of factorials in <expr> with the
     factorials themselves by converting, for example, `(n + 1)*n!'
     into `(n + 1)!'.

     `sumsplitfact'  if set to `false' will cause `minfactorial'  to be
     applied after a `factcomb'.

Function: factorial
Operator: !

     Represents the factorial function.  Maxima treats `factorial (<x>)'
     the same as `<x>!'.

     For any complex number `x', except for negative integers, `x!' is
     defined as `gamma(x+1)'.

     For an integer `x', `x!' simplifies to the product of the integers
     from 1 to `x' inclusive.  `0!' simplifies to 1.  For a real or
     complex number in float or bigfloat precision `x', `x!' simplifies
     to the value of `gamma (x+1)'.  For `x' equal to `n/2' where `n' is
     an odd integer, `x!' simplifies to a rational factor times `sqrt
     (%pi)' (since `gamma (1/2)' is equal to `sqrt (%pi)').

     The option variables `factlim'  and `gammalim'  control the
     numerical evaluation of factorials for integer and rational
     arguments.  The functions `minfactorial'  and `factcomb'
     simplifies expressions containing factorials.

     The functions `gamma',  `bffac',  and `cbffac'  are varieties of
     the gamma function.  `bffac' and `cbffac' are called internally by
     `gamma' to evaluate the gamma function for real and complex
     numbers in bigfloat precision.

     `makegamma'  substitutes `gamma' for factorials and related
     functions.

     Maxima knows the derivative of the factorial function and the
     limits for specific values like negative integers.

     The option variable `factorial_expand'  controls the
     simplification of expressions like `(n+x)!', where `n' is an
     integer.

     See also `binomial'.

     The factorial of an integer is simplified to an exact number
     unless the operand is greater than `factlim'.  The factorial for
     real and complex numbers is evaluated in float or bigfloat
     precision.

          (%i1) factlim:10;
          (%o1)                                 10
          (%i2) [0!, (7/2)!, 8!, 20!];
                                      105 sqrt(%pi)
          (%o2)                   [1, -------------, 40320, 20!]
                                           16
          (%i3) [4.77!, (1.0+%i)!];
          (%o3)    [81.44668037931197,
                    .3430658398165454 %i + .6529654964201665]
          (%i4) [2.86b0!, (1.0b0+%i)!];
          (%o4) [5.046635586910012b0,
                 3.430658398165454b-1 %i + 6.529654964201667b-1]

     The factorial of a known constant, or general expression is not
     simplified.  Even so it may be possible to simplify the factorial
     after evaluating the operand.

          (%i1) [(%i + 1)!, %pi!, %e!, (cos(1) + sin(1))!];
          (%o1)      [(%i + 1)!, %pi!, %e!, (sin(1) + cos(1))!]
          (%i2) ev (%, numer, %enumer);
          (%o2) [.3430658398165454 %i + .6529654964201665,
                 7.188082728976031,
                 4.260820476357003, 1.227580202486819]

     Factorials are simplified, not evaluated.  Thus `x!' may be
     replaced even in a quoted expression.

          (%i1) '([0!, (7/2)!, 4.77!, 8!, 20!]);
                    105 sqrt(%pi)
          (%o1) [1, -------------, 81.44668037931199, 40320,
                         16
                                                       2432902008176640000]

     Maxima knows the derivative of the factorial function.

          (%i1) diff(x!,x);
          (%o1)                           x! psi (x + 1)
                                                0

     The option variable `factorial_expand' controls expansion and
     simplification of expressions with the factorial function.

          (%i1) (n+1)!/n!,factorial_expand:true;
          (%o1)                                n + 1

Option variable: factlim

     Default value: -1

     `factlim' specifies the highest factorial which is automatically
     expanded.  If it is -1 then all integers are expanded.

Option variable: factorial_expand

     Default value: false

     The option variable `factorial_expand' controls the simplification
     of expressions like `(n+1)!', where `n' is an integer.  See
     `factorial'  for an example.

Function: genfact (<x>, <y>, <z>)

     Returns the generalized factorial, defined as `x (x-z) (x - 2 z)
     ... (x - (y - 1) z)'.  Thus, for integral <x>, `genfact (x, x, 1)
     = x!' and `genfact (x, x/2, 2) = x!!'.

Function: minfactorial (<expr>)

     Examines <expr> for occurrences of two factorials which differ by
     an integer.  `minfactorial' then turns one into a polynomial times
     the other.

          (%i1) n!/(n+2)!;
                                         n!
          (%o1)                       --------
                                      (n + 2)!
          (%i2) minfactorial (%);
                                          1
          (%o2)                    ---------------
                                   (n + 1) (n + 2)

Option variable: sumsplitfact

     Default value: `true'

     When `sumsplitfact' is `false', `minfactorial'  is applied after a
     `factcomb'.

10.4 Root, Exponential and Logarithmic Functions

Option variable: %e_to_numlog

     Default value: `false'

     When `true', `r' some rational number, and `x' some expression,
     `%e^(r*log(x))' will be simplified into `x^r' .  It should be noted
     that the `radcan' command also does this transformation, and more
     complicated transformations of this ilk as well.  The `logcontract'
     command "contracts" expressions containing `log'.

Option variable: %emode

     Default value: `true'

     When `%emode' is `true', `%e^(%pi %i x)' is simplified as follows.

     `%e^(%pi %i x)' simplifies to `cos (%pi x) + %i sin (%pi x)' if
     `x' is a floating point number, an integer, or a multiple of 1/2,
     1/3, 1/4, or 1/6, and then further simplified.

     For other numerical `x', `%e^(%pi %i x)' simplifies to `%e^(%pi %i
     y)' where `y' is `x - 2 k' for some integer `k' such that `abs(y)
     < 1'.

     When `%emode' is `false', no special simplification of `%e^(%pi %i
     x)' is carried out.

Option variable: %enumer

     Default value: `false'

     When `%enumer' is `true', `%e' is replaced by its numeric value
     2.718...  whenever `numer' is `true'.

     When `%enumer' is `false', this substitution is carried out only
     if the exponent in `%e^x' evaluates to a number.

     See also `ev'  and `numer'.

Function: exp (<x>)

     Represents the exponential function.  Instances of `exp (<x>)' in
     input are simplified to `%e^<x>'; `exp' does not appear in
     simplified expressions.

     `demoivre' if `true' causes `%e^(a + b %i)' to simplify to `%e^(a
     (cos(b) + %i sin(b)))' if `b' is free of `%i'.  See `demoivre'.

     `%emode', when `true', causes `%e^(%pi %i x)' to be simplified.
     See `%emode'.

     `%enumer', when `true' causes `%e' to be replaced by 2.718...
     whenever `numer' is `true'.  See `%enumer'.

Function: li [<s>] (<z>)

     Represents the polylogarithm function of order <s> and argument
     <z>, defined by the infinite series

                                           inf
                                           ====   k
                                           \     z
                                  Li (z) =  >    --
                                    s      /      s
                                           ====  k
                                           k = 1

     `li [1]' is `- log (1 - z)'.  `li [2]' and `li [3]' are the
     dilogarithm and trilogarithm functions, respectively.

     When the order is 1, the polylogarithm simplifies to `- log (1 -
     z)', which in turn simplifies to a numerical value if <z> is a
     real or complex floating point number or the `numer' evaluation
     flag is present.

     When the order is 2 or 3, the polylogarithm simplifies to a
     numerical value if <z> is a real floating point number or the
     `numer' evaluation flag is present.

     Examples:

          (%i1) assume (x > 0);
          (%o1)                        [x > 0]
          (%i2) integrate ((log (1 - t)) / t, t, 0, x);
          (%o2)                       - li (x)
                                          2
          (%i3) li [2] (7);
          (%o3)                        li (7)
                                         2
          (%i4) li [2] (7), numer;
          (%o4)        1.24827317833392 - 6.113257021832577 %i
          (%i5) li [3] (7);
          (%o5)                        li (7)
                                         3
          (%i6) li [2] (7), numer;
          (%o6)        1.24827317833392 - 6.113257021832577 %i
          (%i7) L : makelist (i / 4.0, i, 0, 8);
          (%o7)   [0.0, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0]
          (%i8) map (lambda ([x], li [2] (x)), L);
          (%o8) [0, .2676526384986274, .5822405249432515,
          .9784693966661848, 1.64493407, 2.190177004178597
           - .7010261407036192 %i, 2.374395264042415
           - 1.273806203464065 %i, 2.448686757245154
           - 1.758084846201883 %i, 2.467401098097648
           - 2.177586087815347 %i]
          (%i9) map (lambda ([x], li [3] (x)), L);
          (%o9) [0, .2584613953442624, 0.537213192678042,
          .8444258046482203, 1.2020569, 1.642866878950322
           - .07821473130035025 %i, 2.060877505514697
           - .2582419849982037 %i, 2.433418896388322
           - .4919260182322965 %i, 2.762071904015935
           - .7546938285978846 %i]

Function: log (<x>)

     Represents the natural (base e) logarithm of <x>.

     Maxima does not have a built-in function for the base 10 logarithm
     or other bases. `log10(x) := log(x) / log(10)' is a useful
     definition.

     Simplification and evaluation of logarithms is governed by several
     global flags:

    ``logexpand''
          causes `log(a^b)' to become `b*log(a)'. If it is set to
          `all', `log(a*b)' will also simplify to `log(a)+log(b)'.  If
          it is set to `super', then `log(a/b)' will also simplify to
          `log(a)-log(b)' for rational numbers `a/b', `a#1'.
          (`log(1/b)', for `b' integer, always simplifies.) If it is
          set to `false', all of these simplifications will be turned
          off.

    ``logsimp''
          if `false' then no simplification of `%e' to a power
          containing `log''s is done.

    ``lognegint''
          if `true' implements the rule `log(-n)' -> `log(n)+%i*%pi' for
          `n' a positive integer.

    ``%e_to_numlog''
          when `true', `r' some rational number, and `x' some
          expression, the expression `%e^(r*log(x))' will be simplified
          into `x^r'.  It should be noted that the `radcan' command
          also does this transformation, and more complicated
          transformations of this as well. The `logcontract' command
          "contracts" expressions containing `log'.

Option variable: logabs

     Default value: `false'

     When doing indefinite integration where logs are generated, e.g.
     `integrate(1/x,x)', the answer is given in terms of `log(abs(...))'
     if `logabs' is `true', but in terms of `log(...)' if `logabs' is
     `false'.  For definite integration, the `logabs:true' setting is
     used, because here "evaluation" of the indefinite integral at the
     endpoints is often needed.

Option variable: logarc
Function: logarc (<expr>)

     When the global variable `logarc' is `true', inverse circular and
     hyperbolic functions are replaced by equivalent logarithmic
     functions.  The default value of `logarc' is `false'.

     The function `logarc(<expr>)' carries out that replacement for an
     expression <expr> without setting the global variable `logarc'.

Option variable: logconcoeffp

     Default value: `false'

     Controls which coefficients are contracted when using
     `logcontract'.  It may be set to the name of a predicate function
     of one argument.  E.g. if you like to generate SQRTs, you can do
     `logconcoeffp:'logconfun$ logconfun(m):=featurep(m,integer) or
     ratnump(m)$' .  Then `logcontract(1/2*log(x));' will give
     `log(sqrt(x))'.

Function: logcontract (<expr>)

     Recursively scans the expression <expr>, transforming
     subexpressions of the form `a1*log(b1) + a2*log(b2) + c' into
     `log(ratsimp(b1^a1 * b2^a2)) + c'

          (%i1) 2*(a*log(x) + 2*a*log(y))$
          (%i2) logcontract(%);
                                           2  4
          (%o2)                     a log(x  y )

     The declaration `declare(n,integer)' causes
     `logcontract(2*a*n*log(x))' to simplify to `a*log(x^(2*n))'.  The
     coefficients that "contract" in this manner are those such as the
     2 and the `n' here which satisfy `featurep(coeff,integer)'.  The
     user can control which coefficients are contracted by setting the
     option `logconcoeffp' to the name of a predicate function of one
     argument.  E.g. if you like to generate SQRTs, you can do
     `logconcoeffp:'logconfun$ logconfun(m):=featurep(m,integer) or
     ratnump(m)$' .  Then `logcontract(1/2*log(x));' will give
     `log(sqrt(x))'.

Option variable: logexpand

     Default value: `true'

     If `true', that is the default value, causes `log(a^b)' to become
     `b*log(a)'.  If it is set to `all', `log(a*b)' will also simplify
     to `log(a)+log(b)'.  If it is set to `super', then `log(a/b)' will
     also simplify to `log(a)-log(b)' for rational numbers `a/b',
     `a#1'.  (`log(1/b)', for integer `b', always simplifies.) If it is
     set to `false', all of these simplifications will be turned off.

Option variable: lognegint

     Default value: `false'

     If `true' implements the rule `log(-n)' -> `log(n)+%i*%pi' for `n'
     a positive integer.

Option variable: logsimp

     Default value: `true'

     If `false' then no simplification of `%e' to a power containing
     `log''s is done.

Function: plog (<x>)

     Represents the principal branch of the complex-valued natural
     logarithm with `-%pi' < `carg(<x>)' <= `+%pi' .

Function: sqrt (<x>)

     The square root of <x>.  It is represented internally by
     `<x>^(1/2)'.  See also `rootscontract'.

     `radexpand' if `true' will cause nth roots of factors of a product
     which are powers of n to be pulled outside of the radical, e.g.
     `sqrt(16*x^2)' will become `4*x' only if `radexpand' is `true'.

10.5 Trigonometric Functions


* Introduction to Trigonometric::
* Functions and Variables for Trigonometric::
Option variable: %piargs

     Default value: `true'

     When `%piargs' is `true', trigonometric functions are simplified
     to algebraic constants when the argument is an integer multiple of
     %pi, %pi/2, %pi/3, %pi/4, or %pi/6.

     Maxima knows some identities which can be applied when %pi, etc.,
     are multiplied by an integer variable (that is, a symbol declared
     to be integer).

     Examples:

          (%i1) %piargs : false$
          (%i2) [sin (%pi), sin (%pi/2), sin (%pi/3)];
                                          %pi       %pi
          (%o2)            [sin(%pi), sin(---), sin(---)]
                                           2         3
          (%i3) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
                                %pi       %pi       %pi
          (%o3)            [sin(---), sin(---), sin(---)]
                                 4         5         6
          (%i4) %piargs : true$
          (%i5) [sin (%pi), sin (%pi/2), sin (%pi/3)];
                                          sqrt(3)
          (%o5)                    [0, 1, -------]
                                             2
          (%i6) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
                                   1         %pi   1
          (%o6)                [-------, sin(---), -]
                                sqrt(2)       5    2
          (%i7) [cos (%pi/3), cos (10*%pi/3), tan (10*%pi/3),
                 cos (sqrt(2)*%pi/3)];
                          1    1               sqrt(2) %pi
          (%o7)          [-, - -, sqrt(3), cos(-----------)]
                          2    2                    3

     Some identities are applied when %pi and %pi/2 are multiplied by
     an integer variable.

          (%i1) declare (n, integer, m, even)$
          (%i2) [sin (%pi * n), cos (%pi * m), sin (%pi/2 * m),
                 cos (%pi/2 * m)];
                                                m/2
          (%o2)                  [0, 1, 0, (- 1)   ]

Option variable: %iargs

     Default value: `true'

     When `%iargs' is `true', trigonometric functions are simplified to
     hyperbolic functions when the argument is apparently a multiple of
     the imaginary unit %i.

     Even when the argument is demonstrably real, the simplification is
     applied; Maxima considers only whether the argument is a literal
     multiple of %i.

     Examples:

          (%i1) %iargs : false$
          (%i2) [sin (%i * x), cos (%i * x), tan (%i * x)];
          (%o2)           [sin(%i x), cos(%i x), tan(%i x)]
          (%i3) %iargs : true$
          (%i4) [sin (%i * x), cos (%i * x), tan (%i * x)];
          (%o4)           [%i sinh(x), cosh(x), %i tanh(x)]

     Even when the argument is demonstrably real, the simplification is
     applied.

          (%i1) declare (x, imaginary)$
          (%i2) [featurep (x, imaginary), featurep (x, real)];
          (%o2)                     [true, false]
          (%i3) sin (%i * x);
          (%o3)                      %i sinh(x)

Function: acos (<x>)

     - Arc Cosine.

Function: acosh (<x>)

     - Hyperbolic Arc Cosine.

Function: acot (<x>)

     - Arc Cotangent.

Function: acoth (<x>)

     - Hyperbolic Arc Cotangent.

Function: acsc (<x>)

     - Arc Cosecant.

Function: acsch (<x>)

     - Hyperbolic Arc Cosecant.

Function: asec (<x>)

     - Arc Secant.

Function: asech (<x>)

     - Hyperbolic Arc Secant.

Function: asin (<x>)

     - Arc Sine.

Function: asinh (<x>)

     - Hyperbolic Arc Sine.

Function: atan (<x>)

     - Arc Tangent.

Function: atan2 (<y>, <x>)

     - yields the value of `atan(<y>/<x>)' in the interval `-%pi' to
     `%pi'.

Function: atanh (<x>)

     - Hyperbolic Arc Tangent.

Package: atrig1

     The `atrig1' package contains several additional simplification
     rules for inverse trigonometric functions.  Together with rules
     already known to Maxima, the following angles are fully
     implemented: `0', `%pi/6', `%pi/4', `%pi/3', and `%pi/2'.
     Corresponding angles in the other three quadrants are also
     available.  Do `load(atrig1);' to use them.

Function: cos (<x>)

     - Cosine.

Function: cosh (<x>)

     - Hyperbolic Cosine.

Function: cot (<x>)

     - Cotangent.

Function: coth (<x>)

     - Hyperbolic Cotangent.

Function: csc (<x>)

     - Cosecant.

Function: csch (<x>)

     - Hyperbolic Cosecant.

Option variable: halfangles

     Default value: `false'

     When `halfangles' is `true', trigonometric functions of arguments
     `<expr>/2' are simplified to functions of <expr>.

     For a real argument <x> in the interval `0 < x < 2*%pi' the sine of
     the half-angle simplifies to a simple formula:

                                   sqrt(1 - cos(x))
                                   ----------------
                                       sqrt(2)

     A complicated factor is needed to make this formula correct for
     all complex arguments <z>:

                realpart(z)
          floor(-----------)
                   2 %pi
     (- 1)                   (1 - unit_step(- imagpart(z))

                                 realpart(z)            realpart(z)
                           floor(-----------) - ceiling(-----------)
                                    2 %pi                  2 %pi
                     ((- 1)                                          + 1))

     Maxima knows this factor and similar factors for the functions
     `sin', `cos', `sinh', and `cosh'.  For special values of the
     argument z these factors simplify accordingly.

     Examples:

          (%i1) halfangles:false;
          (%o1)                                false
          (%i2) sin(x/2);
                                                  x
          (%o2)                               sin(-)
                                                  2
          (%i3) halfangles:true;
          (%o3)                                true
          (%i4) sin(x/2);
                                                              x
                                                      floor(-----)
                                                            2 %pi
                                sqrt(1 - cos(x)) (- 1)
          (%o4)                 ----------------------------------
                                             sqrt(2)
          (%i5) assume(x>0, x<2*%pi)$
          (%i6) sin(x/2);
                                         sqrt(1 - cos(x))
          (%o6)                          ----------------
                                             sqrt(2)

Package: ntrig

     The `ntrig' package contains a set of simplification rules that are
     used to simplify trigonometric function whose arguments are of the
     form `<f>(<n> %pi/10)' where <f> is any of the functions `sin',
     `cos', `tan', `csc', `sec' and `cot'.

Function: sec (<x>)

     - Secant.

Function: sech (<x>)

     - Hyperbolic Secant.

Function: sin (<x>)

     - Sine.

Function: sinh (<x>)

     - Hyperbolic Sine.

Function: tan (<x>)

     - Tangent.

Function: tanh (<x>)

     - Hyperbolic Tangent.

Function: trigexpand (<expr>)

     Expands trigonometric and hyperbolic functions of sums of angles
     and of multiple angles occurring in <expr>.  For best results,
     <expr> should be expanded.  To enhance user control of
     simplification, this function expands only one level at a time,
     expanding sums of angles or multiple angles.  To obtain full
     expansion into sines and cosines immediately, set the switch
     `trigexpand: true'.

     `trigexpand' is governed by the following global flags:

    `trigexpand'
          If `true' causes expansion of all expressions containing
          sin's and cos's occurring subsequently.

    `halfangles'
          If `true' causes half-angles to be simplified away.

    `trigexpandplus'
          Controls the "sum" rule for `trigexpand', expansion of sums
          (e.g. `sin(x + y)') will take place only if `trigexpandplus'
          is `true'.

    `trigexpandtimes'
          Controls the "product" rule for `trigexpand', expansion of
          products (e.g. `sin(2 x)') will take place only if
          `trigexpandtimes' is `true'.

     Examples:

          (%i1) x+sin(3*x)/sin(x),trigexpand=true,expand;
                                   2           2
          (%o1)               - sin (x) + 3 cos (x) + x
          (%i2) trigexpand(sin(10*x+y));
          (%o2)          cos(10 x) sin(y) + sin(10 x) cos(y)

Option variable: trigexpandplus

     Default value: `true'

     `trigexpandplus' controls the "sum" rule for `trigexpand'.  Thus,
     when the `trigexpand' command is used or the `trigexpand' switch
     set to `true', expansion of sums (e.g. `sin(x+y))' will take place
     only if `trigexpandplus' is `true'.

Option variable: trigexpandtimes

     Default value: `true'

     `trigexpandtimes' controls the "product" rule for `trigexpand'.
     Thus, when the `trigexpand' command is used or the `trigexpand'
     switch set to `true', expansion of products (e.g. `sin(2*x)') will
     take place only if `trigexpandtimes' is `true'.

Option variable: triginverses

     Default value: `true'

     `triginverses' controls the simplification of the composition of
     trigonometric and hyperbolic functions with their inverse
     functions.

     If `all', both e.g. `atan(tan(<x>))' and `tan(atan(<x>))' simplify
     to <x>.

     If `true', the `<arcfun>(<fun>(<x>))' simplification is turned off.

     If `false', both the `<arcfun>(<fun>(<x>))' and
     `<fun>(<arcfun>(<x>))' simplifications are turned off.

Function: trigreduce (<expr>, <x>)
Function: trigreduce (<expr>)

     Combines products and powers of trigonometric and hyperbolic sin's
     and cos's of <x> into those of multiples of <x>.  It also tries to
     eliminate these functions when they occur in denominators.  If <x>
     is omitted then all variables in <expr> are used.

     See also `poissimp'.

          (%i1) trigreduce(-sin(x)^2+3*cos(x)^2+x);
                         cos(2 x)      cos(2 x)   1        1
          (%o1)          -------- + 3 (-------- + -) + x - -
                            2             2       2        2

Option variable: trigsign

     Default value: `true'

     When `trigsign' is `true', it permits simplification of negative
     arguments to trigonometric functions.  E.g., `sin(-x)' will become
     `-sin(x)' only if `trigsign' is `true'.

Function: trigsimp (<expr>)

     Employs the identities sin(x)^2 + cos(x)^2 = 1 and cosh(x)^2 -
     sinh(x)^2 = 1 to simplify expressions containing `tan', `sec',
     etc., to `sin', `cos', `sinh', `cosh'.

     `trigreduce', `ratsimp', and `radcan' may be able to further
     simplify the result.

     `demo ("trgsmp.dem")' displays some examples of `trigsimp'.

Function: trigrat (<expr>)

     Gives a canonical simplified quasilinear form of a trigonometrical
     expression; <expr> is a rational fraction of several `sin', `cos'
     or `tan', the arguments of them are linear forms in some variables
     (or kernels) and `%pi/<n>' (<n> integer) with integer coefficients.
     The result is a simplified fraction with numerator and denominator
     linear in `sin' and `cos'.  Thus `trigrat' linearize always when
     it is possible.

          (%i1) trigrat(sin(3*a)/sin(a+%pi/3));
          (%o1)            sqrt(3) sin(2 a) + cos(2 a) - 1

     The following example is taken from Davenport, Siret, and
     Tournier, Calcul Formel, Masson (or in English, Addison-Wesley),
     section 1.5.5, Morley theorem.

          (%i1) c : %pi/3 - a - b$
          (%i2) bc : sin(a)*sin(3*c)/sin(a+b);
                                                    %pi
                            sin(a) sin(3 (- b - a + ---))
                                                     3
          (%o2)             -----------------------------
                                     sin(b + a)
          (%i3) ba : bc, c=a, a=c;
                                                   %pi
                              sin(3 a) sin(b + a - ---)
                                                    3
          (%o3)               -------------------------
                                            %pi
                                    sin(a - ---)
                                             3
          (%i4) ac2 : ba^2 + bc^2 - 2*bc*ba*cos(b);
                   2         2         %pi
                sin (3 a) sin (b + a - ---)
                                        3
          (%o4) ---------------------------
                          2     %pi
                       sin (a - ---)
                                 3
                                                 %pi
           - (2 sin(a) sin(3 a) sin(3 (- b - a + ---)) cos(b)
                                                  3
                       %pi            %pi
           sin(b + a - ---))/(sin(a - ---) sin(b + a))
                        3              3
                2       2              %pi
             sin (a) sin (3 (- b - a + ---))
                                        3
           + -------------------------------
                          2
                       sin (b + a)
          (%i5) trigrat (ac2);
          (%o5) - (sqrt(3) sin(4 b + 4 a) - cos(4 b + 4 a)
           - 2 sqrt(3) sin(4 b + 2 a) + 2 cos(4 b + 2 a)
           - 2 sqrt(3) sin(2 b + 4 a) + 2 cos(2 b + 4 a)
           + 4 sqrt(3) sin(2 b + 2 a) - 8 cos(2 b + 2 a) - 4 cos(2 b - 2 a)
           + sqrt(3) sin(4 b) - cos(4 b) - 2 sqrt(3) sin(2 b) + 10 cos(2 b)
           + sqrt(3) sin(4 a) - cos(4 a) - 2 sqrt(3) sin(2 a) + 10 cos(2 a)
           - 9)/4

10.6 Random Numbers

Function: make_random_state (<n>)
Function: make_random_state (<s>)
Function: make_random_state (true)
Function: make_random_state (false)

     A random state object represents the state of the random number
     generator.  The state comprises 627 32-bit words.

     `make_random_state (<n>)' returns a new random state object
     created from an integer seed value equal to <n> modulo 2^32.  <n>
     may be negative.

     `make_random_state (<s>)' returns a copy of the random state <s>.

     `make_random_state (true)' returns a new random state object,
     using the current computer clock time as the seed.

     `make_random_state (false)' returns a copy of the current state of
     the random number generator.

Function: set_random_state (<s>)

     Copies <s> to the random number generator state.

     `set_random_state' always returns `done'.

Function: random (<x>)

     Returns a pseudorandom number.  If <x> is an integer, `random
     (<x>)' returns an integer from 0 through `<x> - 1' inclusive.  If
     <x> is a floating point number, `random (<x>)' returns a
     nonnegative floating point number less than <x>.  `random'
     complains with an error if <x> is neither an integer nor a float,
     or if <x> is not positive.

     The functions `make_random_state' and `set_random_state' maintain
     the state of the random number generator.

     The Maxima random number generator is an implementation of the
     Mersenne twister MT 19937.

     Examples:

          (%i1) s1: make_random_state (654321)$
          (%i2) set_random_state (s1);
          (%o2)                         done
          (%i3) random (1000);
          (%o3)                          768
          (%i4) random (9573684);
          (%o4)                        7657880
          (%i5) random (2^75);
          (%o5)                11804491615036831636390
          (%i6) s2: make_random_state (false)$
          (%i7) random (1.0);
          (%o7)                   .2310127244107132
          (%i8) random (10.0);
          (%o8)                   4.394553645870825
          (%i9) random (100.0);
          (%o9)                   32.28666704056853
          (%i10) set_random_state (s2);
          (%o10)                        done
          (%i11) random (1.0);
          (%o11)                  .2310127244107132
          (%i12) random (10.0);
          (%o12)                  4.394553645870825
          (%i13) random (100.0);
          (%o13)                  32.28666704056853

11 Maximas Database


Introduction to Maximas Database
Functions and Variables for Properties
Functions and Variables for Facts
Functions and Variables for Predicates

11.1 Introduction to Maximas Database

11.2 Functions and Variables for Properties

Property: alphabetic

     `alphabetic' is a property type recognized by `declare'.

     The expression `declare(<s>, alphabetic)' tells Maxima to recognize
     as alphabetic all of the characters in <s>, which must be a string.

     See also *note Identifiers::.

     Example:

          (%i1) xx\~yy\`\@ : 1729;
          (%o1)                         1729
          (%i2) declare ("~`@", alphabetic);
          (%o2)                         done
          (%i3) xx~yy`@ + @yy`xx + `xx@@yy~;
          (%o3)               `xx@@yy~ + @yy`xx + 1729
          (%i4) listofvars (%);
          (%o4)                  [@yy`xx, `xx@@yy~]

Property: bindtest

     The command `declare(<x>, bindtest' tells Maxima to trigger an
     error when the symbol <x> is evaluated unbound.

          (%i1) aa + bb;
          (%o1)                        bb + aa
          (%i2) declare (aa, bindtest);
          (%o2)                         done
          (%i3) aa + bb;
          aa unbound variable
           -- an error.  Quitting.  To debug this try debugmode(true);
          (%i4) aa : 1234;
          (%o4)                         1234
          (%i5) aa + bb;
          (%o5)                       bb + 1234

Property: constant

     `declare(<a>, constant)' declares <a> to be a constant.  The
     declaration of a symbol to be constant does not prevent the
     assignment of a nonconstant value to the symbol.

     See `constantp'  and `declare'.

     Example:

          (%i1) declare(c, constant);
          (%o1)                         done
          (%i2) constantp(c);
          (%o2)                         true
          (%i3) c : x;
          (%o3)                           x
          (%i4) constantp(c);
          (%o4)                         false

Function: constantp (<expr>)

     Returns `true' if <expr> is a constant expression, otherwise
     returns `false'.

     An expression is considered a constant expression if its arguments
     are numbers (including rational numbers, as displayed with `/R/'),
     symbolic constants such as `%pi',  `%e',  and `%i',

     variables bound to a constant or declared constant by `declare',

     or functions whose arguments are constant.

     `constantp' evaluates its arguments.

     See the property `constant'  which declares a symbol to be
     constant.

     Examples:

          (%i1) constantp (7 * sin(2));
          (%o1)                                true
          (%i2) constantp (rat (17/29));
          (%o2)                                true
          (%i3) constantp (%pi * sin(%e));
          (%o3)                                true
          (%i4) constantp (exp (x));
          (%o4)                                false
          (%i5) declare (x, constant);
          (%o5)                                done
          (%i6) constantp (exp (x));
          (%o6)                                true
          (%i7) constantp (foo (x) + bar (%e) + baz (2));
          (%o7)                                false
          (%i8)

Function: declare (<a_1>, <p_1>, <a_2>, <p_2>, ...)

     Assigns the atom or list of atoms <a_i> the property or list of
     properties <p_i>.  When <a_i> and/or <p_i> are lists, each of the
     atoms gets all of the properties.

     `declare' quotes its arguments.  `declare' always returns `done'.

     As noted in the description for each declaration flag, for some
     flags `featurep(<object>, <feature>)' returns `true' if <object>
     has been declared to have <feature>.

     See also `features'.

     `declare' recognizes the following properties:

    `additive'
          Tells Maxima to simplify <a_i> expressions by the substitution
               `<a_i>(x + y + z + ...)' `-->'       `<a_i>(x) +
          <a_i>(y) + <a_i>(z) + ...'.        The substitution is
          carried out on the first argument only.

    `alphabetic'
          Tells Maxima to recognize all characters in <a_i> (which must
          be a       string) as alphabetic characters.

    `antisymmetric, commutative, symmetric'
          Tells Maxima to recognize <a_i> as a symmetric or
          antisymmetric       function.  `commutative'  is the same as
          `symmetric'.

    `bindtest'
          Tells Maxima to trigger an error when <a_i> is evaluated
          unbound.

    `constant'
          Tells Maxima to consider <a_i> a symbolic constant.

    `even, odd'
          Tells Maxima to recognize <a_i> as an even or odd integer
          variable.

    `evenfun, oddfun'
          Tells Maxima to recognize <a_i> as an odd or even function.

    `evflag'
          Makes <a_i> known to the `ev' function so that <a_i> is bound
               to `true' during the execution of `ev' when <a_i>
          appears as       a flag argument of `ev'.  See `evflag'.

    `evfun'
          Makes <a_i> known to `ev' so that the function named by <a_i>
               is applied when <a_i> appears as a flag argument of `ev'.
               See `evfun'.

    `feature'
          Tells Maxima to recognize <a_i> as the name of a feature.
            Other atoms may then be declared to have the <a_i> property.

    `increasing, decreasing'
          Tells Maxima to recognize <a_i> as an increasing or decreasing
               function.

    `integer, noninteger'
          Tells Maxima to recognize <a_i> as an integer or noninteger
          variable.

    `integervalued'
          Tells Maxima to recognize <a_i> as an integer-valued function.

    `lassociative, rassociative'
          Tells Maxima to recognize <a_i> as a right-associative or
           left-associative function.

    `linear'
          Equivalent to declaring <a_i> both `outative' and
          `additive'.

    `mainvar'
          Tells Maxima to consider <a_i> a "main variable".  A main
          variable       succeeds all other constants and variables in
          the canonical ordering of       Maxima expressions, as
          determined by `ordergreatp'.

    `multiplicative'
          Tells Maxima to simplify <a_i> expressions by the substitution
               `<a_i>(x * y * z * ...)' `-->'       `<a_i>(x) *
          <a_i>(y) * <a_i>(z) * ...'.        The substitution is
          carried out on the first argument only.

    `nary'
          Tells Maxima to recognize <a_i> as an n-ary function.

          The `nary' declaration is not the same as calling the `nary'
              function.  The sole effect of `declare(foo, nary)' is to
          instruct the       Maxima simplifier to flatten nested
          expressions, for example, to simplify       `foo(x, foo(y,
          z))' to `foo(x, y, z)'.

    `nonarray'
          Tells Maxima to consider <a_i> not an array.  This declaration
               prevents multiple evaluation of a subscripted variable
          name.

    `nonscalar'
          Tells Maxima to consider <a_i> a nonscalar variable.  The
          usual       application is to declare a variable as a
          symbolic vector or matrix.

    `noun'
          Tells Maxima to parse <a_i> as a noun.  The effect of this is
          to       replace instances of <a_i> with `'<a_i>' or
          `nounify(<a_i>)', depending on the context.

    `outative'
          Tells Maxima to simplify <a_i> expressions by pulling
          constant factors       out of the first argument.

          When <a_i> has one argument, a factor is considered constant
          if it is       a literal or declared constant.

          When <a_i> has two or more arguments, a factor is considered
          constant       if the second argument is a symbol and the
          factor is free of the second       argument.

    `posfun'
          Tells Maxima to recognize <a_i> as a positive function.

    `rational, irrational'
          Tells Maxima to recognize <a_i> as a rational or irrational
          real       variable.

    `real, imaginary, complex'
          Tells Maxima to recognize <a_i> as a real, pure imaginary, or
          complex       variable.

    `scalar'
          Tells Maxima to consider <a_i> a scalar variable.

Property: decreasing
Property: increasing

     The commands `declare(<f>, decreasing)' or `declare(<f>,
     increasing' tell Maxima to recognize the function <f> as an
     decreasing or increasing function.

     See also `declare'  for more properties.

     Example:

          (%i1) assume(a > b);
          (%o1)                        [a > b]
          (%i2) is(f(a) > f(b));
          (%o2)                        unknown
          (%i3) declare(f, increasing);
          (%o3)                         done
          (%i4) is(f(a) > f(b));
          (%o4)                         true

Property: even
Property: odd

     `declare(<a>, even)' or `declare(<a>, odd)' tells Maxima to
     recognize the symbol <a> as an even or odd integer variable.  The
     properties `even' and `odd' are not recognized by the functions
     `evenp',  `oddp',  and `integerp'.

     See also `declare'  and `askinteger'.

     Example:

          (%i1) declare(n, even);
          (%o1)                         done
          (%i2) askinteger(n, even);
          (%o2)                          yes
          (%i3) askinteger(n);
          (%o3)                          yes
          (%i4) evenp(n);
          (%o4)                         false

Property: feature

     Maxima understands two distinct types of features, system features
     and features which apply to mathematical expressions.  See also
     `status'  for information about system features.  See also
     `features'  and `featurep'  for information about mathematical
     features.

     `feature' itself is not the name of a function or variable.

Function: featurep (<a>, <f>)

     Attempts to determine whether the object <a> has the feature <f>
     on the basis of the facts in the current database.  If so, it
     returns `true', else `false'.

     Note that `featurep' returns `false' when neither <f> nor the
     negation of <f> can be established.

     `featurep' evaluates its argument.

     See also `declare'  and `features'.

          (%i1) declare (j, even)$
          (%i2) featurep (j, integer);
          (%o2)                           true

Declaration: features

     Maxima recognizes certain mathematical properties of functions and
     variables.  These are called "features".

     `declare (<x>, <foo>)' gives the property <foo> to the function or
     variable <x>.

     `declare (<foo>, feature)' declares a new feature <foo>.  For
     example, `declare ([red, green, blue], feature)' declares three
     new features, `red', `green', and `blue'.

     The predicate `featurep (<x>, <foo>)' returns `true' if <x> has
     the <foo> property, and `false' otherwise.

     The infolist `features' is a list of known features.  These are

        integer        noninteger      even
        odd            rational        irrational
        real           imaginary       complex
        analytic       increasing      decreasing
        oddfun         evenfun         posfun
        commutative    lassociative    rassociative
        symmetric      antisymmetric

     plus any user-defined features.

     `features' is a list of mathematical features.  There is also a
     list of non-mathematical, system-dependent features.  See `status'.

     Example:

          (%i1) declare (FOO, feature);
          (%o1)                         done
          (%i2) declare (x, FOO);
          (%o2)                         done
          (%i3) featurep (x, FOO);
          (%o3)                         true

Function: get (<a>, <i>)

     Retrieves the user property indicated by <i> associated with atom
     <a> or returns `false' if <a> doesn't have property <i>.

     `get' evaluates its arguments.

     See also `put'  and `qput'.

          (%i1) put (%e, 'transcendental, 'type);
          (%o1)                    transcendental
          (%i2) put (%pi, 'transcendental, 'type)$
          (%i3) put (%i, 'algebraic, 'type)$
          (%i4) typeof (expr) := block ([q],
                  if numberp (expr)
                  then return ('algebraic),
                  if not atom (expr)
                  then return (maplist ('typeof, expr)),
                  q: get (expr, 'type),
                  if q=false
                  then errcatch (error(expr,"is not numeric.")) else q)$
          (%i5) typeof (2*%e + x*%pi);
          x is not numeric.
          (%o5)  [[transcendental, []], [algebraic, transcendental]]
          (%i6) typeof (2*%e + %pi);
          (%o6)     [transcendental, [algebraic, transcendental]]

Property: integer
Property: noninteger

     `declare(<a>, integer)' or `declare(<a>, noninteger)' tells Maxima
     to recognize <a> as an integer or noninteger variable.

     See also `declare'.

     Example:

          (%i1) declare(n, integer, x, noninteger);
          (%o1)                         done
          (%i2) askinteger(n);
          (%o2)                          yes
          (%i3) askinteger(x);
          (%o3)                          no

Property: integervalued

     `declare(<f>, integervalued)' tells Maxima to recognize <f> as an
     integer-valued function.

     See also `declare'.

     Example:

          (%i1) exp(%i)^f(x);
                                        %i f(x)
          (%o1)                      (%e  )
          (%i2) declare(f, integervalued);
          (%o2)                         done
          (%i3) exp(%i)^f(x);
                                        %i f(x)
          (%o3)                       %e

Property: nonarray

     The command `declare(a, nonarray)' tells Maxima to consider <a> not
     an array.  This declaration prevents multiple evaluation, if <a>
     is a subscripted variable.

     See also `declare'.

     Example:

          (%i1) a:'b$ b:'c$ c:'d$

          (%i4) a[x];
          (%o4)                          d
                                          x
          (%i5) declare(a, nonarray);
          (%o5)                         done
          (%i6) a[x];
          (%o6)                          a
                                          x

Property: nonscalar

     Makes atoms behave as does a list or matrix with respect to the
     dot operator.

     See also `declare'.

Function: nonscalarp (<expr>)

     Returns `true' if <expr> is a non-scalar, i.e., it contains atoms
     declared as non-scalars, lists, or matrices.

     See also the predicate function `scalarp'  and `declare'.

Property: posfun

     `declare (f, posfun)' declares `f' to be a positive function.  `is
     (f(x) > 0)' yields `true'.

     See also `declare'.

Function: printprops (<a>, <i>)
Function: printprops ([<a_1>, ..., <a_n>], <i>)
Function: printprops (all, <i>)

     Displays the property with the indicator <i> associated with the
     atom <a>.  <a> may also be a list of atoms or the atom `all' in
     which case all of the atoms with the given property will be used.
     For example, `printprops ([f, g], atvalue)'.  `printprops' is for
     properties that cannot otherwise be displayed, i.e.  for `atvalue',

     `atomgrad',  `gradef',  and `matchdeclare'.

Function: properties (<a>)

     Returns a list of the names of all the properties associated with
     the atom <a>.

System variable: props

     Default value: `[]'

     `props' are atoms which have any property other than those
     explicitly mentioned in `infolists',  such as specified by
     `atvalue',

     `matchdeclare',  etc., as well as properties specified in the
     `declare'  function.

Function: propvars (<prop>)

     Returns a list of those atoms on the `props'  list which have the
     property indicated by <prop>.  Thus `propvars (atvalue)' returns a
     list of atoms which have atvalues.

Function: put (<atom>, <value>, <indicator>)

     Assigns <value> to the property (specified by <indicator>) of
     <atom>.  <indicator> may be the name of any property, not just a
     system-defined property.

     `rem'  reverses the effect of `put'.

     `put' evaluates its arguments.  `put' returns <value>.

     See also `qput'  and `get'.

     Examples:

          (%i1) put (foo, (a+b)^5, expr);
                                             5
          (%o1)                       (b + a)
          (%i2) put (foo, "Hello", str);
          (%o2)                         Hello
          (%i3) properties (foo);
          (%o3)            [[user properties, str, expr]]
          (%i4) get (foo, expr);
                                             5
          (%o4)                       (b + a)
          (%i5) get (foo, str);
          (%o5)                         Hello

Function: qput (<atom>, <value>, <indicator>)

     Assigns <value> to the property (specified by <indicator>) of
     <atom>.  This is the same as `put',  except that the arguments are
     quoted.

     See also `get'.

     Example:

          (%i1) foo: aa$
          (%i2) bar: bb$
          (%i3) baz: cc$
          (%i4) put (foo, bar, baz);
          (%o4)                          bb
          (%i5) properties (aa);
          (%o5)                [[user properties, cc]]
          (%i6) get (aa, cc);
          (%o6)                          bb
          (%i7) qput (foo, bar, baz);
          (%o7)                          bar
          (%i8) properties (foo);
          (%o8)            [value, [user properties, baz]]
          (%i9) get ('foo, 'baz);
          (%o9)                          bar

Property: rational
Property: irrational

     `declare(<a>, rational)' or `declare(<a>, irrational)' tells
     Maxima to recognize <a> as a rational or irrational real variable.

     See also `declare'.

Property: real
Property: imaginary
Property: complex

     `declare(<a>, real)', `declare(<a>, imaginary)', or `declare(<a>,
     complex)' tells Maxima to recognize <a> as a real, pure imaginary,
     or complex variable.

     See also `declare'.

Function: rem (<atom>, <indicator>)

     Removes the property indicated by <indicator> from <atom>.  `rem'
     reverses the effect of `put'.

     `rem' returns `done' if <atom> had an <indicator> property when
     `rem' was called, or `false' if it had no such property.

Function: remove (<a_1>, <p_1>, ..., <a_n>, <p_n>)
Function: remove ([<a_1>, ..., <a_m>], [<p_1>, ..., <p_n>], ...)
Function: remove ("<a>", operator)
Function: remove (<a>, transfun)
Function: remove (all, <p>)

     Removes properties associated with atoms.

     `remove (<a_1>, <p_1>, ..., <a_n>, <p_n>)' removes property `p_k'
     from atom `a_k'.

     `remove ([<a_1>, ..., <a_m>], [<p_1>, ..., <p_n>], ...)' removes
     properties `<p_1>, ..., <p_n>' from atoms <a_1>, ..., <a_m>.
     There may be more than one pair of lists.

     `remove (all, <p>)' removes the property <p> from all atoms which
     have it.

     The removed properties may be system-defined properties such as
     `function', `macro', or `mode_declare'.  `remove' does not remove
     properties defined by `put'.

     A property may be `transfun' to remove the translated Lisp version
     of a function.  After executing this, the Maxima version of the
     function is executed rather than the translated version.

     `remove ("<a>", operator)' or, equivalently, `remove ("<a>", op)'
     removes from <a> the operator properties declared by `prefix',
     `infix',

     `nary',  `postfix',  `matchfix',  or `nofix'.    Note that the
     name of the operator must be written as a quoted string.

     `remove' always returns `done' whether or not an atom has a
     specified property.  This behavior is unlike the more specific
     remove functions `remvalue',  `remarray',  `remfunction',  and
     `remrule'.

     `remove' quotes its arguments.

Property: scalar

     `declare(<a>, scalar)' tells Maxima to consider <a> a scalar
     variable.

     See also `declare'.

Function: scalarp (<expr>)

     Returns `true' if <expr> is a number, constant, or variable
     declared `scalar'  with `declare',  or composed entirely of
     numbers, constants, and such variables, but not containing
     matrices or lists.

     See also the predicate function `nonscalarp'.

11.3 Functions and Variables for Facts

Function: activate (<context_1>, ..., <context_n>)

     Activates the contexts <context_1>, ..., <context_n>.  The facts
     in these contexts are then available to make deductions and
     retrieve information.  The facts in these contexts are not listed
     by `facts ()'.

     The variable `activecontexts'  is the list of contexts which are
     active by way of the `activate' function.

System variable: activecontexts

     Default value: `[]'

     `activecontexts' is a list of the contexts which are active by way
     of the `activate'  function, as opposed to being active because
     they are subcontexts of the current context.

Function: askinteger (<expr>, integer)
Function: askinteger (<expr>)
Function: askinteger (<expr>, even)
Function: askinteger (<expr>, odd)

     `askinteger (<expr>, integer)' attempts to determine from the
     `assume' database whether <expr> is an integer.  `askinteger'
     prompts the user if it cannot tell otherwise, and attempt to
     install the information in the database if possible.  `askinteger
     (<expr>)' is equivalent to `askinteger (<expr>, integer)'.

     `askinteger (<expr>, even)' and `askinteger (<expr>, odd)'
     likewise attempt to determine if <expr> is an even integer or odd
     integer, respectively.

Function: asksign (<expr>)

     First attempts to determine whether the specified expression is
     positive, negative, or zero.  If it cannot, it asks the user the
     necessary questions to complete its deduction.  The user's answer
     is recorded in the data base for the duration of the current
     computation.  The return value of `asksign' is one of `pos',
     `neg', or `zero'.

Function: assume (<pred_1>, ..., <pred_n>)

     Adds predicates <pred_1>, ..., <pred_n> to the current context.
     If a predicate is inconsistent or redundant with the predicates in
     the current context, it is not added to the context.  The context
     accumulates predicates from each call to `assume'.

     `assume' returns a list whose elements are the predicates added to
     the context or the atoms `redundant' or `inconsistent' where
     applicable.

     The predicates <pred_1>, ..., <pred_n> can only be expressions
     with the relational operators `< <= equal notequal >=' and `>'.
     Predicates cannot be literal equality `=' or literal inequality `#'
     expressions, nor can they be predicate functions such as
     `integerp'.

     Compound predicates of the form `<pred_1> and ... and <pred_n>'
     are recognized, but not `<pred_1> or ... or <pred_n>'.  `not
     <pred_k>' is recognized if <pred_k> is a relational predicate.
     Expressions of the form `not (<pred_1> and <pred_2>)' and `not
     (<pred_1> or <pred_2>)' are not recognized.

     Maxima's deduction mechanism is not very strong; there are many
     obvious consequences which cannot be determined by `is'.

     This is a known weakness.

     `assume' does not handle predicates with complex numbers.  If a
     predicate contains a complex number `assume' returns
     `inconsistent' or `redunant'.

     `assume' evaluates its arguments.

     See also `is',  `facts',  `forget',

     `context',  and `declare'.

     Examples:

          (%i1) assume (xx > 0, yy < -1, zz >= 0);
          (%o1)              [xx > 0, yy < - 1, zz >= 0]
          (%i2) assume (aa < bb and bb < cc);
          (%o2)                  [bb > aa, cc > bb]
          (%i3) facts ();
          (%o3)     [xx > 0, - 1 > yy, zz >= 0, bb > aa, cc > bb]
          (%i4) is (xx > yy);
          (%o4)                         true
          (%i5) is (yy < -yy);
          (%o5)                         true
          (%i6) is (sinh (bb - aa) > 0);
          (%o6)                         true
          (%i7) forget (bb > aa);
          (%o7)                       [bb > aa]
          (%i8) prederror : false;
          (%o8)                         false
          (%i9) is (sinh (bb - aa) > 0);
          (%o9)                        unknown
          (%i10) is (bb^2 < cc^2);
          (%o10)                       unknown

Option variable: assumescalar

     Default value: `true'

     `assumescalar' helps govern whether expressions `expr' for which
     `nonscalarp (expr)' is `false' are assumed to behave like scalars
     for certain transformations.

     Let `expr' represent any expression other than a list or a matrix,
     and let `[1, 2, 3]' represent any list or matrix.  Then `expr .
     [1, 2, 3]' yields `[expr, 2 expr, 3 expr]' if `assumescalar' is
     `true', or `scalarp (expr)' is `true', or `constantp (expr)' is
     `true'.

     If `assumescalar' is `true', such expressions will behave like
     scalars only for commutative operators, but not for noncommutative
     multiplication `.'.

     When `assumescalar' is `false', such expressions will behave like
     non-scalars.

     When `assumescalar' is `all', such expressions will behave like
     scalars for all the operators listed above.

Option variable: assume_pos

     Default value: `false'

     When `assume_pos' is `true' and the sign of a parameter <x> cannot
     be determined from the current context or other considerations,
     `sign' and `asksign (<x>)' return `true'.  This may forestall some
     automatically-generated `asksign' queries, such as may arise from
     `integrate' or other computations.

     By default, a parameter is <x> such that `symbolp (<x>)' or
     `subvarp (<x>)'.  The class of expressions considered parameters
     can be modified to some extent via the variable `assume_pos_pred'.

     `sign' and `asksign' attempt to deduce the sign of expressions
     from the sign of operands within the expression.  For example, if
     `a' and `b' are both positive, then `a + b' is also positive.

     However, there is no way to bypass all `asksign' queries.  In
     particular, when the `asksign' argument is a difference `<x> -
     <y>' or a logarithm `log(<x>)', `asksign' always requests an input
     from the user, even when `assume_pos' is `true' and
     `assume_pos_pred' is a function which returns `true' for all
     arguments.

Option variable: assume_pos_pred

     Default value: `false'

     When `assume_pos_pred' is assigned the name of a function or a
     lambda expression of one argument <x>, that function is called to
     determine whether <x> is considered a parameter for the purpose of
     `assume_pos'.  `assume_pos_pred' is ignored when `assume_pos' is
     `false'.

     The `assume_pos_pred' function is called by `sign' and `asksign'
     with an argument <x> which is either an atom, a subscripted
     variable, or a function call expression.  If the `assume_pos_pred'
     function returns `true', <x> is considered a parameter for the
     purpose of `assume_pos'.

     By default, a parameter is <x> such that `symbolp (<x>)' or
     `subvarp (<x>)'.

     See also `assume'  and `assume_pos'.

     Examples:

          (%i1) assume_pos: true$
          (%i2) assume_pos_pred: symbolp$
          (%i3) sign (a);
          (%o3)                          pos
          (%i4) sign (a[1]);
          (%o4)                          pnz
          (%i5) assume_pos_pred: lambda ([x], display (x), true)$
          (%i6) asksign (a);
                                        x = a

          (%o6)                          pos
          (%i7) asksign (a[1]);
                                       x = a
                                            1

          (%o7)                          pos
          (%i8) asksign (foo (a));
                                     x = foo(a)

          (%o8)                          pos
          (%i9) asksign (foo (a) + bar (b));
                                     x = foo(a)

                                     x = bar(b)

          (%o9)                          pos
          (%i10) asksign (log (a));
                                        x = a

          Is  a - 1  positive, negative, or zero?

          p;
          (%o10)                         pos
          (%i11) asksign (a - b);
                                        x = a

                                        x = b

                                        x = a

                                        x = b

          Is  b - a  positive, negative, or zero?

          p;
          (%o11)                         neg

Option variable: context

     Default value: `initial'

     `context' names the collection of facts maintained by `assume'  and
     `forget'.    `assume' adds facts to the collection named by
     `context', while `forget' removes facts.

     Binding `context' to a name <foo> changes the current context to
     <foo>.  If the specified context <foo> does not yet exist, it is
     created automatically by a call to `newcontext'.

     The specified context is activated automatically.

     See `contexts'  for a general description of the context mechanism.

Option variable: contexts

     Default value: `[initial, global]'

     `contexts' is a list of the contexts which currently exist,
     including the currently active context.

     The context mechanism makes it possible for a user to bind together
     and name a collection of facts, called a context.  Once this is
     done, the user can have Maxima assume or forget large numbers of
     facts merely by activating or deactivating their context.

     Any symbolic atom can be a context, and the facts contained in that
     context will be retained in storage until destroyed one by one by
     calling `forget'  or destroyed as a whole by calling `kill'

     to destroy the context to which they belong.

     Contexts exist in a hierarchy, with the root always being the
     context `global', which contains information about Maxima that some
     functions need.  When in a given context, all the facts in that
     context are "active" (meaning that they are used in deductions and
     retrievals) as are all the facts in any context which is a
     subcontext of the active context.

     When a fresh Maxima is started up, the user is in a context called
     `initial', which has `global' as a subcontext.

     See also `facts',  `newcontext',  `supcontext',

     `killcontext',  `activate',  `deactivate',

     `assume',  and `forget'.

Function: deactivate (<context_1>, ..., <context_n>)

     Deactivates the specified contexts <context_1>, ..., <context_n>.

Function: facts (<item>)
Function: facts ()

     If <item> is the name of a context, `facts (<item>)' returns a
     list of the facts in the specified context.

     If <item> is not the name of a context, `facts (<item>)' returns a
     list of the facts known about <item> in the current context.
     Facts that are active, but in a different context, are not listed.

     `facts ()' (i.e., without an argument) lists the current context.

Function: forget (<pred_1>, ..., <pred_n>)
Function: forget (<L>)

     Removes predicates established by `assume'.

     The predicates may be expressions equivalent to (but not
     necessarily identical to) those previously assumed.

     `forget (<L>)', where <L> is a list of predicates, forgets each
     item on the list.

Function: is (<expr>)

     Attempts to determine whether the predicate <expr> is provable
     from the facts in the `assume' database.

     If the predicate is provably `true' or `false', `is' returns
     `true' or `false', respectively.  Otherwise, the return value is
     governed by the global flag `prederror'.    When `prederror' is
     `true', `is' complains with an error message.  Otherwise, `is'
     returns `unknown'.

     `ev(<expr>, pred)' (which can be written  `<expr>, pred' at the
     interactive prompt) is equivalent to `is(<expr>)'.

     See also `assume',  `facts',  and `maybe'.

     Examples:

     `is' causes evaluation of predicates.

          (%i1) %pi > %e;
          (%o1)                       %pi > %e
          (%i2) is (%pi > %e);
          (%o2)                         true

     `is' attempts to derive predicates from the `assume' database.

          (%i1) assume (a > b);
          (%o1)                        [a > b]
          (%i2) assume (b > c);
          (%o2)                        [b > c]
          (%i3) is (a < b);
          (%o3)                         false
          (%i4) is (a > c);
          (%o4)                         true
          (%i5) is (equal (a, c));
          (%o5)                         false

     If `is' can neither prove nor disprove a predicate from the
     `assume' database, the global flag `prederror' governs the
     behavior of `is'.

          (%i1) assume (a > b);
          (%o1)                        [a > b]
          (%i2) prederror: true$
          (%i3) is (a > 0);
          Maxima was unable to evaluate the predicate:
          a > 0
           -- an error.  Quitting.  To debug this try debugmode(true);
          (%i4) prederror: false$
          (%i5) is (a > 0);
          (%o5)                        unknown

Function: killcontext (<context_1>, ..., <context_n>)

     Kills the contexts <context_1>, ..., <context_n>.

     If one of the contexts is the current context, the new current
     context will become the first available subcontext of the current
     context which has not been killed.  If the first available
     unkilled context is `global' then `initial' is used instead.  If
     the `initial' context is killed, a new, empty `initial' context is
     created.

     `killcontext' refuses to kill a context which is currently active,
     either because it is a subcontext of the current context, or by
     use of the function `activate'.

     `killcontext' evaluates its arguments.  `killcontext' returns
     `done'.

Function: maybe (<expr>)

     Attempts to determine whether the predicate <expr> is provable
     from the facts in the `assume' database.

     If the predicate is provably `true' or `false', `maybe' returns
     `true' or `false', respectively.  Otherwise, `maybe' returns
     `unknown'.

     `maybe' is functionally equivalent to `is' with `prederror:
     false', but the result is computed without actually assigning a
     value to `prederror'.

     See also `assume',  `facts',  and `is'.

     Examples:

          (%i1) maybe (x > 0);
          (%o1)                        unknown
          (%i2) assume (x > 1);
          (%o2)                        [x > 1]
          (%i3) maybe (x > 0);
          (%o3)                         true

Function: newcontext (<name>)

     Creates a new, empty context, called <name>, which has `global' as
     its only subcontext.  The newly-created context becomes the
     currently active context.

     `newcontext' evaluates its argument.  `newcontext' returns <name>.

Function: sign (<expr>)

     Attempts to determine the sign of <expr> on the basis of the facts
     in the current data base.  It returns one of the following
     answers: `pos' (positive), `neg' (negative), `zero', `pz'
     (positive or zero), `nz' (negative or zero), `pn' (positive or
     negative), or `pnz' (positive, negative, or zero, i.e. nothing
     known).

Function: supcontext (<name>, <context>)
Function: supcontext (<name>)

     Creates a new context, called <name>, which has <context> as a
     subcontext.  <context> must exist.

     If <context> is not specified, the current context is assumed.

11.4 Functions and Variables for Predicates

Function: charfun (<p>)

     Return 0 when the predicate <p> evaluates to `false'; return 1 when
     the predicate evaluates to `true'.  When the predicate evaluates to
     something other than `true' or `false' (unknown),  return a noun
     form.

     Examples:

          (%i1) charfun (x < 1);
          (%o1)                    charfun(x < 1)
          (%i2) subst (x = -1, %);
          (%o2)                           1
          (%i3) e : charfun ('"and" (-1 < x, x < 1))$
          (%i4) [subst (x = -1, e), subst (x = 0, e), subst (x = 1, e)];
          (%o4)                       [0, 1, 0]

Function: compare (<x>, <y>)

     Return a comparison operator <op> (`<', `<=', `>', `>=', `=', or
     `#') such that `is (<x> <op> <y>)' evaluates to `true'; when
     either <x> or <y> depends on `%i' and `<x> # <y>', return
     `notcomparable'; when there is no such operator or Maxima isn't
     able to determine the operator, return `unknown'.

     Examples:

          (%i1) compare (1, 2);
          (%o1)                           <
          (%i2) compare (1, x);
          (%o2)                        unknown
          (%i3) compare (%i, %i);
          (%o3)                           =
          (%i4) compare (%i, %i + 1);
          (%o4)                     notcomparable
          (%i5) compare (1/x, 0);
          (%o5)                           #
          (%i6) compare (x, abs(x));
          (%o6)                          <=

     The function `compare' doesn't try to determine whether the real
     domains of its arguments are nonempty; thus

          (%i1) compare (acos (x^2 + 1), acos (x^2 + 1) + 1);
          (%o1)                           <

     The real domain of `acos (x^2 + 1)' is empty.

Function: equal (<a>, <b>)

     Represents equivalence, that is, equal value.

     By itself, `equal' does not evaluate or simplify.  The function
     `is'  attempts to evaluate `equal' to a Boolean value.
     `is(equal(<a>, <b>))' returns `true' (or `false') if and only if
     <a> and <b> are equal (or not equal) for all possible values of
     their variables, as determined by evaluating `ratsimp(<a> - <b>)';
     if `ratsimp'  returns 0, the two expressions are considered
     equivalent.  Two expressions may be equivalent even if they are
     not syntactically equal (i.e., identical).

     When `is' fails to reduce `equal' to `true' or `false', the result
     is governed by the global flag `prederror'.    When `prederror' is
     `true', `is' complains with an error message.  Otherwise, `is'
     returns `unknown'.

     In addition to `is', some other operators evaluate `equal' and
     `notequal' to `true' or `false', namely `if',

     `and',  `or',  and `not'.

     The negation of `equal' is `notequal'.

     Examples:

     By itself, `equal' does not evaluate or simplify.

          (%i1) equal (x^2 - 1, (x + 1) * (x - 1));
                                  2
          (%o1)            equal(x  - 1, (x - 1) (x + 1))
          (%i2) equal (x, x + 1);
          (%o2)                    equal(x, x + 1)
          (%i3) equal (x, y);
          (%o3)                      equal(x, y)

     The function `is' attempts to evaluate `equal' to a Boolean value.
     `is(equal(<a>, <b>))' returns `true' when `ratsimp(<a> - <b>)'
     returns 0.  Two expressions may be equivalent even if they are not
     syntactically equal (i.e., identical).

          (%i1) ratsimp (x^2 - 1 - (x + 1) * (x - 1));
          (%o1)                           0
          (%i2) is (equal (x^2 - 1, (x + 1) * (x - 1)));
          (%o2)                         true
          (%i3) is (x^2 - 1 = (x + 1) * (x - 1));
          (%o3)                         false
          (%i4) ratsimp (x - (x + 1));
          (%o4)                          - 1
          (%i5) is (equal (x, x + 1));
          (%o5)                         false
          (%i6) is (x = x + 1);
          (%o6)                         false
          (%i7) ratsimp (x - y);
          (%o7)                         x - y
          (%i8) is (equal (x, y));
          (%o8)                        unknown
          (%i9) is (x = y);
          (%o9)                         false

     When `is' fails to reduce `equal' to `true' or `false', the result
     is governed by the global flag `prederror'.

          (%i1) [aa : x^2 + 2*x + 1, bb : x^2 - 2*x - 1];
                              2             2
          (%o1)             [x  + 2 x + 1, x  - 2 x - 1]
          (%i2) ratsimp (aa - bb);
          (%o2)                        4 x + 2
          (%i3) prederror : true;
          (%o3)                         true
          (%i4) is (equal (aa, bb));
          Maxima was unable to evaluate the predicate:
                 2             2
          equal(x  + 2 x + 1, x  - 2 x - 1)
           -- an error.  Quitting.  To debug this try debugmode(true);
          (%i5) prederror : false;
          (%o5)                         false
          (%i6) is (equal (aa, bb));
          (%o6)                        unknown

     Some operators evaluate `equal' and `notequal' to `true' or
     `false'.

          (%i1) if equal (y, y - 1) then FOO else BAR;
          (%o1)                          BAR
          (%i2) eq_1 : equal (x, x + 1);
          (%o2)                    equal(x, x + 1)
          (%i3) eq_2 : equal (y^2 + 2*y + 1, (y + 1)^2);
                                   2                   2
          (%o3)             equal(y  + 2 y + 1, (y + 1) )
          (%i4) [eq_1 and eq_2, eq_1 or eq_2, not eq_1];
          (%o4)                  [false, true, true]

     Because `not <expr>' causes evaluation of <expr>, `not equal(<a>,
     <b>)' is equivalent to `is(notequal(<a>, <b>))'.

          (%i1) [notequal (2*z, 2*z - 1), not equal (2*z, 2*z - 1)];
          (%o1)            [notequal(2 z, 2 z - 1), true]
          (%i2) is (notequal (2*z, 2*z - 1));
          (%o2)                         true

Function: notequal (<a>, <b>)

     Represents the negation of `equal(<a>, <b>)'.

     Examples:

          (%i1) equal (a, b);
          (%o1)                      equal(a, b)
          (%i2) maybe (equal (a, b));
          (%o2)                        unknown
          (%i3) notequal (a, b);
          (%o3)                    notequal(a, b)
          (%i4) not equal (a, b);
          (%o4)                    notequal(a, b)
          (%i5) maybe (notequal (a, b));
          (%o5)                        unknown
          (%i6) assume (a > b);
          (%o6)                        [a > b]
          (%i7) equal (a, b);
          (%o7)                      equal(a, b)
          (%i8) maybe (equal (a, b));
          (%o8)                         false
          (%i9) notequal (a, b);
          (%o9)                    notequal(a, b)
          (%i10) maybe (notequal (a, b));
          (%o10)                        true

Function: unknown (<expr>)

     Returns `true' if and only if <expr> contains an operator or
     function not recognized by the Maxima simplifier.

Function: zeroequiv (<expr>, <v>)

     Tests whether the expression <expr> in the variable <v> is
     equivalent to zero, returning `true', `false', or `dontknow'.

     `zeroequiv' has these restrictions:

       1. Do not use functions that Maxima does not know how to
          differentiate and evaluate.

       2. If the expression has poles on the real line, there may be
          errors in the result (but this is unlikely to occur).

       3. If the expression contains functions which are not solutions
          to first order differential equations (e.g. Bessel functions)
          there may be incorrect results.

       4. The algorithm uses evaluation at randomly chosen points for
          carefully selected subexpressions.  This is always a somewhat
          hazardous business, although the algorithm tries to minimize
          the potential for error.

     For example `zeroequiv (sin(2 * x) - 2 * sin(x) * cos(x), x)'
     returns `true' and `zeroequiv (%e^x + x, x)' returns `false'.  On
     the other hand `zeroequiv (log(a * b) - log(a) - log(b), a)'
     returns `dontknow' because of the presence of an extra parameter
     `b'.

12 Plotting


Introduction to Plotting
Plotting Formats
Functions and Variables for Plotting
Plotting Options
Gnuplot Options
Gnuplot_pipes Format Functions

12.1 Introduction to Plotting

Maxima uses an external plotting package to make the plots (see the
section on Plotting formats).  The plotting functions calculate a set of
points and pass them to the plotting package together with a set of
commands.  That information can be passed to the external program either
through a pipe or by calling the program with the name of a file where
the data has been saved.  The data file is given the name
`maxout.interface', where `interface' is the name of the plotting
interface being used (gnuplot, xmaxima, mgnuplot or gnuplot_pipes).

   The `maxout.interface' file, in the cases when it is used, is
created in the directory specified by the system variable
`maxima_tempdir'.    That location can be changed; by assigning to that
variable a string that represents a valid directory where Maxima can
create new files.

   After a plot has been created, the file `maxout.interface' can be
executed again with the appropriate external program.  If a Maxima
plotting command fails to show anything, that file can be inspected to
look for possible sources of problems.

   Together with the plotting functions described in this chapter,
package *note draw:: adds other functionalities.  Note that some
plotting options are named equal in both plotting contexts, but with
different syntax; if you want to access the draw information related to
these options, you have to type `?? opt', where `opt' is the name of
the option.

12.2 Plotting Formats

There are currently two external plotting programs that Maxima use:
Gnuplot and Xmaxima.  There are various different formats for those
programs, which can be selected with the option `plot_format'  (see the
Plotting Options section).

   The plotting formats are the following:

   * *gnuplot* (default on Windows)

     Used to launch the external program gnuplot, which must be
     installed in your system.  All plotting commands and data are
     saved into the file `maxout.gnuplot'.

   * *gnuplot_pipes* (default on non-Windows platforms)

     This format is not available in Windows platforms.  It is similar
     to the `gnuplot' format except that the commands are sent to
     gnuplot through a pipe, while the data are saved into the file
     `maxout.gnuplot_pipes'.  A single gnuplot process is kept open and
     subsequent plot commands will be sent to the same process,
     replacing previous plots, unless the gnuplot pipe is closed with
     the function `gnuplot_close'.    When this format is used, the
     function `gnuplot_replot'  can be used to modify a plot that has
     already displayed on the screen.

     This format should only be used to plot to the screen; for
     plotting to files it is better to use the `gnuplot' format.

   * *mgnuplot*

     Mgnuplot is a Tk-based wrapper around gnuplot.  It is included in
     the Maxima distribution.  Mgnuplot offers a rudimentary GUI for
     gnuplot, but has fewer overall features than the plain gnuplot
     interface.  Mgnuplot requires an external gnuplot installation
     and, in Unix systems, the Tcl/Tk system.

   * *xmaxima*

     Xmaxima is a Tcl/Tk graphical interface for Maxima that can also
     be used to display plots created when Maxima is run from the
     console or from other graphical interfaces.  To use this format,
     the xmaxima program, which is distributed together with Maxima,
     should be installed.  If Maxima is being run from xmaxima itself,
     this format will make the plot functions send the data and
     commands through the same socket used for the communication
     between Maxima and Xmaxima.  When used from the console or from
     other interface, the commands and data will be saved in the file
     `maxout.xmaxima', and the xmaxima program will be launched with the
     name of the location of that file as argument.

     In previous versions this format used to be called `openmath'; that
     old name will still be accepted as a synonym for `xmaxima'.

12.3 Functions and Variables for Plotting

Function: contour_plot (<expr>, <x_range>, <y_range>, <options>, ...)

     It plots the contours (curves of equal value) of <expr> over the
     region <x_range> by <y_range>.  Any additional arguments are
     treated the same as in `plot3d'.

     This function only works when the plot format is either `gnuplot'
     or `gnuplot_pipes'.  The additional package `implicit_plot' can
     also be used to plot contours and it works for any format.  See
     `implicit_plot'.

     Examples:

          (%i1) contour_plot (x^2 + y^2, [x, -4, 4], [y, -4, 4])$

          (%i1) F(x, y) := x^3 + y^2;
                                             3    2
          (%o1)                  F(x, y) := x  + y
          (%i2) contour_plot (F, [u, -4, 4], [v, -4, 4])$

     You can add any options accepted by `plot3d'; for instance, the
     option `legend'  with a value of false, to remove the legend.
     Gnuplot chooses, by default, 3 contours to show.  To increase the
     number of levels, it is necessary to specify a custom gnuplot
     preamble:

          (%i1) contour_plot (u^3 + v^2, [u, -4, 4], [v, -4, 4],
                              [legend,false],
                              [gnuplot_preamble, "set cntrparam levels 12"])$

Function: get_plot_option (<keyword>, <index>)

     Returns a value of the option with name <keyword>, stored in the
     global variable `plot_options'.  A value of 1 for the index will
     return the keyword itself; a value of 2 turn returns the first
     parameter following the keyword, and so on.

     See also `plot_options',  `set_plot_option'  and the section on
     Plotting Options.

Function: implicit_plot (<expr>, <x_range>, <y_range>)
Function: implicit_plot ([<expr_1>, ..., <expr_n>], <x_range>, <y_range>)

     Displays a plot of one or more expressions in implicit form.
     <expr> is the expression to be plotted, <x_range> the range of the
     horizontal axis and <y_range> the range of vertical axis.
     `implicit_plot' respects global setting for the Gnuplot driver set
     by the <set_plot_option> function.  Options can also be passed to
     `implicit_plot' function as optional arguments.

     `implicit_plot' works by tracking sign changes on the area given
     by <x_range> and <y_range> and can fail for complicated
     expressions.

     `load(implicit_plot)' loads this function.

     Example:
          (%i1) implicit_plot (x^2 = y^3 - 3*y + 1, [x, -4, 4], [y, -4, 4],
           [gnuplot_preamble, "set zeroaxis"]);

Function: make_transform ([<var1>, <var2>, <var3>], <fx>, <fy>, <fz>)

     Returns a function suitable to be used in the option `transform_xy'

     of plot3d.  The three variables <var1>, <var2>, <var3> are three
     dummy variable names, which represent the 3 variables given by the
     plot3d command (first the two independent variables and then the
     function that depends on those two variables).  The three functions
     <fx>, <fy>, <fz> must depend only on those 3 variables, and will
     give the corresponding x, y and z coordinates that should be
     plotted.  There are two transformations defined by default:
     `polar_to_xy'  and `spherical_to_xyz'.   See the documentation for
     those two transformations.

System function: polar_to_xy

     It can be given as value for the `transform_xy'  option of plot3d.
     Its effect will be to interpret the two independent variables in
     plot3d as the distance from the z axis and the azimuthal angle
     (polar coordinates), and transform them into x and y coordinates.

Function: plot2d (<plot>, <x_range>, ..., <options>, ...)
Function: plot2d ([<plot_1>, ..., <plot_n>], ..., <options>, ...)
Function: plot2d ([<plot_1>, ..., <plot_n>], <x_range>, ..., <options>, ...)

     Where <plot>, <plot_1>, ..., <plot_n> can be either expressions,
     function names or a list with the any of the forms: `[discrete,
     [<x1>, ..., <xn>], [<y1>, ..., <yn>]]', `[discrete, [[<x1>, <y1>],
     ..., [<xn>, ..., <yn>]]' or `[parametric, <x_expr>, <y_expr>,
     <t_range>]'.

     Displays a plot of one or more expressions as a function of one
     variable or parameter.

     `plot2d' displays one or several plots in two dimensions.  When
     expressions or function name are used to define the plots, they
     should all depend on only one variable <var> and the use of
     <x_range> will be mandatory, to provide the name of the variable
     and its minimum and maximum values; the syntax for <x_range> is:
     `[<variable>, <min>, <max>]'.

     A plot can also be defined in the discrete or parametric forms.
     The discrete form is used to plot a set of points with given
     coordinates.  A discrete plot is defined by a list starting with
     the keyword <discrete>, followed by one or two lists of values.
     If two lists are given, they must have the same length; the first
     list will be interpreted as the x coordinates of the points to be
     plotted and the second list as the y coordinates.  If only one
     list is given after the <discrete> keyword, each element on the
     list should also be a list with two values that correspond to the
     x and y coordinates of a point.

     A parametric plot is defined by a list starting with the keyword
     <parametric>, followed by two expressions or function names and a
     range for the parameter.  The range for the parameter must be a
     list with the name of the parameter followed by its minimum and
     maximum values: `[<param>, <min>, <max>]'.  The plot will show the
     path traced out by the point with coordinates given by the two
     expressions or functions, as <param> increases from <min> to <max>.

     A range for the vertical axis is an optional argument with the
     form: `[y, <min>, <max>]' (the keyword <y> is always used for the
     vertical axis).  If that option is used, the plot will show that
     exact vertical range, independently of the values reached by the
     plot.  If the vertical range is not specified, it will be set up
     according to the minimum and maximum values of the second
     coordinate of the plot points.

     All other options should also be lists, starting with a keyword and
     followed by one or more values.  See `plot_options'.

     If there are several plots to be plotted, a legend will be written
     to identity each of the expressions.  The labels that should be
     used in that legend can be given with the option `legend'.    If
     that option is not used, Maxima will create labels from the
     expressions or function names.

     *Examples:*

     Plot of a common function:

          (%i1) plot2d (sin(x), [x, -%pi, %pi])$

     If the functions grows too fast, it might be necessary to limit the
     values in the vertical axis using the y option:

          (%i1) plot2d (sec(x), [x, -2, 2], [y, -20, 20])$
          plot2d: some values were clipped.

     The aspect of the plot might be different depending on the plotting
     program used.  For instance, when the plot box is disable, Xmaxima
     will plot the axes using arrows:

          (%i1) plot2d ( x^2-1, [x, -3, 3], [y, -2, 10],
                                [box, false], [plot_format, xmaxima])$

     A plot with a logarithmic scale:

          (%i1) plot2d (exp(3*s), [s, -2, 2], [logy])$

     Plotting functions by name:

          (%i1) F(x) := x^2 $
          (%i2) :lisp (defun |$g| (x) (m* x x x))
          $g
          (%i2) H(x) := if x < 0 then x^4 - 1 else 1 - x^5 $
          (%i3) plot2d ([F, G, H], [u, -1, 1], [y, -1.5, 1.5])$

     A plot of the butterfly curve, defined parametrically:

          (%i1) r: (exp(cos(t))-2*cos(4*t)-sin(t/12)^5)$
          (%i2) plot2d([parametric, r*sin(t), r*cos(t),
                       [t, -8*%pi, 8*%pi], [nticks, 2000]])$

     A "circle" with two turns, when plotted with only 7 points:

          (%i1) plot2d ([parametric, cos(t), sin(t),
                        [t, -2*%pi, 2*%pi], [nticks, 8]])$

     Plot of a common function together with the parametric
     representation of a circle.  The size of the plot has been
     adjusted with the x and y options, to make the circle look round
     and not deformed as an ellipse.  These values work well for the
     Postscript terminal used to produce this plot; you might have to
     adjust the values for your screen.

          (%i1) plot2d([[parametric, cos(t), sin(t),
                        [t,0,2*%pi], [nticks, 80]],
                  abs(x)], [x,-2,2], [y, -1.5, 1.5])$
          plot2d: some values were clipped.

     A plot of a discrete set of points, defining x and y coordinates
     separately:

          (%i1) plot2d ([discrete, [10, 20, 30, 40, 50],
                                   [.6, .9, 1.1, 1.3, 1.4]])$

     The same points shown in the previous example, defining each point
     separately and without any lines joining the points:

          (%i1) plot2d([discrete, [[10, .6], [20, .9], [30, 1.1],
                                   [40, 1.3], [50, 1.4]]],
                                  [style, points])$

     In this example, a table with three columns is saved in a file
     "data.txt" which is then read and the second and third column are
     plotted on the two axes:

          (%i1) with_stdout ("data.txt", for x:0 thru 10 do
                                             print (x, x^2, x^3))$
          (%i2) data: read_matrix ("data.txt")$
          (%i3) plot2d ([discrete, transpose(data)[2], transpose(data)[3]],
            [style,points], [point_type,diamond], [color,red])$

     A plot of experimental data points together with the theoretical
     function that predicts the data:

          (%i1) xy: [[10, .6], [20, .9], [30, 1.1], [40, 1.3], [50, 1.4]]$
          (%i2) plot2d([[discrete, xy], 2*%pi*sqrt(l/980)], [l,0,50],
                  [style, points, lines], [color, red, blue],
                  [point_type, asterisk],
                  [legend, "experiment", "theory"],
                  [xlabel, "pendulum's length (cm)"],
                  [ylabel, "period (s)"])$

     See also the section about Plotting Options.

Function: plot3d (<expr>, <x_range>, <y_range>, ..., <options>, ...)
Function: plot3d ([<expr_1>, ..., <expr_n>], <x_range>, <y_range>, ..., <options>, ...)

     Displays a plot of one or more surfaces defined as functions of two
     variables or in parametric form.

     The functions to be plotted may be specified as expressions or
     function names.  The mouse can be used to rotate the plot looking
     at the surface from different sides.

     *Examples:*

     Plot of a common function:

          (%i1) plot3d (2^(-u^2 + v^2), [u, -3, 3], [v, -2, 2])$

     Use of the z option to limit a function that goes to infinity (in
     this case the function is minus infinity on the x and y axes);
     this also shows how to plot with only lines and no shading:

          (%i1) plot3d ( log ( x^2*y^2 ), [x, -2, 2], [y, -2, 2], [z, -8, 4],
                         [palette, false], [color, magenta, blue])$

     The infinite values of z can also be avoided by choosing a grid
     that does not fall on any asymptotes; this example also shows how
     to select one of the predefined palettes, in this case the fourth
     one:

          (%i1) plot3d (log (x^2*y^2), [x, -2, 2], [y, -2, 2],
                   [grid, 29, 29],
                   [palette, get_plot_option(palette,5)])$

     Two surfaces in the same plot, sharing the same domain; in gnuplot
     the two surfaces will use the same palette:

          (%i1) plot3d ([2^(-x^2 + y^2), 4*sin(3*(x^2+y^2))/(x^2+y^2),
                        [x, -3, 3], [y, -2, 2]])$

     The same two surfaces, but now with different domains; in xmaxima
     each surface will use a different palette, chosen from the list
     defined by the option palette:

          (%i1) plot3d ([[2^(-x^2 + y^2),[x,-2,2],[y,-2,2]],
                   4*sin(3*(x^2+y^2))/(x^2+y^2),
                   [x, -3, 3], [y, -2, 2]], [plot_format,xmaxima])$

     Plot of a Klein bottle, defined parametrically:

          (%i1) expr_1:5*cos(x)*(cos(x/2)*cos(y)+sin(x/2)*sin(2*y)+3.0)-10.0$
          (%i2) expr_2:-5*sin(x)*(cos(x/2)*cos(y) + sin(x/2)*sin(2*y) + 3.0)$
          (%i3) expr_3: 5*(-sin(x/2)*cos(y) + cos(x/2)*sin(2*y))$
          (%i4) plot3d ([expr_1, expr_2, expr_3], [x, -%pi, %pi],
                  [y, -%pi, %pi], [grid, 40, 40])$

     Plot of a spherical harmonic, using of the predefined
     transformations, `spherical_to_xyz', to transform from spherical
     to rectangular coordinates.  See the documentation for
     `spherical_to_xyz'.

          (%i1) plot3d (sin(2*theta)*cos(phi), [theta, 0, %pi],
                        [phi, 0, 2*%pi],
                        [transform_xy, spherical_to_xyz], [grid,30,60])$

     Use of the predefined function `polar_to_xy' to transform from
     cylindrical to rectangular coordinates.  See the documentation for
     `polar_to_xy'.    This example also shows how to eliminate the
     bounding box and the legend.

          (%i1) plot3d (r^.33*cos(th/3), [r, 0, 1], [th, 0, 6*%pi],
                   [grid, 12, 80],
                   [transform_xy, polar_to_xy], [box, false],
                   [legend,false])$

     Plot of a sphere using the transformation from spherical to
     rectangular coordinates.  In xmaxima the three axes are scaled in
     the same proportion, maintaining the symmetric shape of the
     sphere.  A palette with different shades of a single color is used:

          (%i1) plot3d ( 5, [theta, 0, %pi], [phi, 0, 2*%pi],
                   [plot_format,xmaxima],
                   [transform_xy, spherical_to_xyz],
                   [palette,[value,0.65,0.7,0.1,0.9]])$

     Definition of a function of two-variables using a matrix.  Notice
     the single quote in the definition of the function, to prevent
     plot3d from failing when it realizes that the matrix will require
     integer indices.

          (%i1) M: matrix([1, 2, 3, 4], [1, 2, 3, 2], [1, 2, 3, 4],
                          [1, 2, 3, 3])$
          (%i2) f(x, y) := float('M [round(x), round(y)])$
          (%i3) plot3d (f(x,y), [x, 1, 4], [y, 1, 4], [grid, 4, 4])$
          apply: subscript must be an integer; found: round(x)

     By setting the elevation equal to zero, a surface can be seen as a
     map in which each color represents a different level.  The option
     `colorbox'  is used to show the correspondence among colors and
     levels, and the mesh lines are disabled to make the colors easier
     to see.

          (%i1) plot3d (cos (-x^2 + y^3/4), [x, -4, 4], [y, -4, 4],
                  [mesh_lines_color, false], [elevation, 0], [azimuth, 0],
                  [colorbox, true], [grid, 150, 150])$

     See also the section about Plotting Options.

System variable: plot_options

     Elements of this list state the default options for plotting.  If
     an option is present in a `plot2d'  or `plot3d'  call, that value
     takes precedence over the default option.  Otherwise, the value in
     `plot_options' is used.  Default options are assigned by
     `set_plot_option'.  There are other local options specific to each
     plotting command, and not included in this list of global options.

     Each element of `plot_options' is a list of two or more items.  The
     first item is the name of the option, and the remainder comprises
     the value or values assigned to the option.  In some cases, the
     assigned value is a list, which may include several items.

     See also `set_plot_option',  `get_plot_option'  and the section on
     Plotting Options.

Function: set_plot_option (<option>)

     Accepts most of the options listed in the section Plotting
     Options, and saves them into the global variable `plot_options'.

     `set_plot_option' evaluates its argument and returns the complete
     list `plot_options' (after modifying the option given).

     See also `plot_options',  `get_plot_option'  and the section on
     Plotting Options.

     Example:

     Modification of the `grid'  values.

          (%i1) set_plot_option ([grid, 30, 40]);
          (%o1) [[t, - 3, 3], [grid, 30, 40], [transform_xy, false],
          [run_viewer, true], [axes, true], [plot_format, gnuplot_pipes],
          [color, blue, red, green, magenta, black, cyan],
          [point_type, bullet, circle, plus, times, asterisk, box, square,
          triangle, delta, wedge, nabla, diamond, lozenge],
          [palette, [hue, 0.25, 0.7, 0.8, 0.5],
          [hue, 0.65, 0.8, 0.9, 0.55], [hue, 0.55, 0.8, 0.9, 0.4],
          [hue, 0.95, 0.7, 0.8, 0.5]], [gnuplot_term, default],
          [gnuplot_out_file, false], [nticks, 29], [adapt_depth, 5],
          [gnuplot_preamble, ], [gnuplot_default_term_command,
          set term pop], [gnuplot_dumb_term_command, set term dumb 79 22],
          [gnuplot_ps_term_command, set size 1.5, 1.5;set term postscript \
          eps enhanced color solid 24], [plot_realpart, false]]

System function: spherical_to_xyz

     It can be given as value for the `transform_xy'  option of
     `plot3d'.    Its effect will be to interpret the two independent
     variables and the function in `plot3d' as the spherical
     coordinates of a point (first, the angle with the z axis, then the
     angle of the xy projection with the x axis and finally the
     distance from the origin) and transform them into x, y and z
     coordinates.

12.4 Plotting Options

All options consist of a list starting with one of the keywords in this
section, followed by one or more values.  Most of the options can be
used in any of the plotting commands (`plot2d',  `plot3d',

   `contour_plot',  `implicit_plot')  or in the function
`set_plot_option'.    The exceptions will be specified in the following
list.
Plot option: adapt_depth [adapt_depth, <integer>]

     Default value: `5'

     The maximum number of splittings used by the adaptive plotting
     routine.

Plot option: axes [axes, <symbol>]

     Default value: `true'

     Where <symbol> can be either `true', `false', `x' or `y'.  If
     `false', no axes will be shown; if equal to `x' or `y' only the x
     or y axis will be shown, and if it is equal to `true', both axes
     will be shown.  This option is used only by plot2d and
     implicit_plot.

Plot option: azimuth [azimuth, <number>]

     Default value: `30'

     A plot3d plot can be thought of as starting with its x and y axis
     in the horizontal and vertical axis, as in plot2d, and the z axis
     coming out of the paper perpendicularly.  The z axis is then
     rotated around the x axis an angle equals to `elevation' and then
     the xy plane is rotated around the new z axis an angle `azimuth'.
     This option sets the value for the azimuth, in degrees.

     See also `elevation'.

Plot option: box [box, <symbol>]

     Default value: `true'

     If set to `true', a bounding box will be drawn for the plot; if set
     to `false', no box will be drawn.

Plot option: color [color, <color_1>, ..., <color_n>]

     Default value: `blue', `red', `green', `magenta', `black', `cyan'

     In `plot2d'  and `implicit_plot',  it defines the color (or colors)
     for the various curves.  In `plot3d',  it defines the colors used
     for the mesh lines of the surfaces, when no palette is being used;
     one side of the surface will have color <color_1> and the other
     <color_2> (or the same color if there is only one color).

     If there are more curves or surfaces than colors, the colors will
     be repeated in sequence.  When using gnuplot, the colors could be:
     `blue', `red', `green', `magenta', `black', or `cyan'; in xmaxima
     the colors can be those or a string starting with the character #
     and followed by six hexadecimal digits: two for the red component,
     two for green component and two for the blue component.  If given
     the name of an unknown color, black will be used instead.

Plot option: colorbox [colorbox, <symbol>]

     Default value: `false'

     Where <symbol> can be either `true' or `false'.  If `true',
     whenever `plot3d'  uses a palette of different colors to represent
     the different values of z, a box will be shown on the right,
     indicating the colors used according to the scale of values of z.
     This option does not work in xmaxima.

Plot option: elevation [elevation, <number>]

     Default value: `60'

     A `plot3d'  plot can be thought of as starting with its x and y
     axis in the horizontal and vertical axis, as in `plot2d',  and the
     z axis coming out of the paper perpendicularly.  The z axis is
     then rotated around the x axis an angle equals to `elevation' and
     then the xy plane is rotated around the new z axis an angle
     `azimuth'.  This option sets the value for the elevation, in
     degrees.

     See also `azimuth'.

Plot option: grid [grid, <integer>, <integer>]

     Default value: `30', `30'

     Sets the number of grid points to use in the x- and y-directions
     for three-dimensional plotting.

Plot option: legend [legend, <string_1>, ..., <string_n>]
Plot option: legend [legend, <false>]

     It specifies the labels for the plots when various plots are
     shown.  If there are more plots than the number of labels given,
     they will be repeated.  If given the value `false', no legends
     will be shown.  By default, the names of the expressions or
     functions will be used, or the words discrete1, discrete2, ...,
     for discrete sets of points.  This option can not be set with
     `set_plot_option'.

Plot option: logx [logx]

     Makes the horizontal axes to be scaled logarithmically.  It can
     not be used with `set_plot_option'.

Plot option: logy [logy]

     Makes the vertical axes to be scaled logarithmically.  It can not
     be used with `set_plot_option'.

Plot option: mesh_lines_color [mesh_lines_color, <color>]

     Default value: `black'

     It sets the color used by plot3d to draw the mesh lines, when a
     palette is being used.  It accepts the same colors as for the
     option `color'

     (see the list of allowed colors in `color').  It can also be given
     a value `false' to eliminate completely the mesh lines.

Plot option: nticks [nticks, <integer>]

     Default value: `29'

     When plotting functions with `plot2d',  it is gives the initial
     number of points used by the adaptive plotting routine for
     plotting functions.  When plotting parametric functions with
     plot2d or `plot3d',  it sets the number of points that will be
     shown for the plot.

Plot option: palette [palette, [<palette_1>], ..., [<palette_n>]]
Plot option: palette [palette, <false>]

     Default value: `[hue, 0.25, 0.7, 0.8, 0.5]', `[hue, 0.65, 0.8,
     0.9, 0.55]', `[hue, 0.55, 0.8, 0.9, 0.4]', `[hue, 0.95, 0.7, 0.8,
     0.5]'

     It can consist of one palette or a list of several palettes.  Each
     palette is a list with a keyword followed by four numbers.  The
     first three numbers, which must be between 0 and 1, define the
     hue, saturation and value of a basic color to be assigned to the
     minimum value of z.  The keyword specifies which of the three
     attributes (hue, saturation or value) will be increased according
     to the values of z.  The last number indicates the increase
     corresponding to the maximum value of z.  That last number can be
     bigger than 1 or negative; the corresponding values of the
     modified attribute will be rounded modulo 1.

     Gnuplot only uses the first palette in the list; xmaxima will use
     the palettes in the list sequentially, when several surfaces are
     plotted together; if the number of palettes is exhausted, they
     will be repeated sequentially.

     The color of the mesh lines will be given by the option
     `mesh_lines_color'.    If `palette' is given the value `false',
     the surfaces will not be shaded but represented with a mesh of
     curves only.  In that case, the colors of the lines will be
     determined by the option `color'.

Plot option: plot_format [plot_format, <format>]

     Default value: `gnuplot', in Windows systems, or `gnuplot_pipes' in
     other systems.

     Where <format> is one of the following: gnuplot, xmaxima, mgnuplot
     or gnuplot_pipes.

     It sets the format to be used for plotting.

Plot option: plot_realpart [plot_realpart, <symbol>]

     Default value: `false'

     If set to `true', the functions to be plotted will be considered
     as complex functions whose real value should be plotted; this is
     equivalent to plotting `realpart(<function>)'.  If set to `false',
     nothing will be plotted when the function does not give a real
     value.  For instance, when `x' is negative, `log(x)' gives a
     complex value, with real value equal to `log(abs(x))'; if
     `plot_realpart' were `true', `log(-5)' would be plotted as
     `log(5)', while nothing would be plotted if `plot_realpart' were
     `false'.

Plot option: point_type [point_type, <type_1>, ..., <type_n>]

     Default value: `bullet', `circle', `plus', `times', `asterisk',
     `box', `square', `triangle', `delta', `wedge', `nabla', `diamond',
     `lozenge'

     In gnuplot, each set of points to be plotted with the style
     "points" or "linespoints" will be represented with objects taken
     from this list, in sequential order.  If there are more sets of
     points than objects in this list, they will be repeated
     sequentially.  The possible objects that can be used are:
     `bullet', `circle', `plus', `times', `asterisk', `box', `square',
     `triangle', `delta', `wedge', `nabla', `diamond', `lozenge'

Plot option: psfile [psfile, <string>]

     Saves the plot into a Postscript file with name equal to <string>,
     rather than showing it in the screen.  By default, the file will be
     created in the directory defined by the variable `maxima_tempdir'.

     The value of that variable can be changed to save the file in a
     different directory.

Plot option: run_viewer [run_viewer, <symbol>]

     Default value: `true'

     Controls whether or not the appropriate viewer for the plot format
     should be run.

Plot option: style [style, <type_1>, ..., <type1_n>]
Plot option: style [style, [<style_1>], ..., [<style_n>]]

     Default value: `lines' (will plot all sets of points joined with
     lines of thickness 1 and the first color given by the option
     `color' ).

     The styles that will be used for the various functions or sets of
     data in a 2d plot.  The word <style> must be followed by one or
     more styles.  If there are more functions and data sets than the
     styles given, the styles will be repeated.  Each style can be
     either <lines> for line segments, <points> for isolated points,
     <linespoints> for segments and points, or <dots> for small
     isolated dots.  Gnuplot accepts also an <impulses> style.

     Each of the styles can be enclosed inside a list with some
     additional parameters.  <lines> accepts one or two numbers: the
     width of the line and an integer that identifies a color.  The
     default color codes are: 1: blue, 2: red, 3: magenta, 4: orange,
     5: brown, 6: lime and 7: aqua.  If you use Gnuplot with a terminal
     different than X11, those colors might be different; for example,
     if you use the option [<gnuplot_term>, <ps>], color index 4 will
     correspond to black, instead of orange.

     <points> accepts one two or three parameters; the first parameter
     is the radius of the points, the second parameter is an integer
     that selects the color, using the same code used for <lines> and
     the third parameter is currently used only by Gnuplot and it
     corresponds to several objects instead of points.  The default
     types of objects are: 1: filled circles, 2: open circles, 3: plus
     signs, 4: x, 5: *, 6: filled squares, 7: open squares, 8: filled
     triangles, 9: open triangles, 10: filled inverted triangles, 11:
     open inverted triangles, 12: filled lozenges and 13: open lozenges.

     <linesdots> accepts up to four parameters: line width, points
     radius, color and type of object to replace the points.

     See also `color'  and `point_type'.

Plot option: t [t, <min>, <max>]

     Default value: `-3', `3'

     Default range for parametric plots.

Plot option: transform_xy [transform_xy, <symbol>]

     Default value: `false'

     Where <symbol> is either `false' or the result obtained by using
     the function `transform_xy'.  If different from `false', it will
     be used to transform the 3 coordinates in plot3d.

     See `make_transform',  `polar_to_xy'  and `spherical_to_xyz'.

Plot option: x [x, <min>, <max>]

     When used as the first option in a `plot2d'  command (or any of the
     first two in `plot3d' ), it indicates that the first independent
     variable is x and it sets its range.  It can also be used again
     after the first option (or after the second option in plot3d) to
     define the effective horizontal domain that will be shown in the
     plot.

Plot option: xlabel [xlabel, <string>]

     Specifies the <string> that will label the first axis; if this
     option is not used, that label will be the name of the independent
     variable, when plotting functions with `plot2d'  or
     `implicit_plot',  or the name of the first variable, when plotting
     surfaces with `plot3d'  or contours with `contour_plot',  or the
     first expression in the case of a parametric plot.  It can not be
     used with `set_plot_option'.

Plot option: y [y, <min>, <max>]

     When used as one of the first two options in `plot3d',  it
     indicates that one of the independent variables is y and it sets
     its range.  Otherwise, it defines the effective domain of the
     second variable that will be shown in the plot.

Plot option: ylabel [ylabel, <string>]

     Specifies the <string> that will label the second axis; if this
     option is not used, that label will be "y", when plotting functions
     with `plot2d'  or `implicit_plot',  or the name of the second
     variable, when plotting surfaces with `plot3d'  or contours with
     `contour_plot',  or the second expression in the case of a
     parametric plot.  It can not be used with `set_plot_option'.

Plot option: z [z, <min>, <max>]

     Used in `plot3d'  to set the effective range of values of z that
     will be shown in the plot.

Plot option: zlabel [zlabel, <string>]

     Specifies the <string> that will label the third axis, when using
     `plot3d'.    If this option is not used, that label will be "z",
     when plotting surfaces, or the third expression in the case of a
     parametric plot.  It can not be used with `set_plot_option'  and
     it will be ignored by `plot2d'  and `implicit_plot'.

12.5 Gnuplot Options

There are several plot options specific to gnuplot.  All of them consist
of a keyword (the name of the option), followed by a string that should
be a valid gnuplot command, to be passed directly to gnuplot.  In most
cases, there exist a corresponding plotting option that will produce a
similar result and whose use is more recommended than the gnuplot
specific option.
Plot option: gnuplot_term

     Sets the output terminal type for gnuplot.
        * *default* (default value)

          Gnuplot output is displayed in a separate graphical window.

        * *dumb*

          Gnuplot output is displayed in the Maxima console by an
          "ASCII art" approximation to graphics.

        * *ps*

          Gnuplot generates commands in the PostScript page description
          language.  If the option `gnuplot_out_file'  is set to
          <filename>, gnuplot writes the PostScript commands to
          <filename>.  Otherwise, it is saved as `maxplot.ps' file.

        * any other valid gnuplot term specification

          Gnuplot can generate output in many other graphical formats
          such as png, jpeg, svg etc.  To create plot in all these
          formats the `gnuplot_term' can be set to any supported
          gnuplot term name (symbol) or even full gnuplot term
          specification with any valid options (string).  For example
          `[gnuplot_term, png]' creates output in PNG (Portable Network
          Graphics) format while `[gnuplot_term, "png size 1000,1000"]'
          creates PNG of 1000 x 1000 pixels size.  If the option
          `gnuplot_out_file'  is set to <filename>, gnuplot writes the
          output to <filename>.  Otherwise, it is saved as
          `maxplot.<term>' file, where <term> is gnuplot terminal name.

Plot option: gnuplot_out_file

     When used in conjunction with the `gnuplot_term'  option, it can be
     used to save the plot in a file, in one of the graphic formats
     supported by Gnuplot.  If you want to create a Postscript file,
     you can use the option `psfile'  instead, which will also work in
     Openmath, and does the same thing with just one option.

          [gnuplot_term, png], [gnuplot_out_file, "graph3.png"]

Plot option: gnuplot_pm3d

     With a value of `false', it can be used to prevent the usage of
     PM3D mode, which is enabled by default.

Plot option: gnuplot_preamble

     Inserts gnuplot commands before the plot is drawn.  Any valid
     gnuplot commands may be used.  Multiple commands should be
     separated with a semi-colon.  The example shown produces a log
     scale plot.  The default value for `gnuplot_preamble' is the empty
     string `""'.

Plot option: gnuplot_curve_titles

     This is an old option that has been replaced by `legend'  described
     above.

Plot option: gnuplot_curve_styles

     This is an obsolete option that has been replaced by `style'.

Plot option: gnuplot_default_term_command

     The gnuplot command to set the terminal type for the default
     terminal.  The default value is `"set term pop"'.

Plot option: gnuplot_dumb_term_command

     The gnuplot command to set the terminal type for the dumb
     terminal.  The default value is `"set term dumb 79 22"', which
     makes the text output 79 characters by 22 characters.

Plot option: gnuplot_ps_term_command

     The gnuplot command to set the terminal type for the PostScript
     terminal.  The default value is `"set size 1.5, 1.5; set term
     postscript eps enhanced color solid 24"', which sets the size to
     1.5 times gnuplot's default, and the font size to 24, among other
     things.  See the gnuplot documentation for `set term postscript'
     for more information.

12.6 Gnuplot_pipes Format Functions

Function: gnuplot_start ()

     Opens the pipe to gnuplot used for plotting with the
     `gnuplot_pipes' format.  Is not necessary to manually open the
     pipe before plotting.

Function: gnuplot_close ()

     Closes the pipe to gnuplot which is used with the `gnuplot_pipes'
     format.

Function: gnuplot_restart ()

     Closes the pipe to gnuplot which is used with the `gnuplot_pipes'
     format and opens a new pipe.

Function: gnuplot_replot ()
Function: gnuplot_replot (<s>)

     Updates the gnuplot window.  If `gnuplot_replot' is called with a
     gnuplot command in a string <s>, then `s' is sent to gnuplot
     before reploting the window.

Function: gnuplot_reset ()

     Resets the state of gnuplot used with the `gnuplot_pipes' format
     To update the gnuplot window call `gnuplot_replot'  after
     `gnuplot_reset'.

13 File Input and Output


Comments
Files
Functions and Variables for File Input and Output
Functions and Variables for TeX Output
Functions and Variables for Fortran Output

13.1 Comments

A comment in Maxima input is any text between `/*' and `*/'.

   The Maxima parser treats a comment as whitespace for the purpose of
finding tokens in the input stream; a token always ends at a comment.
An input such as `a/* foo */b' contains two tokens, `a' and `b', and
not a single token `ab'.  Comments are otherwise ignored by Maxima;
neither the content nor the location of comments is stored in parsed
input expressions.

   Comments can be nested to arbitrary depth.  The `/*' and `*/'
delimiters form matching pairs.  There must be the same number of `/*'
as there are `*/'.

   Examples:

     (%i1) /* aa is a variable of interest */  aa : 1234;
     (%o1)                         1234
     (%i2) /* Value of bb depends on aa */  bb : aa^2;
     (%o2)                        1522756
     (%i3) /* User-defined infix operator */  infix ("b");
     (%o3)                           b
     (%i4) /* Parses same as a b c, not abc */  a/* foo */b/* bar */c;
     (%o4)                         a b c
     (%i5) /* Comments /* can be nested /* to any depth */ */ */  1 + xyz;
     (%o5)                        xyz + 1

13.2 Files

A file is simply an area on a particular storage device which contains
data or text.  Files on the disks are figuratively grouped into
"directories".  A directory is just a list of files.  Commands which
deal with files are:

   appendfile           batch                 batchload
   closefile            file_output_append    filename_merge
   file_search          file_search_maxima    file_search_lisp
   file_search_demo     file_search_usage     file_search_tests
   file_type            file_type_lisp        file_type_maxima
   load                 load_pathname         loadfile
   loadprint            pathname_directory    pathname_name
   pathname_type        printfile             save
   stringout            with_stdout           writefile

   When a file name is passed to functions like `plot2d',

   `save',  or `writefile'  and the file name does not include a path,
Maxima stores the file in the current working directory.  The current
working directory depends on the system like Windows or Linux and on
the installation.

13.3 Functions and Variables for File Input and Output

Function: appendfile (<filename>)

     Appends a console transcript to <filename>.  `appendfile' is the
     same as `writefile',  except that the transcript file, if it
     exists, is always appended.

     `closefile'  closes the transcript file opened by `appendfile' or
     `writefile'.

Function: batch (<filename>)
Function: batch (<filename>, `option')

     `batch(<filename>)' reads Maxima expressions from <filename> and
     evaluates them.  `batch' searches for <filename> in the list
     `file_search_maxima'.    See also `file_search'.

     `batch(<filename>, `demo')' is like `demo(<filename>)'.  In this
     case `batch' searches for <filename> in the list
     `file_search_demo'.    See `demo'.

     `batch(<filename>, `test')' is like `run_testsuite'  with the
     option `display_all=true'.  For this case `batch' searches
     <filename> in the list `file_search_maxima' and not in the list
     `file_search_tests'  like `run_testsuite'.  Furthermore,
     `run_testsuite' runs tests which are in the list `testsuite_files'.
      With `batch' it is possible to run any file in a test mode, which
     can be found in the list `file_search_maxima'.  This is useful,
     when writing a test file.

     <filename> comprises a sequence of Maxima expressions, each
     terminated with `;' or `$'.  The special variable `%'  and the
     function `%th'  refer to previous results within the file.  The
     file may include `:lisp' constructs.  Spaces, tabs, and newlines
     in the file are ignored.  A suitable input file may be created by
     a text editor or by the `stringout'  function.

     `batch' reads each input expression from <filename>, displays the
     input to the console, computes the corresponding output
     expression, and displays the output expression.  Input labels are
     assigned to the input expressions and output labels are assigned
     to the output expressions.  `batch' evaluates every input
     expression in the file unless there is an error.  If user input is
     requested (by `asksign'  or `askinteger',  for example) `batch'
     pauses to collect the requisite input and then continue.

     It may be possible to halt `batch' by typing `control-C' at the
     console.  The effect of `control-C' depends on the underlying Lisp
     implementation.

     `batch' has several uses, such as to provide a reservoir for
     working command lines, to give error-free demonstrations, or to
     help organize one's thinking in solving complex problems.

     `batch' evaluates its argument.  `batch' returns the path of
     <filename> as a string, when called with no second argument or
     with the option `demo'.  When called with the option `test', the
     return value is a an empty list `[]' or a list with <filename> and
     the numbers of the tests which have failed.

     See also `load',  `batchload',  and `demo'.

Function: batchload (<filename>)

     Reads Maxima expressions from <filename> and evaluates them,
     without displaying the input or output expressions and without
     assigning labels to output expressions.  Printed output (such as
     produced by `print'  or `describe')  is displayed, however.

     The special variable `%'  and the function `%th'  refer to previous
     results from the interactive interpreter, not results within the
     file.  The file cannot include `:lisp' constructs.

     `batchload' returns the path of <filename>, as a string.
     `batchload' evaluates its argument.

     See also `batch',  and `load'.

Function: closefile ()

     Closes the transcript file opened by `writefile'  or `appendfile'.

Option variable: file_output_append

     Default value: `false'

     `file_output_append' governs whether file output functions append
     or truncate their output file.  When `file_output_append' is
     `true', such functions append to their output file.  Otherwise,
     the output file is truncated.

     `save',  `stringout',  and `with_stdout'  respect
     `file_output_append'.  Other functions which write output files do
     not respect `file_output_append'.  In particular, plotting and
     translation functions always truncate their output file, and `tex'
     and `appendfile'  always append.

Function: filename_merge (<path>, <filename>)

     Constructs a modified path from <path> and <filename>.  If the
     final component of <path> is of the form `###.<something>', the
     component is replaced with `<filename>.<something>'.  Otherwise,
     the final component is simply replaced by <filename>.

     The result is a Lisp pathname object.

Function: file_search (<filename>)
Function: file_search (<filename>, <pathlist>)

     `file_search' searches for the file <filename> and returns the
     path to the file (as a string) if it can be found; otherwise
     `file_search' returns `false'.  `file_search (<filename>)'
     searches in the default search directories, which are specified by
     the `file_search_maxima',  `file_search_lisp',  and
     `file_search_demo'  variables.

     `file_search' first checks if the actual name passed exists,
     before attempting to match it to "wildcard" file search patterns.
     See `file_search_maxima'  concerning file search patterns.

     The argument <filename> can be a path and file name, or just a
     file name, or, if a file search directory includes a file search
     pattern, just the base of the file name (without an extension).
     For example,

          file_search ("/home/wfs/special/zeta.mac");
          file_search ("zeta.mac");
          file_search ("zeta");

     all find the same file, assuming the file exists and
     `/home/wfs/special/###.mac' is in `file_search_maxima'.

     `file_search (<filename>, <pathlist>)' searches only in the
     directories specified by <pathlist>, which is a list of strings.
     The argument <pathlist> supersedes the default search directories,
     so if the path list is given, `file_search' searches only the ones
     specified, and not any of the default search directories.  Even if
     there is only one directory in <pathlist>, it must still be given
     as a one-element list.

     The user may modify the default search directories.  See
     `file_search_maxima'.

     `file_search' is invoked by `load'  with `file_search_maxima' and
     `file_search_lisp'  as the search directories.

Option variable: file_search_maxima
Option variable: file_search_lisp
Option variable: file_search_demo
Option variable: file_search_usage
Option variable: file_search_tests

     These variables specify lists of directories to be searched by
     `load',  `demo',  and some other Maxima functions.  The default
     values of these variables name various directories in the Maxima
     installation.

     The user can modify these variables, either to replace the default
     values or to append additional directories.  For example,

          file_search_maxima: ["/usr/local/foo/###.mac",
              "/usr/local/bar/###.mac"]$

     replaces the default value of `file_search_maxima', while

          file_search_maxima: append (file_search_maxima,
              ["/usr/local/foo/###.mac", "/usr/local/bar/###.mac"])$

     appends two additional directories.  It may be convenient to put
     such an expression in the file `maxima-init.mac' so that the file
     search path is assigned automatically when Maxima starts.  See
     also *note Introduction for Runtime Environment::.

     Multiple filename extensions and multiple paths can be specified
     by special "wildcard" constructions.  The string `###' expands
     into the sought-after name, while a comma-separated list enclosed
     in curly braces `{foo,bar,baz}' expands into multiple strings.
     For example, supposing the sought-after name is `neumann',

          "/home/{wfs,gcj}/###.{lisp,mac}"

     expands into `/home/wfs/neumann.lisp', `/home/gcj/neumann.lisp',
     `/home/wfs/neumann.mac', and `/home/gcj/neumann.mac'.

Function: file_type (<filename>)

     Returns a guess about the content of <filename>, based on the
     filename extension.  <filename> need not refer to an actual file;
     no attempt is made to open the file and inspect the content.

     The return value is a symbol, either `object', `lisp', or
     `maxima'.  If the extension is matches one of the values in
     `file_type_maxima', `file_type' returns `maxima'.  If the
     extension matches one of the values in `file_type_lisp',
     `file_type' returns `lisp'.  If none of the above, `file_type'
     returns `object'.

     See also `pathname_type'.

     See `file_type_maxima'  and `file_type_lisp'  for the default
     values.

     Examples:

          (%i2) map('file_type,
                    ["test.lisp", "test.mac", "test.dem", "test.txt"]);
          (%o2)            [lisp, maxima, maxima, object]

Option variable: file_type_lisp

     Default value:  `[l, lsp, lisp]'

     `file_type_lisp' is a list of file extensions that maxima
     recognizes as denoting a Lisp source file.

     See also `file_type'.

Option variable: file_type_maxima

     Default value:  `[mac, mc, demo, dem, dm1, dm2, dm3, dmt]'

     `file_type_maxima' is a list of file extensions that maxima
     recognizes as denoting a Maxima source file.

     See also `file_type'.

Function: load (<filename>)

     Evaluates expressions in <filename>, thus bringing variables,
     functions, and other objects into Maxima.  The binding of any
     existing object is clobbered by the binding recovered from
     <filename>.  To find the file, `load' calls `file_search'  with
     `file_search_maxima'  and `file_search_lisp'  as the search
     directories.  If `load' succeeds, it returns the name of the file.
     Otherwise `load' prints an error message.

     `load' works equally well for Lisp code and Maxima code.  Files
     created by `save',  `translate_file',  and `compile_file',  which
     create Lisp code, and `stringout',  which creates Maxima code, can
     all be processed by `load'.  `load' calls `loadfile'  to load Lisp
     files and `batchload'  to load Maxima files.

     `load' does not recognize `:lisp' constructs in Maxima files, and
     while processing <filename>, the global variables `_', `__', `%',
     and `%th' have whatever bindings they had when `load' was called.

     See also `loadfile',  `batch',  `batchload',  and `demo'.
     `loadfile' processes Lisp files; `batch', `batchload', and `demo'
     process Maxima files.

     See `file_search'  for more detail about the file search mechanism.

     `load' evaluates its argument.

System variable: load_pathname

     Default value: `false'

     When a file is loaded with the functions `load',  `loadfile'  or
     `batchload'  the system variable `load_pathname' is bound to the
     pathname of the file which is processed.

     The variable `load_pathname' can be accessed from the file during
     the loading.

     Example:

     Suppose we have a batchfile `test.mac' in the directory
     `"/home/dieter/workspace/mymaxima/temp/"' with the following commands

          print("The value of load_pathname is: ", load_pathname)$
          print("End of batchfile")$

     then we get the following output

          (%i1) load("/home/dieter/workspace/mymaxima/temp/test.mac")$
          The value of load_pathname is:
                             /home/dieter/workspace/mymaxima/temp/test.mac
          End of batchfile

Function: loadfile (<filename>)

     Evaluates Lisp expressions in <filename>.  `loadfile' does not
     invoke `file_search',  so `filename' must include the file
     extension and as much of the path as needed to find the file.

     `loadfile' can process files created by `save',

     `translate_file',  and `compile_file'.    The user may find it
     more convenient to use `load'  instead of `loadfile'.

Option variable: loadprint

     Default value: `true'

     `loadprint' tells whether to print a message when a file is loaded.

        * When `loadprint' is `true', always print a message.

        * When `loadprint' is `'loadfile', print a message only if a
          file is loaded by the function `loadfile'.

        * When `loadprint' is `'autoload', print a message only if a
          file is automatically loaded.  See `setup_autoload'.

        * When `loadprint' is `false', never print a message.

Function: pathname_directory (<pathname>)
Function: pathname_name (<pathname>)
Function: pathname_type (<pathname>)

     These functions return the components of <pathname>.

     Examples:

          (%i1) pathname_directory("/home/dieter/maxima/changelog.txt");
          (%o1)                 /home/dieter/maxima/
          (%i2) pathname_name("/home/dieter/maxima/changelog.txt");
          (%o2)                       changelog
          (%i3) pathname_type("/home/dieter/maxima/changelog.txt");
          (%o3)                          txt

Function: printfile (<path>)

     Prints the file named by <path> to the console.  <path> may be a
     string or a symbol; if it is a symbol, it is converted to a string.

     If <path> names a file which is accessible from the current working
     directory, that file is printed to the console.  Otherwise,
     `printfile' attempts to locate the file by appending <path> to
     each of the elements of `file_search_usage'  via `filename_merge'.

     `printfile' returns <path> if it names an existing file, or
     otherwise the result of a successful filename merge.

Function: save (<filename>, <name_1>, <name_2>, <name_3>, ...)
Function: save (<filename>, values, functions, labels, ...)
Function: save (<filename>, [<m>, <n>])
Function: save (<filename>, <name_1>=<expr_1>, ...)
Function: save (<filename>, all)
Function: save (<filename>, <name_1>=<expr_1>, <name_2>=<expr_2>, ...)

     Stores the current values of <name_1>, <name_2>, <name_3>, ..., in
     <filename>.  The arguments are the names of variables, functions,
     or other objects.  If a name has no value or function associated
     with it, it is ignored.  `save' returns <filename>.

     `save' stores data in the form of Lisp expressions.  The data
     stored by `save' may be recovered by `load (<filename>)'.  See
     `load'.

     The global flag `file_output_append'  governs whether `save'
     appends or truncates the output file.  When `file_output_append'
     is `true', `save' appends to the output file.  Otherwise, `save'
     truncates the output file.  In either case, `save' creates the
     file if it does not yet exist.

     The special form `save (<filename>, values, functions, labels,
     ...)'  stores the items named by `values',  `functions',

     `labels',  etc.  The names may be any specified by the variable
     `infolists'.    `values' comprises all user-defined variables.

     The special form `save (<filename>, [<m>, <n>])' stores the values
     of input and output labels <m> through <n>.  Note that <m> and <n>
     must be literal integers.  Input and output labels may also be
     stored one by one, e.g., `save ("foo.1", %i42, %o42)'.  `save
     (<filename>, labels)' stores all input and output labels.  When
     the stored labels are recovered, they clobber existing labels.

     The special form `save (<filename>, <name_1>=<expr_1>,
     <name_2>=<expr_2>, ...)' stores the values of <expr_1>, <expr_2>,
     ..., with names <name_1>, <name_2>, ...  It is useful to apply
     this form to input and output labels, e.g., `save ("foo.1",
     aa=%o88)'.  The right-hand side of the equality in this form may
     be any expression, which is evaluated.  This form does not
     introduce the new names into the current Maxima environment, but
     only stores them in <filename>.

     These special forms and the general form of `save' may be mixed at
     will.  For example, `save (<filename>, aa, bb, cc=42, functions,
     [11, 17])'.

     The special form `save (<filename>, all)' stores the current state
     of Maxima.  This includes all user-defined variables, functions,
     arrays, etc., as well as some automatically defined items.  The
     saved items include system variables, such as `file_search_maxima'
     or `showtime',  if they have been assigned new values by the user;
     see `myoptions'.

     `save' evaluates <filename> and quotes all other arguments.

Function: stringout (<filename>, <expr_1>, <expr_2>, <expr_3>, ...)
Function: stringout (<filename>, [<m>, <n>])
Function: stringout (<filename>, input)
Function: stringout (<filename>, functions)
Function: stringout (<filename>, values)

     `stringout' writes expressions to a file in the same form the
     expressions would be typed for input.  The file can then be used
     as input for the `batch'  or `demo'  commands, and it may be
     edited for any purpose.  `stringout' can be executed while
     `writefile'  is in progress.

     The global flag `file_output_append'  governs whether `stringout'
     appends or truncates the output file.  When `file_output_append' is
     `true', `stringout' appends to the output file.  Otherwise,
     `stringout' truncates the output file.  In either case, `stringout'
     creates the file if it does not yet exist.

     The general form of `stringout' writes the values of one or more
     expressions to the output file.  Note that if an expression is a
     variable, only the value of the variable is written and not the
     name of the variable.  As a useful special case, the expressions
     may be input labels (`%i1', `%i2', `%i3', ...) or output labels
     (`%o1', `%o2', `%o3', ...).

     If `grind'  is `true', `stringout' formats the output using the
     `grind' format.  Otherwise the `string' format is used.  See
     `grind' and `string'.

     The special form `stringout (<filename>, [<m>, <n>])' writes the
     values of input labels m through n, inclusive.

     The special form `stringout (<filename>, input)' writes all input
     labels to the file.

     The special form `stringout (<filename>, functions)' writes all
     user-defined functions (named by the global list `functions')  to
     the file.

     The special form `stringout (<filename>, values)' writes all
     user-assigned variables (named by the global list `values')  to
     the file.  Each variable is printed as an assignment statement,
     with the name of the variable, a colon, and its value.  Note that
     the general form of `stringout' does not print variables as
     assignment statements.

Function: with_stdout (<f>, <expr_1>, <expr_2>, <expr_3>, ...)
Function: with_stdout (<s>, <expr_1>, <expr_2>, <expr_3>, ...)

     Evaluates <expr_1>, <expr_2>, <expr_3>, ... and writes any output
     thus generated to a file <f> or output stream <s>.  The evaluated
     expressions are not written to the output.  Output may be
     generated by `print',  `display',  `grind',  among other functions.

     The global flag `file_output_append'  governs whether `with_stdout'
     appends or truncates the output file <f>.  When
     `file_output_append' is `true', `with_stdout' appends to the
     output file.  Otherwise, `with_stdout' truncates the output file.
     In either case, `with_stdout' creates the file if it does not yet
     exist.

     `with_stdout' returns the value of its final argument.

     See also `writefile'.

          (%i1) with_stdout ("tmp.out", for i:5 thru 10 do
                print (i, "! yields", i!))$
          (%i2) printfile ("tmp.out")$
          5 ! yields 120
          6 ! yields 720
          7 ! yields 5040
          8 ! yields 40320
          9 ! yields 362880
          10 ! yields 3628800

Function: writefile (<filename>)

     Begins writing a transcript of the Maxima session to <filename>.
     All interaction between the user and Maxima is then recorded in
     this file, just as it appears on the console.

     As the transcript is printed in the console output format, it
     cannot be reloaded into Maxima.  To make a file containing
     expressions which can be reloaded, see `save'  and `stringout'.
     `save' stores expressions in Lisp form, while `stringout' stores
     expressions in Maxima form.

     The effect of executing `writefile' when <filename> already exists
     depends on the underlying Lisp implementation; the transcript file
     may be clobbered, or the file may be appended.  `appendfile'
     always appends to the transcript file.

     It may be convenient to execute `playback'  after `writefile' to
     save the display of previous interactions.  As `playback' displays
     only the input and output variables (`%i1', `%o1', etc.), any
     output generated by a print statement in a function (as opposed to
     a return value) is not displayed by `playback'.

     `closefile'  closes the transcript file opened by `writefile' or
     `appendfile'.

13.4 Functions and Variables for TeX Output

Function: tex (<expr>)
Function: tex (<expr>, <destination>)
Function: tex (<expr>, false)
Function: tex (<label>)
Function: tex (<label>, <destination>)
Function: tex (<label>, false)

     Prints a representation of an expression suitable for the TeX
     document preparation system.  The result is a fragment of a
     document, which can be copied into a larger document but not
     processed by itself.

     `tex (<expr>)' prints a TeX representation of <expr> on the
     console.

     `tex (<label>)' prints a TeX representation of the expression
     named by <label> and assigns it an equation label (to be displayed
     to the left of the expression).  The TeX equation label is the
     same as the Maxima label.

     <destination> may be an output stream or file name.  When
     <destination> is a file name, `tex' appends its output to the
     file.  The functions `openw' and `opena' create output streams.

     `tex (<expr>, false)' and `tex (<label>, false)' return their TeX
     output as a string.

     `tex' evaluates its first argument after testing it to see if it
     is a label.  Quote-quote `''' forces evaluation of the argument,
     thereby defeating the test and preventing the label.

     See also `texput'.

     Examples:

          (%i1) integrate (1/(1+x^3), x);
                                              2 x - 1
                            2            atan(-------)
                       log(x  - x + 1)        sqrt(3)    log(x + 1)
          (%o1)      - --------------- + ------------- + ----------
                              6             sqrt(3)          3
          (%i2) tex (%o1);
          $$-{{\log \left(x^2-x+1\right)}\over{6}}+{{\arctan \left({{2\,x-1
           }\over{\sqrt{3}}}\right)}\over{\sqrt{3}}}+{{\log \left(x+1\right)
           }\over{3}}\leqno{\tt (\%o1)}$$
          (%o2)                          (\%o1)
          (%i3) tex (integrate (sin(x), x));
          $$-\cos x$$
          (%o3)                           false
          (%i4) tex (%o1, "foo.tex");
          (%o4)                          (\%o1)

     `tex (<expr>, false)' returns its TeX output as a string.

          (%i1) S : tex (x * y * z, false);
          (%o1) $$x\,y\,z$$
          (%i2) S;
          (%o2) $$x\,y\,z$$

Function: tex1 (<e>)

     Returns a string which represents the TeX output for the
     expressions <e>.  The TeX output is not enclosed in delimiters for
     an equation or any other environment.

     Examples:

          (%i1) tex1 (sin(x) + cos(x));
          (%o1)                     \sin x+\cos x

Function: texput (<a>, <s>)
Function: texput (<a>, <f>)
Function: texput (<a>, <s>, <operator_type>)
Function: texput (<a>, [<s_1>, <s_2>], matchfix)
Function: texput (<a>, [<s_1>, <s_2>, <s_3>], matchfix)

     Assign the TeX output for the atom <a>, which can be a symbol or
     the name of an operator.

     `texput (<a>, <s>)' causes the `tex' function to interpolate the
     string <s> into the TeX output in place of <a>.

     `texput (<a>, <f>)' causes the `tex' function to call the function
     <f> to generate TeX output.  <f> must accept one argument, which
     is an expression which has operator <a>, and must return a string
     (the TeX output).  <f> may call `tex1' to generate TeX output for
     the arguments of the input expression.

     `texput (<a>, <s>, <operator_type>)', where <operator_type> is
     `prefix', `infix', `postfix', `nary', or `nofix', causes the `tex'
     function to interpolate <s> into the TeX output in place of <a>,
     and to place the interpolated text in the appropriate position.

     `texput (<a>, [<s_1>, <s_2>], matchfix)' causes the `tex' function
     to interpolate <s_1> and <s_2> into the TeX output on either side
     of the arguments of <a>.  The arguments (if more than one) are
     separated by commas.

     `texput (<a>, [<s_1>, <s_2>, <s_3>], matchfix)' causes the `tex'
     function to interpolate <s_1> and <s_2> into the TeX output on
     either side of the arguments of <a>, with <s_3> separating the
     arguments.

     Examples:

     Assign TeX output for a variable.

          (%i1) texput (me,"\\mu_e");
          (%o1)                         \mu_e
          (%i2) tex (me);
          $$\mu_e$$
          (%o2)                         false

     Assign TeX output for an ordinary function (not an operator).

          (%i1) texput (lcm, "\\mathrm{lcm}");
          (%o1)                     \mathrm{lcm}
          (%i2) tex (lcm (a, b));
          $$\mathrm{lcm}\left(a , b\right)$$
          (%o2)                         false

     Call a function to generate TeX output.

          (%i1) texfoo (e) := block ([a, b], [a, b] : args (e),
            concat("\\left[\\stackrel{",tex1(b),"}{",tex1(a),"}\\right]"))$
          (%i2) texput (foo, texfoo);
          (%o2)                        texfoo
          (%i3) tex (foo (2^x, %pi));
          $$\left[\stackrel{\pi}{2^{x}}\right]$$
          (%o3)                         false

     Assign TeX output for a prefix operator.

          (%i1) prefix ("grad");
          (%o1)                         grad
          (%i2) texput ("grad", " \\nabla ", prefix);
          (%o2)                        \nabla
          (%i3) tex (grad f);
          $$ \nabla f$$
          (%o3)                         false

     Assign TeX output for an infix operator.

          (%i1) infix ("~");
          (%o1)                           ~
          (%i2) texput ("~", " \\times ", infix);
          (%o2)                        \times
          (%i3) tex (a ~ b);
          $$a \times b$$
          (%o3)                         false

     Assign TeX output for a postfix operator.

          (%i1) postfix ("##");
          (%o1)                          ##
          (%i2) texput ("##", "!!", postfix);
          (%o2)                          !!
          (%i3) tex (x ##);
          $$x!!$$
          (%o3)                         false

     Assign TeX output for a nary operator.

          (%i1) nary ("@@");
          (%o1)                          @@
          (%i2) texput ("@@", " \\circ ", nary);
          (%o2)                         \circ
          (%i3) tex (a @@ b @@ c @@ d);
          $$a \circ b \circ c \circ d$$
          (%o3)                         false

     Assign TeX output for a nofix operator.

          (%i1) nofix ("foo");
          (%o1)                          foo
          (%i2) texput ("foo", "\\mathsc{foo}", nofix);
          (%o2)                     \mathsc{foo}
          (%i3) tex (foo);
          $$\mathsc{foo}$$
          (%o3)                         false

     Assign TeX output for a matchfix operator.

          (%i1) matchfix ("<<", ">>");
          (%o1)                          <<
          (%i2) texput ("<<", [" \\langle ", " \\rangle "], matchfix);
          (%o2)                [ \langle ,  \rangle ]
          (%i3) tex (<<a>>);
          $$ \langle a \rangle $$
          (%o3)                         false
          (%i4) tex (<<a, b>>);
          $$ \langle a , b \rangle $$
          (%o4)                         false
          (%i5) texput ("<<", [" \\langle ", " \\rangle ", " \\, | \\,"],
                matchfix);
          (%o5)           [ \langle ,  \rangle ,  \, | \,]
          (%i6) tex (<<a>>);
          $$ \langle a \rangle $$
          (%o6)                         false
          (%i7) tex (<<a, b>>);
          $$ \langle a \, | \,b \rangle $$
          (%o7)                         false

Function: get_tex_environment (<op>)
Function: set_tex_environment (<op>, <before>, <after>)

     Customize the TeX environment output by `tex'.  As maintained by
     these functions, the TeX environment comprises two strings: one is
     printed before any other TeX output, and the other is printed
     after.

     Only the TeX environment of the top-level operator in an expression
     is output; TeX environments associated with other operators are
     ignored.

     `get_tex_environment' returns the TeX enviroment which is applied
     to the operator <op>; returns the default if no other environment
     has been assigned.

     `set_tex_environment' assigns the TeX environment for the operator
     <op>.

     Examples:

          (%i1) get_tex_environment (":=");
          (%o1) [
          \begin{verbatim}
          , ;
          \end{verbatim}
          ]
          (%i2) tex (f (x) := 1 - x);

          \begin{verbatim}
          f(x):=1-x;
          \end{verbatim}

          (%o2)                         false
          (%i3) set_tex_environment (":=", "$$", "$$");
          (%o3)                       [$$, $$]
          (%i4) tex (f (x) := 1 - x);
          $$f(x):=1-x$$
          (%o4)                         false

Function: get_tex_environment_default ()
Function: set_tex_environment_default (<before>, <after>)

     Customize the TeX environment output by `tex'.  As maintained by
     these functions, the TeX environment comprises two strings: one is
     printed before any other TeX output, and the other is printed
     after.

     `get_tex_environment_default' returns the TeX environment which is
     applied to expressions for which the top-level operator has no
     specific TeX environment (as assigned by `set_tex_environment').

     `set_tex_environment_default' assigns the default TeX environment.

     Examples:

          (%i1) get_tex_environment_default ();
          (%o1)                       [$$, $$]
          (%i2) tex (f(x) + g(x));
          $$g\left(x\right)+f\left(x\right)$$
          (%o2)                         false
          (%i3) set_tex_environment_default ("\\begin{equation}
          ", "
          \\end{equation}");
          (%o3) [\begin{equation}
          ,
          \end{equation}]
          (%i4) tex (f(x) + g(x));
          \begin{equation}
          g\left(x\right)+f\left(x\right)
          \end{equation}
          (%o4)                         false

13.5 Functions and Variables for Fortran Output

Option variable: fortindent

     Default value: `0'

     `fortindent' controls the left margin indentation of expressions
     printed out by the `fortran'  command.  `0' gives normal printout
     (i.e., 6 spaces), and positive values will causes the expressions
     to be printed farther to the right.

Function: fortran (<expr>)

     Prints <expr> as a Fortran statement.  The output line is indented
     with spaces.  If the line is too long, `fortran' prints
     continuation lines.  `fortran' prints the exponentiation operator
     `^' as `**', and prints a complex number `a + b %i' in the form
     `(a,b)'.

     <expr> may be an equation.  If so, `fortran' prints an assignment
     statement, assigning the right-hand side of the equation to the
     left-hand side.  In particular, if the right-hand side of <expr>
     is the name of a matrix, then `fortran' prints an assignment
     statement for each element of the matrix.

     If <expr> is not something recognized by `fortran', the expression
     is printed in `grind'  format without complaint.  `fortran' does
     not know about lists, arrays, or functions.

     `fortindent'  controls the left margin of the printed lines.  `0'
     is the normal margin (i.e., indented 6 spaces).  Increasing
     `fortindent' causes expressions to be printed further to the right.

     When `fortspaces'  is `true', `fortran' fills out each printed
     line with spaces to 80 columns.

     `fortran' evaluates its arguments; quoting an argument defeats
     evaluation.  `fortran' always returns `done'.

     See also the function `f90'  for printing one or more expressions
     as a Fortran 90 program.

     Examples:

     (%i1) expr: (a + b)^12$
     (%i2) fortran (expr);
           (b+a)**12
     (%o2)                         done
     (%i3) fortran ('x=expr);
           x = (b+a)**12
     (%o3)                         done
     (%i4) fortran ('x=expand (expr));
           x = b**12+12*a*b**11+66*a**2*b**10+220*a**3*b**9+495*a**4*b**8+792
          1   *a**5*b**7+924*a**6*b**6+792*a**7*b**5+495*a**8*b**4+220*a**9*b
          2   **3+66*a**10*b**2+12*a**11*b+a**12
     (%o4)                         done
     (%i5) fortran ('x=7+5*%i);
           x = (7,5)
     (%o5)                         done
     (%i6) fortran ('x=[1,2,3,4]);
           x = [1,2,3,4]
     (%o6)                         done
     (%i7) f(x) := x^2$
     (%i8) fortran (f);
           f
     (%o8)                         done

Option variable: fortspaces

     Default value: `false'

     When `fortspaces' is `true', `fortran' fills out each printed line
     with spaces to 80 columns.

14 Polynomials


Introduction to Polynomials
Functions and Variables for Polynomials

14.1 Introduction to Polynomials

Polynomials are stored in Maxima either in General Form or as
Cannonical Rational Expressions (CRE) form.  The latter is a standard
form, and is used internally by operations such as factor, ratsimp, and
so on.

   Canonical Rational Expressions constitute a kind of representation
which is especially suitable for expanded polynomials and rational
functions (as well as for partially factored polynomials and rational
functions when RATFAC is set to `true').  In this CRE form an ordering
of variables (from most to least main) is assumed for each expression.
Polynomials are represented recursively by a list consisting of the
main variable followed by a series of pairs of expressions, one for
each term of the polynomial.  The first member of each pair is the
exponent of the main variable in that term and the second member is the
coefficient of that term which could be a number or a polynomial in
another variable again represented in this form.  Thus the principal
part of the CRE form of 3*X^2-1 is (X 2 3 0 -1) and that of 2*X*Y+X-3
is (Y 1 (X 1 2) 0 (X 1 1 0 -3)) assuming Y is the main variable, and is
(X 1 (Y 1 2 0 1) 0 -3) assuming X is the main variable.  "Main"-ness is
usually determined by reverse alphabetical order.  The "variables" of a
CRE expression needn't be atomic.  In fact any subexpression whose main
operator is not + - * / or ^ with integer power will be considered a
"variable" of the expression (in CRE form) in which it occurs.  For
example the CRE variables of the expression X+SIN(X+1)+2*SQRT(X)+1 are
X, SQRT(X), and SIN(X+1).  If the user does not specify an ordering of
variables by using the RATVARS function Maxima will choose an
alphabetic one.  In general, CRE's represent rational expressions, that
is, ratios of polynomials, where the numerator and denominator have no
common factors, and the denominator is positive.  The internal form is
essentially a pair of polynomials (the numerator and denominator)
preceded by the variable ordering list.  If an expression to be
displayed is in CRE form or if it contains any subexpressions in CRE
form, the symbol /R/ will follow the line label.  See the RAT function
for converting an expression to CRE form.  An extended CRE form is used
for the representation of Taylor series.  The notion of a rational
expression is extended so that the exponents of the variables can be
positive or negative rational numbers rather than just positive
integers and the coefficients can themselves be rational expressions as
described above rather than just polynomials.  These are represented
internally by a recursive polynomial form which is similar to and is a
generalization of CRE form, but carries additional information such as
the degree of truncation.  As with CRE form, the symbol /T/ follows the
line label of such expressions.

14.2 Functions and Variables for Polynomials

Option variable: algebraic

     Default value: `false'

     `algebraic' must be set to `true' in order for the simplification
     of algebraic integers to take effect.

Option variable: berlefact

     Default value: `true'

     When `berlefact' is `false' then the Kronecker factoring algorithm
     will be used otherwise the Berlekamp algorithm, which is the
     default, will be used.

Function: bezout (<p1>, <p2>, <x>)

     an alternative to the `resultant'  command.  It returns a matrix.
     `determinant' of this matrix is the desired resultant.

     Examples:

          (%i1) bezout(a*x+b, c*x^2+d, x);
                                   [ b c  - a d ]
          (%o1)                    [            ]
                                   [  a     b   ]
          (%i2) determinant(%);
                                      2      2
          (%o2)                      a  d + b  c
          (%i3) resultant(a*x+b, c*x^2+d, x);
                                      2      2
          (%o3)                      a  d + b  c

Function: bothcoef (<expr>, <x>)

     Returns a list whose first member is the coefficient of <x> in
     <expr> (as found by `ratcoef' if <expr> is in CRE form otherwise
     by `coeff') and whose second member is the remaining part of
     <expr>.  That is, `[A, B]' where `<expr> = A*<x> + B'.

     Example:

          (%i1) islinear (expr, x) := block ([c],
                  c: bothcoef (rat (expr, x), x),
                  is (freeof (x, c) and c[1] # 0))$
          (%i2) islinear ((r^2 - (x - r)^2)/x, x);
          (%o2)                         true

Function: coeff (<expr>, <x>, <n>)
Function: coeff (<expr>, <x>)

     Returns the coefficient of `<x>^<n>' in <expr>, where <expr> is a
     polynomial or a monomial term in <x>.

     `coeff(<expr>, <x>^<n>)' is equivalent to `coeff(<expr>, <x>,
     <n>)'.  `coeff(<expr>, <x>, 0)' returns the remainder of <expr>
     which is free of <x>.  If omitted, <n> is assumed to be 1.

     <x> may be a simple variable or a subscripted variable, or a
     subexpression of <expr> which comprises an operator and all of its
     arguments.

     It may be possible to compute coefficients of expressions which
     are equivalent to <expr> by applying `expand' or `factor'.
     `coeff' itself does not apply `expand' or `factor' or any other
     function.

     `coeff' distributes over lists, matrices, and equations.

     Examples:

     `coeff' returns the coefficient `<x>^<n>' in <expr>.

          (%i1) coeff (b^3*a^3 + b^2*a^2 + b*a + 1, a^3);
                                          3
          (%o1)                          b

     `coeff(<expr>, <x>^<n>)' is equivalent to `coeff(<expr>, <x>,
     <n>)'.

          (%i1) coeff (c[4]*z^4 - c[3]*z^3 - c[2]*z^2 + c[1]*z, z, 3);
          (%o1)                         - c
                                           3
          (%i2) coeff (c[4]*z^4 - c[3]*z^3 - c[2]*z^2 + c[1]*z, z^3);
          (%o2)                         - c
                                           3

     `coeff(<expr>, <x>, 0)' returns the remainder of <expr> which is
     free of <x>.

          (%i1) coeff (a*u + b^2*u^2 + c^3*u^3, b, 0);
                                      3  3
          (%o1)                      c  u  + a u

     <x> may be a simple variable or a subscripted variable, or a
     subexpression of <expr> which comprises an operator and all of its
     arguments.

          (%i1) coeff (h^4 - 2*%pi*h^2 + 1, h, 2);
          (%o1)                        - 2 %pi
          (%i2) coeff (v[1]^4 - 2*%pi*v[1]^2 + 1, v[1], 2);
          (%o2)                        - 2 %pi
          (%i3) coeff (sin(1+x)*sin(x) + sin(1+x)^3*sin(x)^3, sin(1+x)^3);
                                          3
          (%o3)                        sin (x)
          (%i4) coeff ((d - a)^2*(b + c)^3 + (a + b)^4*(c - d), a + b, 4);
          (%o4)                         c - d

     `coeff' itself does not apply `expand' or `factor' or any other
     function.

          (%i1) coeff (c*(a + b)^3, a);
          (%o1)                           0
          (%i2) expand (c*(a + b)^3);
                           3          2        2        3
          (%o2)           b  c + 3 a b  c + 3 a  b c + a  c
          (%i3) coeff (%, a);
                                          2
          (%o3)                        3 b  c
          (%i4) coeff (b^3*c + 3*a*b^2*c + 3*a^2*b*c + a^3*c, (a + b)^3);
          (%o4)                           0
          (%i5) factor (b^3*c + 3*a*b^2*c + 3*a^2*b*c + a^3*c);
                                            3
          (%o5)                      (b + a)  c
          (%i6) coeff (%, (a + b)^3);
          (%o6)                           c

     `coeff' distributes over lists, matrices, and equations.

          (%i1) coeff ([4*a, -3*a, 2*a], a);
          (%o1)                      [4, - 3, 2]
          (%i2) coeff (matrix ([a*x, b*x], [-c*x, -d*x]), x);
                                    [  a    b  ]
          (%o2)                     [          ]
                                    [ - c  - d ]
          (%i3) coeff (a*u - b*v = 7*u + 3*v, u);
          (%o3)                         a = 7

Function: content (<p_1>, <x_1>, ..., <x_n>)

     Returns a list whose first element is the greatest common divisor
     of the coefficients of the terms of the polynomial <p_1> in the
     variable <x_n> (this is the content) and whose second element is
     the polynomial <p_1> divided by the content.

     Examples:

          (%i1) content (2*x*y + 4*x^2*y^2, y);
                                             2
          (%o1)                   [2 x, 2 x y  + y]

Function: denom (<expr>)

     Returns the denominator of the rational expression <expr>.

Function: divide (<p_1>, <p_2>, <x_1>, ..., <x_n>)

     computes the quotient and remainder of the polynomial <p_1>
     divided by the polynomial <p_2>, in a main polynomial variable,
     <x_n>.  The other variables are as in the `ratvars' function.  The
     result is a list whose first element is the quotient and whose
     second element is the remainder.

     Examples:

          (%i1) divide (x + y, x - y, x);
          (%o1)                       [1, 2 y]
          (%i2) divide (x + y, x - y);
          (%o2)                      [- 1, 2 x]

     Note that `y' is the main variable in the second example.

Function: eliminate ([<eqn_1>, ..., <eqn_n>], [<x_1>, ..., <x_k>])

     Eliminates variables from equations (or expressions assumed equal
     to zero) by taking successive resultants. This returns a list of
     `<n> - <k>' expressions with the <k> variables <x_1>, ..., <x_k>
     eliminated.  First <x_1> is eliminated yielding `<n> - 1'
     expressions, then `x_2' is eliminated, etc.  If `<k> = <n>' then a
     single expression in a list is returned free of the variables
     <x_1>, ..., <x_k>.  In this case `solve' is called to solve the
     last resultant for the last variable.

     Example:

          (%i1) expr1: 2*x^2 + y*x + z;
                                                2
          (%o1)                    z + x y + 2 x
          (%i2) expr2: 3*x + 5*y - z - 1;
          (%o2)                  - z + 5 y + 3 x - 1
          (%i3) expr3: z^2 + x - y^2 + 5;
                                    2    2
          (%o3)                    z  - y  + x + 5
          (%i4) eliminate ([expr3, expr2, expr1], [y, z]);
                       8         7         6          5          4
          (%o4) [7425 x  - 1170 x  + 1299 x  + 12076 x  + 22887 x

                                              3         2
                                      - 5154 x  - 1291 x  + 7688 x + 15376]

Function: ezgcd (<p_1>, <p_2>, <p_3>, ...)

     Returns a list whose first element is the greatest common divisor
     of the polynomials <p_1>, <p_2>, <p_3>, ... and whose remaining
     elements are the polynomials divided by the greatest common
     divisor.  This always uses the `ezgcd' algorithm.

     See also `gcd',  `gcdex',  `gcdivide',  and `poly_gcd'.

     Examples:

     The three polynomials have the greatest common divisor `2*x-3'.
     The gcd is first calculated with the function `gcd' and then with
     the function `ezgcd'.

          (%i1) p1 : 6*x^3-17*x^2+14*x-3;
                                  3       2
          (%o1)                6 x  - 17 x  + 14 x - 3
          (%i2) p2 : 4*x^4-14*x^3+12*x^2+2*x-3;
                              4       3       2
          (%o2)            4 x  - 14 x  + 12 x  + 2 x - 3
          (%i3) p3 : -8*x^3+14*x^2-x-3;
                                    3       2
          (%o3)                - 8 x  + 14 x  - x - 3

          (%i4) gcd(p1, gcd(p2, p3));
          (%o4)                        2 x - 3

          (%i5) ezgcd(p1, p2, p3);
                             2               3      2           2
          (%o5) [2 x - 3, 3 x  - 4 x + 1, 2 x  - 4 x  + 1, - 4 x  + x + 1]

Option variable: facexpand

     Default value: `true'

     `facexpand' controls whether the irreducible factors returned by
     `factor' are in expanded (the default) or recursive (normal CRE)
     form.

Function: factor (<expr>)
Function: factor (<expr>, <p>)

     Factors the expression <expr>, containing any number of variables
     or functions, into factors irreducible over the integers.  `factor
     (<expr>, <p>)' factors <expr> over the field of rationals with an
     element adjoined whose minimum polynomial is <p>.

     `factor' uses `ifactors' function for factoring integers.

     `factorflag' if `false' suppresses the factoring of integer factors
     of rational expressions.

     `dontfactor' may be set to a list of variables with respect to
     which factoring is not to occur.  (It is initially empty).
     Factoring also will not take place with respect to any variables
     which are less important (using the variable ordering assumed for
     CRE form) than those on the `dontfactor' list.

     `savefactors' if `true' causes the factors of an expression which
     is a product of factors to be saved by certain functions in order
     to speed up later factorizations of expressions containing some of
     the same factors.

     `berlefact' if `false' then the Kronecker factoring algorithm will
     be used otherwise the Berlekamp algorithm, which is the default,
     will be used.

     `intfaclim' if `true' maxima will give up factorization of
     integers if no factor is found after trial divisions and Pollard's
     rho method.  If set to `false' (this is the case when the user
     calls `factor' explicitly), complete factorization of the integer
     will be attempted.  The user's setting of `intfaclim' is used for
     internal calls to `factor'.  Thus, `intfaclim' may be reset to
     prevent Maxima from taking an inordinately long time factoring
     large integers.

     Examples:

          (%i1) factor (2^63 - 1);
                              2
          (%o1)              7  73 127 337 92737 649657
          (%i2) factor (-8*y - 4*x + z^2*(2*y + x));
          (%o2)               (2 y + x) (z - 2) (z + 2)
          (%i3) -1 - 2*x - x^2 + y^2 + 2*x*y^2 + x^2*y^2;
                          2  2        2    2    2
          (%o3)          x  y  + 2 x y  + y  - x  - 2 x - 1
          (%i4) block ([dontfactor: [x]], factor (%/36/(1 + 2*y + y^2)));
                                 2
                               (x  + 2 x + 1) (y - 1)
          (%o4)                ----------------------
                                     36 (y + 1)
          (%i5) factor (1 + %e^(3*x));
                                x         2 x     x
          (%o5)              (%e  + 1) (%e    - %e  + 1)
          (%i6) factor (1 + x^4, a^2 - 2);
                              2              2
          (%o6)             (x  - a x + 1) (x  + a x + 1)
          (%i7) factor (-y^2*z^2 - x*z^2 + x^2*y^2 + x^3);
                                 2
          (%o7)              - (y  + x) (z - x) (z + x)
          (%i8) (2 + x)/(3 + x)/(b + x)/(c + x)^2;
                                       x + 2
          (%o8)               ------------------------
                                                     2
                              (x + 3) (x + b) (x + c)
          (%i9) ratsimp (%);
                          4                  3
          (%o9) (x + 2)/(x  + (2 c + b + 3) x

               2                       2             2                   2
           + (c  + (2 b + 6) c + 3 b) x  + ((b + 3) c  + 6 b c) x + 3 b c )
          (%i10) partfrac (%, x);
                     2                   4                3
          (%o10) - (c  - 4 c - b + 6)/((c  + (- 2 b - 6) c

               2              2         2                2
           + (b  + 12 b + 9) c  + (- 6 b  - 18 b) c + 9 b ) (x + c))

                           c - 2
           - ---------------------------------
               2                             2
             (c  + (- b - 3) c + 3 b) (x + c)

                                   b - 2
           + -------------------------------------------------
                       2             2       3      2
             ((b - 3) c  + (6 b - 2 b ) c + b  - 3 b ) (x + b)

                                   1
           - ----------------------------------------------
                       2
             ((b - 3) c  + (18 - 6 b) c + 9 b - 27) (x + 3)
          (%i11) map ('factor, %);
                        2
                       c  - 4 c - b + 6                 c - 2
          (%o11) - ------------------------- - ------------------------
                          2        2                                  2
                   (c - 3)  (c - b)  (x + c)   (c - 3) (c - b) (x + c)

                                 b - 2                        1
                      + ------------------------ - ------------------------
                                       2                          2
                        (b - 3) (c - b)  (x + b)   (b - 3) (c - 3)  (x + 3)
          (%i12) ratsimp ((x^5 - 1)/(x - 1));
                                 4    3    2
          (%o12)                x  + x  + x  + x + 1
          (%i13) subst (a, x, %);
                                 4    3    2
          (%o13)                a  + a  + a  + a + 1
          (%i14) factor (%th(2), %);
                                 2        3        3    2
          (%o14)   (x - a) (x - a ) (x - a ) (x + a  + a  + a + 1)
          (%i15) factor (1 + x^12);
                                 4        8    4
          (%o15)               (x  + 1) (x  - x  + 1)
          (%i16) factor (1 + x^99);
                           2            6    3
          (%o16) (x + 1) (x  - x + 1) (x  - x  + 1)

             10    9    8    7    6    5    4    3    2
           (x   - x  + x  - x  + x  - x  + x  - x  + x  - x + 1)

             20    19    17    16    14    13    11    10    9    7    6
           (x   + x   - x   - x   + x   + x   - x   - x   - x  + x  + x

              4    3            60    57    51    48    42    39    33
           - x  - x  + x + 1) (x   + x   - x   - x   + x   + x   - x

              30    27    21    18    12    9    3
           - x   - x   + x   + x   - x   - x  + x  + 1)

Option variable: factorflag

     Default value: `false'

     When `factorflag' is `false', suppresses the factoring of integer
     factors of rational expressions.

Function: factorout (<expr>, <x_1>, <x_2>, ...)

     Rearranges the sum <expr> into a sum of terms of the form `f
     (<x_1>, <x_2>, ...)*g' where `g' is a product of expressions not
     containing any <x_i> and `f' is factored.

     Note that the option variable `keepfloat' is ignored by
     `factorout'.

     Example:

          (%i1) expand (a*(x+1)*(x-1)*(u+1)^2);
                       2  2          2      2      2
          (%o1)     a u  x  + 2 a u x  + a x  - a u  - 2 a u - a
          (%i2) factorout(%,x);
                   2
          (%o2) a u  (x - 1) (x + 1) + 2 a u (x - 1) (x + 1)
                                                        + a (x - 1) (x + 1)

Function: factorsum (<expr>)

     Tries to group terms in factors of <expr> which are sums into
     groups of terms such that their sum is factorable.  `factorsum'
     can recover the result of `expand ((x + y)^2 + (z + w)^2)' but it
     can't recover `expand ((x + 1)^2 + (x + y)^2)' because the terms
     have variables in common.

     Example:

          (%i1) expand ((x + 1)*((u + v)^2 + a*(w + z)^2));
                     2      2                            2      2
          (%o1) a x z  + a z  + 2 a w x z + 2 a w z + a w  x + v  x

                                               2        2    2            2
                                  + 2 u v x + u  x + a w  + v  + 2 u v + u
          (%i2) factorsum (%);
                                             2          2
          (%o2)            (x + 1) (a (z + w)  + (v + u) )

Function: fasttimes (<p_1>, <p_2>)

     Returns the product of the polynomials <p_1> and <p_2> by using a
     special algorithm for multiplication of polynomials.  `p_1' and
     `p_2' should be multivariate, dense, and nearly the same size.
     Classical multiplication is of order `n_1 n_2' where `n_1' is the
     degree of `p_1' and `n_2' is the degree of `p_2'.  `fasttimes' is
     of order `max (n_1, n_2)^1.585'.

Function: fullratsimp (<expr>)

     `fullratsimp' repeatedly applies `ratsimp' followed by
     non-rational simplification to an expression until no further
     change occurs, and returns the result.

     When non-rational expressions are involved, one call to `ratsimp'
     followed as is usual by non-rational ("general") simplification
     may not be sufficient to return a simplified result.  Sometimes,
     more than one such call may be necessary.  `fullratsimp' makes
     this process convenient.

     `fullratsimp (<expr>, <x_1>, ..., <x_n>)' takes one or more
     arguments similar to `ratsimp' and `rat'.

     Example:

          (%i1) expr: (x^(a/2) + 1)^2*(x^(a/2) - 1)^2/(x^a - 1);
                                 a/2     2   a/2     2
                               (x    - 1)  (x    + 1)
          (%o1)                -----------------------
                                        a
                                       x  - 1
          (%i2) ratsimp (expr);
                                    2 a      a
                                   x    - 2 x  + 1
          (%o2)                    ---------------
                                        a
                                       x  - 1
          (%i3) fullratsimp (expr);
                                        a
          (%o3)                        x  - 1
          (%i4) rat (expr);
                                 a/2 4       a/2 2
                               (x   )  - 2 (x   )  + 1
          (%o4)/R/             -----------------------
                                        a
                                       x  - 1

Function: fullratsubst (<a>, <b>, <c>)

     is the same as `ratsubst' except that it calls itself recursively
     on its result until that result stops changing.  This function is
     useful when the replacement expression and the replaced expression
     have one or more variables in common.

     `fullratsubst' will also accept its arguments in the format of
     `lratsubst'.  That is, the first argument may be a single
     substitution equation or a list of such equations, while the
     second argument is the expression being processed.

     `load ("lrats")' loads `fullratsubst' and `lratsubst'.

     Examples:

          (%i1) load ("lrats")$

        * `subst' can carry out multiple substitutions.  `lratsubst' is
          analogous to `subst'.

          (%i2) subst ([a = b, c = d], a + c);
          (%o2)                         d + b
          (%i3) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c));
          (%o3)                (d + a c) e + a d + b c

        * If only one substitution is desired, then a single equation
          may be given as first argument.

          (%i4) lratsubst (a^2 = b, a^3);
          (%o4)                          a b

        * `fullratsubst' is equivalent to `ratsubst' except that it
          recurses until its result stops changing.

          (%i5) ratsubst (b*a, a^2, a^3);
                                         2
          (%o5)                         a  b
          (%i6) fullratsubst (b*a, a^2, a^3);
                                           2
          (%o6)                         a b

        * `fullratsubst' also accepts a list of equations or a single
          equation as first argument.

          (%i7) fullratsubst ([a^2 = b, b^2 = c, c^2 = a], a^3*b*c);
          (%o7)                           b
          (%i8) fullratsubst (a^2 = b*a, a^3);
                                           2
          (%o8)                         a b

        * `fullratsubst' may cause an indefinite recursion.

          (%i9) errcatch (fullratsubst (b*a^2, a^2, a^3));

          *** - Lisp stack overflow. RESET

Function: gcd (<p_1>, <p_2>, <x_1>, ...)

     Returns the greatest common divisor of <p_1> and <p_2>.  The flag
     `gcd' determines which algorithm is employed.  Setting `gcd' to
     `ez', `subres', `red', or `spmod' selects the `ezgcd',
     subresultant `prs', reduced, or modular algorithm, respectively.
     If `gcd' `false' then `gcd (<p_1>, <p_2>, <x>)' always returns 1
     for all <x>.  Many functions (e.g. `ratsimp',

     `factor',  etc.) cause gcd's to be taken implicitly.  For
     homogeneous polynomials it is recommended that `gcd' equal to
     `subres' be used.  To take the gcd when an algebraic is present,
     e.g., `gcd (<x>^2 - 2*sqrt(2)* <x> + 2, <x> - sqrt(2))', the option
     variable `algebraic'  must be `true' and `gcd' must not be `ez'.

     The `gcd' flag, default: `spmod', if `false' will also prevent the
     greatest common divisor from being taken when expressions are
     converted to canonical rational expression (CRE) form.  This will
     sometimes speed the calculation if gcds are not required.

     See also `ezgcd',  `gcdex',  `gcdivide',  and `poly_gcd'.

     Example:

          (%i1) p1:6*x^3+19*x^2+19*x+6;
                                  3       2
          (%o1)                6 x  + 19 x  + 19 x + 6
          (%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                            5       4       3       2
          (%o2)          6 x  + 13 x  + 12 x  + 13 x  + 6 x
          (%i3) gcd(p1, p2);
                                      2
          (%o3)                    6 x  + 13 x + 6
          (%i4) p1/gcd(p1, p2), ratsimp;
          (%o4)                         x + 1
          (%i5) p2/gcd(p1, p2), ratsimp;
                                        3
          (%o5)                        x  + x

     `ezgcd'  returns a list whose first element is the greatest common
     divisor of the polynomials <p_1> and <p_2>, and whose remaining
     elements are the polynomials divided by the greatest common
     divisor.

          (%i6) ezgcd(p1, p2);
                              2                     3
          (%o6)           [6 x  + 13 x + 6, x + 1, x  + x]

Function: gcdex (<f>, <g>)
Function: gcdex (<f>, <g>, <x>)

     Returns a list `[<a>, <b>, <u>]' where <u> is the greatest common
     divisor (gcd) of <f> and <g>, and <u> is equal to `<a> <f> + <b>
     <g>'.  The arguments <f> and <g> should be univariate polynomials,
     or else polynomials in <x> a supplied main variable since we need
     to be in a principal ideal domain for this to work.  The gcd means
     the gcd regarding <f> and <g> as univariate polynomials with
     coefficients being rational functions in the other variables.

     `gcdex' implements the Euclidean algorithm, where we have a
     sequence of `L[i]: [a[i], b[i], r[i]]' which are all perpendicular
     to `[f, g, -1]' and the next one is built as if `q =
     quotient(r[i]/r[i+1])' then `L[i+2]: L[i] - q L[i+1]', and it
     terminates at `L[i+1]' when the remainder `r[i+2]' is zero.

     The arguments <f> and <g> can be integers.  For this case the
     function `igcdex'  is called by `gcdex'.

     See also `ezgcd',  `gcd',  `gcdivide',  and `poly_gcd'.

     Examples:

          (%i1) gcdex (x^2 + 1, x^3 + 4);
                                 2
                                x  + 4 x - 1  x + 4
          (%o1)/R/           [- ------------, -----, 1]
                                     17        17
          (%i2) % . [x^2 + 1, x^3 + 4, -1];
          (%o2)/R/                        0

     Note that the gcd in the following is `1' since we work in
     `k(y)[x]', not the  `y+1' we would expect in `k[y, x]'.

          (%i1) gcdex (x*(y + 1), y^2 - 1, x);
                                         1
          (%o1)/R/                 [0, ------, 1]
                                        2
                                       y  - 1

Function: gcfactor (<n>)

     Factors the Gaussian integer <n> over the Gaussian integers, i.e.,
     numbers of the form `<a> + <b> `%i'' where <a> and <b> are
     rational integers (i.e.,  ordinary integers).  Factors are
     normalized by making <a> and <b> non-negative.

Function: gfactor (<expr>)

     Factors the polynomial <expr> over the Gaussian integers (that is,
     the integers with the imaginary unit `%i' adjoined).  This is like
     `factor (<expr>, <a>^2+1)' where <a> is `%i'.

     Example:

          (%i1) gfactor (x^4 - 1);
          (%o1)           (x - 1) (x + 1) (x - %i) (x + %i)

Function: gfactorsum (<expr>)

     is similar to `factorsum' but applies `gfactor' instead of
     `factor'.

Function: hipow (<expr>, <x>)

     Returns the highest explicit exponent of <x> in <expr>.  <x> may
     be a variable or a general expression.  If <x> does not appear in
     <expr>, `hipow' returns `0'.

     `hipow' does not consider expressions equivalent to `expr'.  In
     particular, `hipow' does not expand `expr', so `hipow (<expr>,
     <x>)' and `hipow (expand (<expr>, <x>))' may yield different
     results.

     Examples:

          (%i1) hipow (y^3 * x^2 + x * y^4, x);
          (%o1)                           2
          (%i2) hipow ((x + y)^5, x);
          (%o2)                           1
          (%i3) hipow (expand ((x + y)^5), x);
          (%o3)                           5
          (%i4) hipow ((x + y)^5, x + y);
          (%o4)                           5
          (%i5) hipow (expand ((x + y)^5), x + y);
          (%o5)                           0

Option variable: intfaclim

     Default value: true

     If `true', maxima will give up factorization of integers if no
     factor is found after trial divisions and Pollard's rho method and
     factorization will not be complete.

     When `intfaclim' is `false' (this is the case when the user calls
     `factor' explicitly), complete factorization will be attempted.
     `intfaclim' is set to `false' when factors are computed in
     `divisors', `divsum' and `totient'.

     Internal calls to `factor' respect the user-specified value of
     `intfaclim'.  Setting `intfaclim' to `true' may reduce the time
     spent factoring large integers.

Option variable: keepfloat

     Default value: `false'

     When `keepfloat' is `true', prevents floating point numbers from
     being rationalized when expressions which contain them are
     converted to canonical rational expression (CRE) form.

     Note that the function `solve' and those functions calling it
     (`eigenvalues', for example) currently ignore this flag, converting
     floating point numbers anyway.

     Examples:

          (%i1) rat(x/2.0);

          `rat' replaced 0.5 by 1/2 = 0.5
                                                 x
          (%o1)/R/                               -
                                                 2
          (%i2) rat(x/2.0), keepfloat;

          (%o2)/R/                             0.5 x

     `solve' ignores `keepfloat':

          (%i3) solve(1.0-x,x), keepfloat;

          `rat' replaced 1.0 by 1/1 = 1.0
          (%o3)                               [x = 1]

Function: lopow (<expr>, <x>)

     Returns the lowest exponent of <x> which explicitly appears in
     <expr>.  Thus

          (%i1) lopow ((x+y)^2 + (x+y)^a, x+y);
          (%o1)                       min(a, 2)

Function: lratsubst (<L>, <expr>)

     is analogous to `subst (<L>, <expr>)' except that it uses
     `ratsubst' instead of `subst'.

     The first argument of `lratsubst' is an equation or a list of
     equations identical in format to that accepted by `subst'.  The
     substitutions are made in the order given by the list of equations,
     that is, from left to right.

     `load ("lrats")' loads `fullratsubst' and `lratsubst'.

     Examples:

          (%i1) load ("lrats")$

        * `subst' can carry out multiple substitutions.  `lratsubst' is
          analogous to `subst'.

          (%i2) subst ([a = b, c = d], a + c);
          (%o2)                         d + b
          (%i3) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c));
          (%o3)                (d + a c) e + a d + b c

        * If only one substitution is desired, then a single equation
          may be given as first argument.

          (%i4) lratsubst (a^2 = b, a^3);
          (%o4)                          a b

Option variable: modulus

     Default value: `false'

     When `modulus' is a positive number <p>, operations on rational
     numbers (as returned by `rat' and related functions) are carried
     out modulo <p>, using the so-called "balanced" modulus system in
     which `<n> modulo <p>' is defined as an integer <k> in
     `[-(<p>-1)/2, ..., 0, ..., (<p>-1)/2]' when <p> is odd, or
     `[-(<p>/2 - 1), ..., 0, ...., <p>/2]' when <p> is even, such that
     `<a> <p> + <k>' equals <n> for some integer <a>.

     If <expr> is already in canonical rational expression (CRE) form
     when `modulus' is reset, then you may need to re-rat <expr>, e.g.,
     `expr: rat (ratdisrep (expr))', in order to get correct results.

     Typically `modulus' is set to a prime number.  If `modulus' is set
     to a positive non-prime integer, this setting is accepted, but a
     warning message is displayed.  Maxima signals an error, when zero
     or a negative integer is assigned to `modulus'.

     Examples:

          (%i1) modulus:7;
          (%o1)                           7
          (%i2) polymod([0,1,2,3,4,5,6,7]);
          (%o2)            [0, 1, 2, 3, - 3, - 2, - 1, 0]
          (%i3) modulus:false;
          (%o3)                         false
          (%i4) poly:x^6+x^2+1;
                                      6    2
          (%o4)                      x  + x  + 1
          (%i5) factor(poly);
                                      6    2
          (%o5)                      x  + x  + 1
          (%i6) modulus:13;
          (%o6)                          13
          (%i7) factor(poly);
                                2        4      2
          (%o7)               (x  + 6) (x  - 6 x  - 2)
          (%i8) polymod(%);
                                      6    2
          (%o8)                      x  + x  + 1

Function: num (<expr>)

     Returns the numerator of <expr> if it is a ratio.  If <expr> is
     not a ratio, <expr> is returned.

     `num' evaluates its argument.

Function: polydecomp (<p>, <x>)

     Decomposes the polynomial <p> in the variable <x> into the
     functional composition of polynomials in <x>.  `polydecomp'
     returns a list `[<p_1>, ..., <p_n>]' such that

          lambda ([x], p_1) (lambda ([x], p_2) (... (lambda ([x], p_n) (x))
            ...))

     is equal to <p>.  The degree of <p_i> is greater than 1 for <i>
     less than <n>.

     Such a decomposition is not unique.

     Examples:

          (%i1) polydecomp (x^210, x);
                                    7   5   3   2
          (%o1)                   [x , x , x , x ]
          (%i2) p : expand (subst (x^3 - x - 1, x, x^2 - a));
                          6      4      3    2
          (%o2)          x  - 2 x  - 2 x  + x  + 2 x - a + 1
          (%i3) polydecomp (p, x);
                                  2       3
          (%o3)                 [x  - a, x  - x - 1]

     The following function composes `L = [e_1, ..., e_n]' as functions
     in `x'; it is the inverse of polydecomp:

          compose (L, x) :=
            block ([r : x], for e in L do r : subst (e, x, r), r) $

     Re-express above example using `compose':

          (%i3) polydecomp (compose ([x^2 - a, x^3 - x - 1], x), x);
                                  2       3
          (%o3)                 [x  - a, x  - x - 1]

     Note that though `compose (polydecomp (<p>, <x>), <x>)' always
     returns <p> (unexpanded), `polydecomp (compose ([<p_1>, ...,
     <p_n>], <x>), <x>)' does not necessarily return `[<p_1>, ...,
     <p_n>]':

          (%i4) polydecomp (compose ([x^2 + 2*x + 3, x^2], x), x);
                                    2       2
          (%o4)                   [x  + 2, x  + 1]
          (%i5) polydecomp (compose ([x^2 + x + 1, x^2 + x + 1], x), x);
                                2       2
                               x  + 3  x  + 5
          (%o5)               [------, ------, 2 x + 1]
                                 4       2

Function: polymod (<p>)
Function: polymod (<p>, <m>)

     Converts the polynomial <p> to a modular representation with
     respect to the current modulus which is the value of the variable
     `modulus'.

     `polymod (<p>, <m>)' specifies a modulus <m> to be used instead of
     the current value of `modulus'.

     See `modulus'.

Function: powers (<expr>, <x>)

     Gives the powers of <x> occuring in <expr>.

     `load (powers)' loads this function.

Function: quotient (<p_1>, <p_2>)
Function: quotient (<p_1>, <p_2>, <x_1>, ..., <x_n>)

     Returns the polynomial <p_1> divided by the polynomial <p_2>.  The
     arguments <x_1>, ..., <x_n> are interpreted as in `ratvars'.

     `quotient' returns the first element of the two-element list
     returned by `divide'.

Function: rat (<expr>)
Function: rat (<expr>, <x_1>, ..., <x_n>)

     Converts <expr> to canonical rational expression (CRE) form by
     expanding and combining all terms over a common denominator and
     cancelling out the greatest common divisor of the numerator and
     denominator, as well as converting floating point numbers to
     rational numbers within a tolerance of `ratepsilon'.  The
     variables are ordered according to the <x_1>, ..., <x_n>, if
     specified, as in `ratvars'.

     `rat' does not generally simplify functions other than addition
     `+', subtraction `-', multiplication `*', division `/', and
     exponentiation to an integer power, whereas `ratsimp' does handle
     those cases.  Note that atoms (numbers and variables) in CRE form
     are not the same as they are in the general form.  For example,
     `rat(x)- x' yields `rat(0)' which has a different internal
     representation than 0.

     When `ratfac' is `true', `rat' yields a partially factored form
     for CRE.  During rational operations the expression is maintained
     as fully factored as possible without an actual call to the factor
     package.  This should always save space and may save some time in
     some computations.  The numerator and denominator are still made
     relatively prime (e.g.  `rat ((x^2 - 1)^4/(x + 1)^2)' yields `(x -
     1)^4 (x + 1)^2)', but the factors within each part may not be
     relatively prime.

     `ratprint' if `false' suppresses the printout of the message
     informing the user of the conversion of floating point numbers to
     rational numbers.

     `keepfloat' if `true' prevents floating point numbers from being
     converted to rational numbers.

     See also `ratexpand' and  `ratsimp'.

     Examples:
          (%i1) ((x - 2*y)^4/(x^2 - 4*y^2)^2 + 1)*(y + a)*(2*y + x) /
                (4*y^2 + x^2);
                                                     4
                                            (x - 2 y)
                        (y + a) (2 y + x) (------------ + 1)
                                             2      2 2
                                           (x  - 4 y )
          (%o1)         ------------------------------------
                                        2    2
                                     4 y  + x
          (%i2) rat (%, y, a, x);
                                      2 a + 2 y
          (%o2)/R/                    ---------
                                       x + 2 y

Option variable: ratalgdenom

     Default value: `true'

     When `ratalgdenom' is `true', allows rationalization of
     denominators with respect to radicals to take effect.
     `ratalgdenom' has an effect only when canonical rational
     expressions (CRE) are used in algebraic mode.

Function: ratcoef (<expr>, <x>, <n>)
Function: ratcoef (<expr>, <x>)

     Returns the coefficient of the expression `<x>^<n>' in the
     expression <expr>.  If omitted, <n> is assumed to be 1.

     The return value is free (except possibly in a non-rational sense)
     of the variables in <x>.  If no coefficient of this type exists, 0
     is returned.

     `ratcoef' expands and rationally simplifies its first argument and
     thus it may produce answers different from those of `coeff' which
     is purely syntactic.  Thus `ratcoef ((x + 1)/y + x, x)' returns
     `(y + 1)/y' whereas `coeff' returns 1.

     `ratcoef (<expr>, <x>, 0)', viewing <expr> as a sum, returns a sum
     of those terms which do not contain <x>.  Therefore if <x> occurs
     to any negative powers, `ratcoef' should not be used.

     Since <expr> is rationally simplified before it is examined,
     coefficients may not appear quite the way they were envisioned.

     Example:

          (%i1) s: a*x + b*x + 5$
          (%i2) ratcoef (s, a + b);
          (%o2)                           x

Function: ratdenom (<expr>)

     Returns the denominator of <expr>, after coercing <expr> to a
     canonical rational expression (CRE).  The return value is a CRE.

     <expr> is coerced to a CRE by `rat' if it is not already a CRE.
     This conversion may change the form of <expr> by putting all terms
     over a common denominator.

     `denom' is similar, but returns an ordinary expression instead of
     a CRE.  Also, `denom' does not attempt to place all terms over a
     common denominator, and thus some expressions which are considered
     ratios by `ratdenom' are not considered ratios by `denom'.

Option variable: ratdenomdivide

     Default value: `true'

     When `ratdenomdivide' is `true', `ratexpand' expands a ratio in
     which the numerator is a sum into a sum of ratios, all having a
     common denominator.  Otherwise, `ratexpand' collapses a sum of
     ratios into a single ratio, the numerator of which is the sum of
     the numerators of each ratio.

     Examples:

          (%i1) expr: (x^2 + x + 1)/(y^2 + 7);
                                      2
                                     x  + x + 1
          (%o1)                      ----------
                                        2
                                       y  + 7
          (%i2) ratdenomdivide: true$
          (%i3) ratexpand (expr);
                                 2
                                x        x        1
          (%o3)               ------ + ------ + ------
                               2        2        2
                              y  + 7   y  + 7   y  + 7
          (%i4) ratdenomdivide: false$
          (%i5) ratexpand (expr);
                                      2
                                     x  + x + 1
          (%o5)                      ----------
                                        2
                                       y  + 7
          (%i6) expr2: a^2/(b^2 + 3) + b/(b^2 + 3);
                                               2
                                     b        a
          (%o6)                    ------ + ------
                                    2        2
                                   b  + 3   b  + 3
          (%i7) ratexpand (expr2);
                                            2
                                       b + a
          (%o7)                        ------
                                        2
                                       b  + 3

Function: ratdiff (<expr>, <x>)

     Differentiates the rational expression <expr> with respect to <x>.
     <expr> must be a ratio of polynomials or a polynomial in <x>.  The
     argument <x> may be a variable or a subexpression of <expr>.

     The result is equivalent to `diff', although perhaps in a
     different form.  `ratdiff' may be faster than `diff', for rational
     expressions.

     `ratdiff' returns a canonical rational expression (CRE) if `expr'
     is a CRE.  Otherwise, `ratdiff' returns a general expression.

     `ratdiff' considers only the dependence of <expr> on <x>, and
     ignores any dependencies established by `depends'.

     Example:

          (%i1) expr: (4*x^3 + 10*x - 11)/(x^5 + 5);
                                     3
                                  4 x  + 10 x - 11
          (%o1)                   ----------------
                                        5
                                       x  + 5
          (%i2) ratdiff (expr, x);
                              7       5       4       2
                           8 x  + 40 x  - 55 x  - 60 x  - 50
          (%o2)          - ---------------------------------
                                    10       5
                                   x   + 10 x  + 25
          (%i3) expr: f(x)^3 - f(x)^2 + 7;
                                   3       2
          (%o3)                   f (x) - f (x) + 7
          (%i4) ratdiff (expr, f(x));
                                     2
          (%o4)                   3 f (x) - 2 f(x)
          (%i5) expr: (a + b)^3 + (a + b)^2;
                                        3          2
          (%o5)                  (b + a)  + (b + a)
          (%i6) ratdiff (expr, a + b);
                              2                    2
          (%o6)            3 b  + (6 a + 2) b + 3 a  + 2 a

Function: ratdisrep (<expr>)

     Returns its argument as a general expression.  If <expr> is a
     general expression, it is returned unchanged.

     Typically `ratdisrep' is called to convert a canonical rational
     expression (CRE) into a general expression.  This is sometimes
     convenient if one wishes to stop the "contagion", or use rational
     functions in non-rational contexts.

     See also `totaldisrep'.

Function: ratexpand (<expr>)
Option variable: ratexpand

     Expands <expr> by multiplying out products of sums and
     exponentiated sums, combining fractions over a common denominator,
     cancelling the greatest common divisor of the numerator and
     denominator, then splitting the numerator (if a sum) into its
     respective terms divided by the denominator.

     The return value of `ratexpand' is a general expression, even if
     <expr> is a canonical rational expression (CRE).

     The switch `ratexpand' if `true' will cause CRE expressions to be
     fully expanded when they are converted back to general form or
     displayed, while if it is `false' then they will be put into a
     recursive form.  See also `ratsimp'.

     When `ratdenomdivide' is `true', `ratexpand' expands a ratio in
     which the numerator is a sum into a sum of ratios, all having a
     common denominator.  Otherwise, `ratexpand' collapses a sum of
     ratios into a single ratio, the numerator of which is the sum of
     the numerators of each ratio.

     When `keepfloat' is `true', prevents floating point numbers from
     being rationalized when expressions which contain them are
     converted to canonical rational expression (CRE) form.

     Examples:

          (%i1) ratexpand ((2*x - 3*y)^3);
                               3         2       2        3
          (%o1)          - 27 y  + 54 x y  - 36 x  y + 8 x
          (%i2) expr: (x - 1)/(x + 1)^2 + 1/(x - 1);
                                   x - 1       1
          (%o2)                   -------- + -----
                                         2   x - 1
                                  (x + 1)
          (%i3) expand (expr);
                              x              1           1
          (%o3)          ------------ - ------------ + -----
                          2              2             x - 1
                         x  + 2 x + 1   x  + 2 x + 1
          (%i4) ratexpand (expr);
                                  2
                               2 x                 2
          (%o4)           --------------- + ---------------
                           3    2            3    2
                          x  + x  - x - 1   x  + x  - x - 1

Option variable: ratfac

     Default value: `false'

     When `ratfac' is `true', canonical rational expressions (CRE) are
     manipulated in a partially factored form.

     During rational operations the expression is maintained as fully
     factored as possible without calling `factor'.  This should always
     save space and may save time in some computations.  The numerator
     and denominator are made relatively prime, for example `rat ((x^2
     - 1)^4/(x + 1)^2)' yields `(x - 1)^4 (x + 1)^2)', but the factors
     within each part may not be relatively prime.

     In the `ctensr' (Component Tensor Manipulation) package, Ricci,
     Einstein, Riemann, and Weyl tensors and the scalar curvature are
     factored automatically when `ratfac' is `true'.  `ratfac' should
     only be set for cases where the tensorial components are known to
     consist of few terms.

     The `ratfac' and `ratweight' schemes are incompatible and may not
     both be used at the same time.

Function: ratnumer (<expr>)

     Returns the numerator of <expr>, after coercing <expr> to a
     canonical rational expression (CRE).  The return value is a CRE.

     <expr> is coerced to a CRE by `rat' if it is not already a CRE.
     This conversion may change the form of <expr> by putting all terms
     over a common denominator.

     `num' is similar, but returns an ordinary expression instead of a
     CRE.  Also, `num' does not attempt to place all terms over a
     common denominator, and thus some expressions which are considered
     ratios by `ratnumer' are not considered ratios by `num'.

Function: ratp (<expr>)

     Returns `true' if <expr> is a canonical rational expression (CRE)
     or extended CRE, otherwise `false'.

     CRE are created by `rat' and related functions.  Extended CRE are
     created by `taylor' and related functions.

Option variable: ratprint

     Default value: `true'

     When `ratprint' is `true', a message informing the user of the
     conversion of floating point numbers to rational numbers is
     displayed.

Function: ratsimp (<expr>)
Function: ratsimp (<expr>, <x_1>, ..., <x_n>)

     Simplifies the expression <expr> and all of its subexpressions,
     including the arguments to non-rational functions.  The result is
     returned as the quotient of two polynomials in a recursive form,
     that is, the coefficients of the main variable are polynomials in
     the other variables.  Variables may include non-rational functions
     (e.g., `sin (x^2 + 1)') and the arguments to any such functions
     are also rationally simplified.

     `ratsimp (<expr>, <x_1>, ..., <x_n>)' enables rational
     simplification with the specification of variable ordering as in
     `ratvars'.

     When `ratsimpexpons' is `true', `ratsimp' is applied to the
     exponents of expressions during simplification.

     See also `ratexpand'.  Note that `ratsimp' is affected by some of
     the flags which affect `ratexpand'.

     Examples:

          (%i1) sin (x/(x^2 + x)) = exp ((log(x) + 1)^2 - log(x)^2);
                                                   2      2
                             x         (log(x) + 1)  - log (x)
          (%o1)        sin(------) = %e
                            2
                           x  + x
          (%i2) ratsimp (%);
                                       1          2
          (%o2)                  sin(-----) = %e x
                                     x + 1
          (%i3) ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1));
                                 3/2
                          (x - 1)    - sqrt(x - 1) (x + 1)
          (%o3)           --------------------------------
                               sqrt((x - 1) (x + 1))
          (%i4) ratsimp (%);
                                     2 sqrt(x - 1)
          (%o4)                    - -------------
                                           2
                                     sqrt(x  - 1)
          (%i5) x^(a + 1/a), ratsimpexpons: true;
                                         2
                                        a  + 1
                                        ------
                                          a
          (%o5)                        x

Option variable: ratsimpexpons

     Default value: `false'

     When `ratsimpexpons' is `true', `ratsimp' is applied to the
     exponents of expressions during simplification.

Option variable: radsubstflag

     Default value: `false'

     `radsubstflag', if `true', permits `ratsubst' to make
     substitutions such as `u' for `sqrt (x)' in `x'.

Function: ratsubst (<a>, <b>, <c>)

     Substitutes <a> for <b> in <c> and returns the resulting
     expression.  <b> may be a sum, product, power, etc.

     `ratsubst' knows something of the meaning of expressions whereas
     `subst' does a purely syntactic substitution.  Thus `subst (a, x +
     y, x + y + z)' returns `x + y + z' whereas `ratsubst' returns `z +
     a'.

     When `radsubstflag' is `true', `ratsubst' makes substitutions for
     radicals in expressions which don't explicitly contain them.

     `ratsubst' ignores the value `true' of the option variable
     `keepfloat'.

     Examples:

          (%i1) ratsubst (a, x*y^2, x^4*y^3 + x^4*y^8);
                                        3      4
          (%o1)                      a x  y + a
          (%i2) cos(x)^4 + cos(x)^3 + cos(x)^2 + cos(x) + 1;
                         4         3         2
          (%o2)       cos (x) + cos (x) + cos (x) + cos(x) + 1
          (%i3) ratsubst (1 - sin(x)^2, cos(x)^2, %);
                      4           2                     2
          (%o3)    sin (x) - 3 sin (x) + cos(x) (2 - sin (x)) + 3
          (%i4) ratsubst (1 - cos(x)^2, sin(x)^2, sin(x)^4);
                                  4           2
          (%o4)                cos (x) - 2 cos (x) + 1
          (%i5) radsubstflag: false$
          (%i6) ratsubst (u, sqrt(x), x);
          (%o6)                           x
          (%i7) radsubstflag: true$
          (%i8) ratsubst (u, sqrt(x), x);
                                          2
          (%o8)                          u

Function: ratvars (<x_1>, ..., <x_n>)
Function: ratvars ()
System variable: ratvars

     Declares main variables <x_1>, ..., <x_n> for rational expressions.
     <x_n>, if present in a rational expression, is considered the main
     variable.  Otherwise, <x_[n-1]> is considered the main variable if
     present, and so on through the preceding variables to <x_1>, which
     is considered the main variable only if none of the succeeding
     variables are present.

     If a variable in a rational expression is not present in the
     `ratvars' list, it is given a lower priority than <x_1>.

     The arguments to `ratvars' can be either variables or non-rational
     functions such as `sin(x)'.

     The variable `ratvars' is a list of the arguments of the function
     `ratvars' when it was called most recently.  Each call to the
     function `ratvars' resets the list.  `ratvars ()' clears the list.

Option variable: ratvarswitch

     Default value: `true'

     Maxima keeps an internal list in the Lisp variable `VARLIST' of
     the main variables for rational expressions.  If `ratvarswitch' is
     `true', every evaluation starts with a fresh list `VARLIST'.  This
     is the default behavior.  Otherwise, the main variables from
     previous evaluations are not removed from the internal list
     `VARLIST'.

     The main variables, which are declared with the function `ratvars'
     are not affected by the option variable `ratvarswitch'.

     Examples:

     If `ratvarswitch' is `true', every evaluation starts with a fresh
     list `VARLIST'.

          (%i1) ratvarswitch:true$

          (%i2) rat(2*x+y^2);
                                       2
          (%o2)/R/                    y  + 2 x
          (%i3) :lisp varlist
          ($X $Y)

          (%i3) rat(2*a+b^2);
                                       2
          (%o3)/R/                    b  + 2 a

          (%i4) :lisp varlist
          ($A $B)

     If `ratvarswitch' is `false', the main variables from the last
     evaluation are still present.

          (%i4) ratvarswitch:false$

          (%i5) rat(2*x+y^2);
                                       2
          (%o5)/R/                    y  + 2 x
          (%i6) :lisp varlist
          ($X $Y)

          (%i6) rat(2*a+b^2);
                                       2
          (%o6)/R/                    b  + 2 a

          (%i7) :lisp varlist
          ($A $B $X $Y)

Function: ratweight (<x_1>, <w_1>, ..., <x_n>, <w_n>)
Function: ratweight ()

     Assigns a weight <w_i> to the variable <x_i>.  This causes a term
     to be replaced by 0 if its weight exceeds the value of the
     variable `ratwtlvl' (default yields no truncation).  The weight of
     a term is the sum of the products of the weight of a variable in
     the term times its power.  For example, the weight of `3 x_1^2
     x_2' is `2 w_1 + w_2'.  Truncation according to `ratwtlvl' is
     carried out only when multiplying or exponentiating canonical
     rational expressions (CRE).

     `ratweight ()' returns the cumulative list of weight assignments.

     Note: The `ratfac' and `ratweight' schemes are incompatible and may
     not both be used at the same time.

     Examples:

          (%i1) ratweight (a, 1, b, 1);
          (%o1)                     [a, 1, b, 1]
          (%i2) expr1: rat(a + b + 1)$
          (%i3) expr1^2;
                            2                  2
          (%o3)/R/         b  + (2 a + 2) b + a  + 2 a + 1
          (%i4) ratwtlvl: 1$
          (%i5) expr1^2;
          (%o5)/R/                  2 b + 2 a + 1

System variable: ratweights

     Default value: `[]'

     `ratweights' is the list of weights assigned by `ratweight'.  The
     list is cumulative: each call to `ratweight' places additional
     items in the list.

     `kill (ratweights)' and `save (ratweights)' both work as expected.

Option variable: ratwtlvl

     Default value: `false'

     `ratwtlvl' is used in combination with the `ratweight' function to
     control the truncation of canonical rational expressions (CRE).
     For the default value of `false', no truncation occurs.

Function: remainder (<p_1>, <p_2>)
Function: remainder (<p_1>, <p_2>, <x_1>, ..., <x_n>)

     Returns the remainder of the polynomial <p_1> divided by the
     polynomial <p_2>.  The arguments <x_1>, ..., <x_n> are interpreted
     as in `ratvars'.

     `remainder' returns the second element of the two-element list
     returned by `divide'.

Function: resultant (<p_1>, <p_2>, <x>)

     The function `resultant' computes the resultant of the two
     polynomials <p_1> and <p_2>, eliminating the variable <x>.  The
     resultant is a determinant of the coefficients of <x> in <p_1> and
     <p_2>, which equals zero if and only if <p_1> and <p_2> have a
     non-constant factor in common.

     If <p_1> or <p_2> can be factored, it may be desirable to call
     `factor'  before calling `resultant'.

     The option variable `resultant' controls which algorithm will be
     used to compute the resultant.  See the option variable
     `resultant'.

     The function `bezout'  takes the same arguments as `resultant' and
     returns a matrix.  The determinant of the return value is the
     desired resultant.

     Examples:

          (%i1) resultant(2*x^2+3*x+1, 2*x^2+x+1, x);
          (%o1)                           8
          (%i2) resultant(x+1, x+1, x);
          (%o2)                           0
          (%i3) resultant((x+1)*x, (x+1), x);
          (%o3)                           0
          (%i4) resultant(a*x^2+b*x+1, c*x + 2, x);
                                   2
          (%o4)                   c  - 2 b c + 4 a

          (%i5) bezout(a*x^2+b*x+1, c*x+2, x);
                                  [ 2 a  2 b - c ]
          (%o5)                   [              ]
                                  [  c      2    ]
          (%i6) determinant(%);
          (%o6)                   4 a - (2 b - c) c

Option variable: resultant

     Default value: `subres'

     The option variable `resultant' controls which algorithm will be
     used to compute the resultant with the function `resultant'.
     The possible values are:

    `subres'
          for the subresultant polynomial remainder sequence (PRS)
          algorithm,

    `mod'
          for the modular resultant algorithm, and

    `red'
          for the reduced polynomial remainder sequence (PRS) algorithm.

     On most problems the default value `subres' should be best.  On
     some large degree univariate or bivariate problems `mod' may be
     better.

Option variable: savefactors

     Default value: `false'

     When `savefactors' is `true', causes the factors of an expression
     which is a product of factors to be saved by certain functions in
     order to speed up later factorizations of expressions containing
     some of the same factors.

Function: showratvars (<expr>)

     Returns a list of the canonical rational expression (CRE)
     variables in expression `expr'.

     See also `ratvars'.

Function: sqfr (<expr>)

     is similar to `factor' except that the polynomial factors are
     "square-free."  That is, they have factors only of degree one.
     This algorithm, which is also used by the first stage of `factor',
     utilizes the fact that a polynomial has in common with its n'th
     derivative all its factors of degree greater than n.  Thus by
     taking greatest common divisors with the polynomial of the
     derivatives with respect to each variable in the polynomial, all
     factors of degree greater than 1 can be found.

     Example:

          (%i1) sqfr (4*x^4 + 4*x^3 - 3*x^2 - 4*x - 1);
                                          2   2
          (%o1)                  (2 x + 1)  (x  - 1)

Function: tellrat (<p_1>, ..., <p_n>)
Function: tellrat ()

     Adds to the ring of algebraic integers known to Maxima the
     elements which are the solutions of the polynomials <p_1>, ...,
     <p_n>.  Each argument <p_i> is a polynomial with integer
     coefficients.

     `tellrat (<x>)' effectively means substitute 0 for <x> in rational
     functions.

     `tellrat ()' returns a list of the current substitutions.

     `algebraic' must be set to `true' in order for the simplification
     of algebraic integers to take effect.

     Maxima initially knows about the imaginary unit `%i' and all roots
     of integers.

     There is a command `untellrat' which takes kernels and removes
     `tellrat' properties.

     When `tellrat''ing a multivariate polynomial, e.g., `tellrat (x^2
     - y^2)', there would be an ambiguity as to whether to substitute
     `<y>^2' for `<x>^2' or vice versa.  Maxima picks a particular
     ordering, but if the user wants to specify which, e.g.  `tellrat
     (y^2 = x^2)' provides a syntax which says replace `<y>^2' by
     `<x>^2'.

     Examples:

          (%i1) 10*(%i + 1)/(%i + 3^(1/3));
                                     10 (%i + 1)
          (%o1)                      -----------
                                            1/3
                                      %i + 3
          (%i2) ev (ratdisrep (rat(%)), algebraic);
                       2/3      1/3              2/3      1/3
          (%o2)    (4 3    - 2 3    - 4) %i + 2 3    + 4 3    - 2
          (%i3) tellrat (1 + a + a^2);
                                      2
          (%o3)                     [a  + a + 1]
          (%i4) 1/(a*sqrt(2) - 1) + a/(sqrt(3) + sqrt(2));
                                1                 a
          (%o4)           ------------- + -----------------
                          sqrt(2) a - 1   sqrt(3) + sqrt(2)
          (%i5) ev (ratdisrep (rat(%)), algebraic);
                   (7 sqrt(3) - 10 sqrt(2) + 2) a - 2 sqrt(2) - 1
          (%o5)    ----------------------------------------------
                                         7
          (%i6) tellrat (y^2 = x^2);
                                  2    2   2
          (%o6)                 [y  - x , a  + a + 1]

Function: totaldisrep (<expr>)

     Converts every subexpression of <expr> from canonical rational
     expressions (CRE) to general form and returns the result.  If
     <expr> is itself in CRE form then `totaldisrep' is identical to
     `ratdisrep'.

     `totaldisrep' may be useful for ratdisrepping expressions such as
     equations, lists, matrices, etc., which have some subexpressions
     in CRE form.

Function: untellrat (<x_1>, ..., <x_n>)

     Removes `tellrat' properties from <x_1>, ..., <x_n>.

15 Special Functions


Introduction to Special Functions
Bessel Functions
Airy Functions
Gamma and factorial Functions
Exponential Integrals
Error Function
Struve Functions
Hypergeometric Functions
Parabolic Cylinder Functions
Functions and Variables for Special Functions

15.1 Introduction to Special Functions

Special function notation follows:

     bessel_j (index, expr)         Bessel function, 1st kind
     bessel_y (index, expr)         Bessel function, 2nd kind
     bessel_i (index, expr)         Modified Bessel function, 1st kind
     bessel_k (index, expr)         Modified Bessel function, 2nd kind

     hankel_1 (v,z)                 Hankel function of the 1st kind
     hankel_2 (v,z)                 Hankel function of the 2nd kind
     struve_h (v,z)                 Struve H function
     struve_l (v,z)                 Struve L function

     assoc_legendre_p[v,u] (z)      Legendre function of degree v and order u
     assoc_legendre_q[v,u] (z)      Legendre function, 2nd kind

     %f[p,q] ([], [], expr)         Generalized Hypergeometric function
     gamma()                        Gamma function
     gammagreek(a,z)                Incomplete gamma function
     gammaincomplete(a,z)           Tail of incomplete gamma function
     hypergeometric(l1, l2, z)      Hypergeometric function
     slommel
     %m[u,k] (z)                    Whittaker function, 1st kind
     %w[u,k] (z)                    Whittaker function, 2nd kind
     erfc (z)                       Complement of the erf function

     expintegral_e (v,z)            Exponential integral E
     expintegral_e1 (z)             Exponential integral E1
     expintegral_ei (z)             Exponential integral Ei
     expintegral_li (z)             Logarithmic integral Li
     expintegral_si (z)             Exponential integral Si
     expintegral_ci (z)             Exponential integral Ci
     expintegral_shi (z)            Exponential integral Shi
     expintegral_chi (z)            Exponential integral Chi

     kelliptic (z)                  Complete elliptic integral of the first
                                    kind (K)
     parabolic_cylinder_d (v,z)     Parabolic cylinder D function

15.2 Bessel Functions

Function: bessel_j (<v>, <z>)

     The Bessel function of the first kind of order v and argument z.

     `bessel_j' is defined as

                          inf
                          ====       k  - v - 2 k  v + 2 k
                          \     (- 1)  2          z
                           >    --------------------------
                          /        k! gamma(v + k + 1)
                          ====
                          k = 0

     although the infinite series is not used for computations.

Function: bessel_y (<v>, <z>)

     The Bessel function of the second kind of order v and argument z.

     `bessel_y' is defined as
                        cos(%pi v) bessel_j(v, z) - bessel_j(-v, z)
                        -------------------------------------------
                                       sin(%pi v)

     when v is not an integer.  When v is an integer n, the limit as v
     approaches n is taken.

Function: bessel_i (<v>, <z>)

     The modified Bessel function of the first kind of order v and
     argument z.

     `bessel_i' is defined as
                              inf
                              ====   - v - 2 k  v + 2 k
                              \     2          z
                               >    -------------------
                              /     k! gamma(v + k + 1)
                              ====
                              k = 0

     although the infinite series is not used for computations.

Function: bessel_k (<v>, <z>)

     The modified Bessel function of the second kind of order v and
     argument z.

     `bessel_k' is defined as
                     %pi csc(%pi v) (bessel_i(-v, z) - bessel_i(v, z))
                     -------------------------------------------------
                                            2

     when v is not an integer.  If v is an integer n, then the limit as
     v approaches n is taken.

Function: hankel_1 (<v>, <z>)

     The Hankel function of the first kind of order v and argument z
     (A&S 9.1.3). `hankel_1' is defined as

             bessel_j(v,z) + %i * bessel_y(v,z)

     Maxima evaluates `hankel_1' numerically for a real order v and
     complex argument z in float precision. The numerical evaluation in
     bigfloat precision and for a complex order v is not supported.

     When `besselexpand' is `true', `hankel_1' is expanded in terms of
     elementary functions when the order v is half of an odd integer.
     See `besselexpand'.

     Maxima knows the derivative of `hankel_1' wrt the argument z.

     Examples:

     Numerical evaluation:

          (%i1) hankel_1(1,0.5);
          (%o1)              .2422684576748738 - 1.471472392670243 %i
          (%i2) hankel_1(1,0.5+%i);
          (%o2)             - .2558287994862166 %i - 0.239575601883016

     A complex order v is not supported. Maxima returns a noun form:

          (%i3) hankel_1(%i,0.5+%i);
          (%o3)                       hankel_1(%i, %i + 0.5)

     Expansion of `hankel_1' when `besselexpand' is `true':

          (%i4) hankel_1(1/2,z),besselexpand:true;
                                sqrt(2) sin(z) - sqrt(2) %i cos(z)
          (%o4)                 ----------------------------------
                                        sqrt(%pi) sqrt(z)

     Derivative of `hankel_1' wrt the argument z. The derivative wrt the
     order v is not supported. Maxima returns a noun form:

          (%i5) diff(hankel_1(v,z),z);
                              hankel_1(v - 1, z) - hankel_1(v + 1, z)
          (%o5)               ---------------------------------------
                                                 2
          (%i6) diff(hankel_1(v,z),v);
                                       d
          (%o6)                        -- (hankel_1(v, z))
                                       dv

Function: hankel_2 (<v>, <z>)

     The Hankel function of the second kind of order v and argument z
     (A&S 9.1.4). `hankel_2' is defined as

             bessel_j(v,z) - %i * bessel_y(v,z)

     Maxima evaluates `hankel_2' numerically for a real order v and
     complex argument z in float precision. The numerical evaluation in
     bigfloat precision and for a complex order v is not supported.

     When `besselexpand' is `true', `hankel_2' is expanded in terms of
     elementary functions when the order v is half of an odd integer.
     See `besselexpand'.

     Maxima knows the derivative of `hankel_2' wrt the argument z.

     For examples see `hankel_1'.

Option variable: besselexpand

     Default value: `false'

     Controls expansion of the Bessel functions when the order is half
     of an odd integer.  In this case, the Bessel functions can be
     expanded in terms of other elementary functions.  When
     `besselexpand' is `true', the Bessel function is expanded.

          (%i1) besselexpand: false$
          (%i2) bessel_j (3/2, z);
                                              3
          (%o2)                      bessel_j(-, z)
                                              2
          (%i3) besselexpand: true$
          (%i4) bessel_j (3/2, z);
                                                  sin(z)   cos(z)
                                 sqrt(2) sqrt(z) (------ - ------)
                                                     2       z
                                                    z
          (%o4)                  ---------------------------------
                                             sqrt(%pi)

Function: scaled_bessel_i (<v>, <z>)

     The scaled modified Bessel function of the first kind of order v
     and argument z.  That is, scaled_bessel_i(v,z) =
     exp(-abs(z))*bessel_i(v, z).  This function is particularly useful
     for calculating bessel_i for large z, which is large.  However,
     maxima does not otherwise know much about this function.  For
     symbolic work, it is probably preferable to work with the
     expression `exp(-abs(z))*bessel_i(v, z)'.

Function: scaled_bessel_i0 (<z>)

     Identical to `scaled_bessel_i(0,z)'.

Function: scaled_bessel_i1 (<z>)

     Identical to `scaled_bessel_i(1,z)'.

Function: %s [<u>,<v>] (<z>)

     Lommel's little s[u,v](z) function.  Probably Gradshteyn & Ryzhik
     8.570.1.

15.3 Airy Functions

The Airy functions Ai(x) and Bi(x) are defined in Abramowitz and Stegun,
Handbook of Mathematical Functions, Section 10.4.

   `y = Ai(x)' and `y = Bi(x)' are two linearly independent solutions
of the Airy differential equation `diff (y(x), x, 2) - x y(x) = 0'.

   If the argument `x' is a real or complex floating point number, the
numerical value of the function is returned.
Function: airy_ai (<x>)

     The Airy function Ai(x).  (A&S 10.4.2)

     The derivative `diff (airy_ai(x), x)' is `airy_dai(x)'.

     See also `airy_bi', `airy_dai', `airy_dbi'.

Function: airy_dai (<x>)

     The derivative of the Airy function Ai `airy_ai(x)'.

     See `airy_ai'.

Function: airy_bi (<x>)

     The Airy function Bi(x).  (A&S 10.4.3)

     The derivative `diff (airy_bi(x), x)' is `airy_dbi(x)'.

     See `airy_ai', `airy_dbi'.

Function: airy_dbi (<x>)

     The derivative of the Airy Bi function `airy_bi(x)'.

     See `airy_ai' and `airy_bi'.

15.4 Gamma and factorial Functions

The gamma function and the related beta, psi and incomplete gamma
functions are defined in Abramowitz and Stegun, Handbook of
Mathematical Functions, Chapter 6.
Function: bffac (<expr>, <n>)

     Bigfloat version of the factorial (shifted gamma) function.  The
     second argument is how many digits to retain and return, it's a
     good idea to request a couple of extra.

Function: bfpsi (<n>, <z>, <fpprec>)
Function: bfpsi0 (<z>, <fpprec>)

     `bfpsi' is the polygamma function of real argument <z> and integer
     order <n>.  `bfpsi0' is the digamma function.  `bfpsi0 (<z>,
     <fpprec>)' is equivalent to `bfpsi (0, <z>, <fpprec>)'.

     These functions return bigfloat values.  <fpprec> is the bigfloat
     precision of the return value.

Function: cbffac (<z>, <fpprec>)

     Complex bigfloat factorial.

     `load ("bffac")' loads this function.

Function: gamma (<z>)

     The basic definition of the gamma function (A&S 6.1.1) is

                                   inf
                                  /
                                  [     z - 1   - t
                       gamma(z) = I    t      %e    dt
                                  ]
                                  /
                                   0

     Maxima simplifies `gamma' for positive integer and positive and
     negative rational numbers. For half integral values the result is
     a rational number times `sqrt(%pi)'. The simplification for
     integer values is controlled by `factlim'. For integers greater
     than `factlim' the numerical result of the factorial function,
     which is used to calculate `gamma', will overflow.  The
     simplification for rational numbers is controlled by `gammalim' to
     avoid internal overflow. See `factlim' and `gammalim'.

     For negative integers `gamma' is not definied.

     Maxima can evalute `gamma' numerically for real and complex values
     in float and bigfloat precision.

     `gamma' has mirror symmetry.

     When `gamma_expand' is `true', Maxima expands `gamma' for
     arguments `z+n' and `z-n' where `n' is an integer.

     Maxima knows the derivate of `gamma'.

     Examples:

     Simplification for integer, half integral, and rational numbers:

          (%i1) map('gamma,[1,2,3,4,5,6,7,8,9]);
          (%o1)        [1, 1, 2, 6, 24, 120, 720, 5040, 40320]
          (%i2) map('gamma,[1/2,3/2,5/2,7/2]);
                              sqrt(%pi)  3 sqrt(%pi)  15 sqrt(%pi)
          (%o2)   [sqrt(%pi), ---------, -----------, ------------]
                                  2           4            8
          (%i3) map('gamma,[2/3,5/3,7/3]);
                                            2           1
                                    2 gamma(-)  4 gamma(-)
                                2           3           3
          (%o3)          [gamma(-), ----------, ----------]
                                3       3           9

     Numerical evaluation for real and complex values:

          (%i4) map('gamma,[2.5,2.5b0]);
          (%o4)     [1.329340388179137, 1.3293403881791370205b0]
          (%i5) map('gamma,[1.0+%i,1.0b0+%i]);
          (%o5) [0.498015668118356 - .1549498283018107 %i,
                    4.9801566811835604272b-1 - 1.5494982830181068513b-1 %i]

     `gamma' has mirror symmetry:

          (%i6) declare(z,complex)$
          (%i7) conjugate(gamma(z));
          (%o7)                  gamma(conjugate(z))

     Maxima expands `gamma(z+n)' and `gamma(z-n)', when `gamma_expand'
     is `true':

          (%i8) gamma_expand:true$

          (%i9) [gamma(z+1),gamma(z-1),gamma(z+2)/gamma(z+1)];
                                         gamma(z)
          (%o9)             [z gamma(z), --------, z + 1]
                                          z - 1

     The deriviative of `gamma':

          (%i10) diff(gamma(z),z);
          (%o10)                  psi (z) gamma(z)
                                     0

     See also `makegamma'.

     The Euler-Mascheroni constant is `%gamma'.

Function: log_gamma (<z>)

     The natural logarithm of the gamma function.

Function: gamma_incomplete (<a>, <z>)

     The incomplete upper gamma function A&S 6.5.2:

                                        inf
                                       /
                                       [     a - 1   - t
              gamma_incomplete(a, z) = I    t      %e    dt
                                       ]
                                       /
                                        z

Function: gamma_incomplete_regularized (<a>, <z>)

     The regularized incomplete upper gamma function A&S 6.5.1:

          gamma_incomplete_regularized(a, z) =
                                                  gamma_incomplete(a, z)
                                                  ----------------------
                                                         gamma(a)

Function: gamma_incomplete_generalized (<a>, <z1>, <z1>)

     The generalized incomplete gamma function.

          gamma_incomplete_generalized(a, z1, z2) =
                                                         z2
                                                        /
                                                        [    a - 1   - t
                                                        I   t      %e    dt
                                                        ]
                                                        /
                                                         z1

Option variable: gammalim

     Default value: 1000000

     `gammalim' controls simplification of the gamma function for
     integral and rational number arguments.  If the absolute value of
     the argument is not greater than `gammalim', then simplification
     will occur.  Note that the `factlim' switch controls
     simplification of the result of `gamma' of an integer argument as
     well.

Function: makegamma (<expr>)

     Transforms instances of binomial, factorial, and beta functions in
     <expr> into gamma functions.

     See also `makefact'.

Function: beta (<a>, <b>)

     The beta function is defined as `gamma(a) gamma(b)/gamma(a+b)'
     (A&S 6.2.1).

     Maxima simplifies the beta function for positive integers and
     rational numbers, which sum to an integer. When
     `beta_args_sum_to_integer' is `true', Maxima simplifies also
     general expressions which sum to an integer.

     For <a> or <b> equal to zero the beta function is not defined.

     In general the beta function is not defined for negative integers
     as an argument. The exception is for <a=-n>, <n> a positive integer
     and <b> a positive integer with <b<=n>, it is possible to define an
     analytic continuation. Maxima gives for this case a result.

     When `beta_expand' is `true', expressions like `beta(a+n,b)' and
     `beta(a-n,b)' or `beta(a,b+n)' and `beta(a,b-n)' with `n' an
     integer are simplified.

     Maxima can evaluate the beta function for real and complex values
     in float and bigfloat precision. For numerical evaluation Maxima
     uses `log_gamma':

                     - log_gamma(b + a) + log_gamma(b) + log_gamma(a)
                   %e

     Maxima knows that the beta function is symmetric and has mirror
     symmetry.

     Maxima knows the derivatives of the beta function with respect to
     <a> or <b>.

     To express the beta function as a ratio of gamma functions see
     `makegamma'.

     Examples:

     Simplification, when one of the arguments is an integer:

          (%i1) [beta(2,3),beta(2,1/3),beta(2,a)];
                                         1   9      1
          (%o1)                         [--, -, ---------]
                                         12  4  a (a + 1)

     Simplification for two rational numbers as arguments which sum to
     an integer:

          (%i2) [beta(1/2,5/2),beta(1/3,2/3),beta(1/4,3/4)];
                                    3 %pi   2 %pi
          (%o2)                    [-----, -------, sqrt(2) %pi]
                                      8    sqrt(3)

     When setting `beta_args_sum_to_integer' to `true' more general
     expression are simplified, when the sum of the arguments is an
     integer:

          (%i3) beta_args_sum_to_integer:true$
          (%i4) beta(a+1,-a+2);
                                          %pi (a - 1) a
          (%o4)                         ------------------
                                        2 sin(%pi (2 - a))

     The possible results, when one of the arguments is a negative
     integer:

          (%i5) [beta(-3,1),beta(-3,2),beta(-3,3)];
                                              1  1    1
          (%o5)                            [- -, -, - -]
                                              3  6    3

     `beta(a+n,b)' or `beta(a-n)' with `n' an integer simplifies when
     `beta_expand' is `true':

          (%i6) beta_expand:true$
          (%i7) [beta(a+1,b),beta(a-1,b),beta(a+1,b)/beta(a,b+1)];
                              a beta(a, b)  beta(a, b) (b + a - 1)  a
          (%o7)              [------------, ----------------------, -]
                                 b + a              a - 1           b

     Beta is not definied, when one of the arguments is zero:

          (%i7) beta(0,b);
          beta: expected nonzero arguments; found 0, b
           -- an error.  To debug this try debugmode(true);

     Numercial evaluation for real and complex arguments in float or
     bigfloat precision:

          (%i8) beta(2.5,2.3);
          (%o8) .08694748611299981

          (%i9) beta(2.5,1.4+%i);
          (%o9) 0.0640144950796695 - .1502078053286415 %i

          (%i10) beta(2.5b0,2.3b0);
          (%o10) 8.694748611299969b-2

          (%i11) beta(2.5b0,1.4b0+%i);
          (%o11) 6.401449507966944b-2 - 1.502078053286415b-1 %i

     Beta is symmetric and has mirror symmetry:

          (%i14) beta(a,b)-beta(b,a);
          (%o14)                                 0
          (%i15) declare(a,complex,b,complex)$
          (%i16) conjugate(beta(a,b));
          (%o16)                 beta(conjugate(a), conjugate(b))

     The derivative of the beta function wrt `a':
          (%i17) diff(beta(a,b),a);
          (%o17)               - beta(a, b) (psi (b + a) - psi (a))
                                                0             0

Function: beta_incomplete (<a>, <b>, <z>)

     The basic definition of the incomplete beta function (A&S 6.6.1) is

                  z
                 /
                 [         b - 1  a - 1
                 I  (1 - t)      t      dt
                 ]
                 /
                  0

     This definition is possible for realpart(a)>0 and realpart(b)>0
     and abs(z)<1. For other values the incomplete beta function can be
     defined through a generalized hypergeometric function:

             gamma(a) hypergeometric_generalized([a, 1 - b], [a + 1], z) z

     (See functions.wolfram.com for a complete definition of the
     incomplete beta function.)

     For negative integers a = -n and positive integers b=m with m<=n
     the incomplete beta function is defined through

                                      m - 1           k
                                      ====  (1 - m)  z
                                n - 1 \            k
                               z       >    -----------
                                      /     k! (n - k)
                                      ====
                                      k = 0

     Maxima uses this definition to simplify `beta_incomplete' for <a> a
     negative integer.

     For <a> a positive integer, `beta_incomplete' simplifies for any
     argument <b> and <z> and for <b> a positive integer for any
     argument <a> and <z>, with the exception of <a> a negative integer.

     For z=0 and realpart(a)>0, `beta_incomplete' has the specific
     value zero. For <z=1> and realpart(b)>0, `beta_incomplete'
     simplifies to the beta function `beta(a,b)'.

     Maxima evaluates `beta_incomplete' numerically for real and
     complex values in float or bigfloat precision. For the numerical
     evaluation an expansion of the incomplete beta function in
     continued fractions is used.

     When the option variable `beta_expand' is `true', Maxima expands
     expressions like `beta_incomplete(a+n,b,z)' and
     `beta_incomplete(a-n,b,z)' where n is a positive integer.

     Maxima knows the derivatives of `beta_incomplete' with respect to
     the variables <a>, <b> and <z> and the integral with respect to the
     variable <z>.

     Examples:

     Simplification for <a> a positive integer:

          (%i1) beta_incomplete(2,b,z);
                                                 b
                                      1 - (1 - z)  (b z + 1)
          (%o1)                       ----------------------
                                            b (b + 1)

     Simplification for <b> a positive integer:

          (%i2) beta_incomplete(a,2,z);
                                                         a
                                        (a (1 - z) + 1) z
          (%o2)                         ------------------
                                            a (a + 1)

     Simplification for <a> and <b> a positive integer:

          (%i3) beta_incomplete(3,2,z);
                                                         3
                                        (3 (1 - z) + 1) z
          (%o3)                         ------------------
                                                12

     <a> is a negative integer and b<=(-a), Maxima simplifies:

          (%i4) beta_incomplete(-3,1,z);
                                                 1
          (%o4)                              - ----
                                                  3
                                               3 z

     For the specific values z=0 and z=1, Maxima simplifies:

          (%i5) assume(a>0,b>0)$
          (%i6) beta_incomplete(a,b,0);
          (%o6)                                 0
          (%i7) beta_incomplete(a,b,1);
          (%o7)                            beta(a, b)

     Numerical evaluation in float or bigfloat precision:

          (%i8) beta_incomplete(0.25,0.50,0.9);
          (%o8)                          4.594959440269333
          (%i9)  fpprec:25$
          (%i10) beta_incomplete(0.25,0.50,0.9b0);
          (%o10)                    4.594959440269324086971203b0

     For abs(z)>1 `beta_incomplete' returns a complex result:

          (%i11) beta_incomplete(0.25,0.50,1.7);
          (%o11)              5.244115108584249 - 1.45518047787844 %i

     Results for more general complex arguments:

          (%i14) beta_incomplete(0.25+%i,1.0+%i,1.7+%i);
          (%o14)             2.726960675662536 - .3831175704269199 %i
          (%i15) beta_incomplete(1/2,5/4*%i,2.8+%i);
          (%o15)             13.04649635168716 %i - 5.802067956270001
          (%i16)

     Expansion, when `beta_expand' is `true':

          (%i23) beta_incomplete(a+1,b,z),beta_expand:true;
                                                                 b  a
                             a beta_incomplete(a, b, z)   (1 - z)  z
          (%o23)             -------------------------- - -----------
                                       b + a                 b + a

          (%i24) beta_incomplete(a-1,b,z),beta_expand:true;
                                                                     b  a - 1
                     beta_incomplete(a, b, z) (- b - a + 1)   (1 - z)  z
          (%o24)     -------------------------------------- - ---------------
                                     1 - a                         1 - a

     Derivative and integral for `beta_incomplete':

          (%i34) diff(beta_incomplete(a, b, z), z);
                                        b - 1  a - 1
          (%o34)                 (1 - z)      z
          (%i35) integrate(beta_incomplete(a, b, z), z);
                        b  a
                 (1 - z)  z
          (%o35) ----------- + beta_incomplete(a, b, z) z
                    b + a
                                                 a beta_incomplete(a, b, z)
                                               - --------------------------
                                                           b + a
          (%i36) factor(diff(%, z));
          (%o36)              beta_incomplete(a, b, z)

Function: beta_incomplete_regularized (<a>, <b>, <z>)

     The regularized incomplete beta function A&S 6.6.2, defined as

          beta_incomplete_regularized(a, b, z) =
                                                beta_incomplete(a, b, z)
                                                ------------------------
                                                       beta(a, b)

     As for `beta_incomplete' this definition is not complete. See
     functions.wolfram.com for a complete definition of
     `beta_incomplete_regularized'.

     `beta_incomplete_regularized' simplifies <a> or <b> a positive
     integer.

     For z=0 and realpart(a)>0, `beta_incomplete_regularized' has the
     specific value 0. For <z=1> and realpart(b)>0,
     `beta_incomplete_regularized' simplifies to 1.

     Maxima can evaluate `beta_incomplete_regularized' for real and
     complex arguments in float and bigfloat precision.

     When `beta_expand' is `true', Maxima expands
     `beta_incomplete_regularized' for arguments a+n or a-n, where n is
     an integer.

     Maxima knows the derivatives of `beta_incomplete_regularized' with
     respect to the variables <a>, <b>, and <z> and the integral with
     respect to the variable <z>.

     Examples:

     Simplification for <a> or <b> a positive integer:

          (%i1) beta_incomplete_regularized(2,b,z);
                                                 b
          (%o1)                       1 - (1 - z)  (b z + 1)

          (%i2) beta_incomplete_regularized(a,2,z);
                                                         a
          (%o2)                         (a (1 - z) + 1) z

          (%i3) beta_incomplete_regularized(3,2,z);
                                                         3
          (%o3)                         (3 (1 - z) + 1) z

     For the specific values z=0 and z=1, Maxima simplifies:

          (%i4) assume(a>0,b>0)$
          (%i5) beta_incomplete_regularized(a,b,0);
          (%o5)                                 0
          (%i6) beta_incomplete_regularized(a,b,1);
          (%o6)                                 1

     Numerical evaluation for real and complex arguments in float and
     bigfloat precision:

          (%i7) beta_incomplete_regularized(0.12,0.43,0.9);
          (%o7)                         .9114011367359802
          (%i8) fpprec:32$
          (%i9) beta_incomplete_regularized(0.12,0.43,0.9b0);
          (%o9)               9.1140113673598075519946998779975b-1
          (%i10) beta_incomplete_regularized(1+%i,3/3,1.5*%i);
          (%o10)             .2865367499935403 %i - 0.122995963334684
          (%i11) fpprec:20$
          (%i12) beta_incomplete_regularized(1+%i,3/3,1.5b0*%i);
          (%o12)      2.8653674999354036142b-1 %i - 1.2299596333468400163b-1

     Expansion, when `beta_expand' is `true':

          (%i13) beta_incomplete_regularized(a+1,b,z);
                                                               b  a
                                                        (1 - z)  z
          (%o13) beta_incomplete_regularized(a, b, z) - ------------
                                                        a beta(a, b)
          (%i14) beta_incomplete_regularized(a-1,b,z);
          (%o14) beta_incomplete_regularized(a, b, z)
                                                               b  a - 1
                                                        (1 - z)  z
                                                   - ----------------------
                                                     beta(a, b) (b + a - 1)

     The derivative and the integral wrt <z>:

          (%i15) diff(beta_incomplete_regularized(a,b,z),z);
                                        b - 1  a - 1
                                 (1 - z)      z
          (%o15)                 -------------------
                                     beta(a, b)
          (%i16) integrate(beta_incomplete_regularized(a,b,z),z);
          (%o16) beta_incomplete_regularized(a, b, z) z
                                                                     b  a
                                                              (1 - z)  z
                    a (beta_incomplete_regularized(a, b, z) - ------------)
                                                              a beta(a, b)
                  - -------------------------------------------------------
                                             b + a

Function: beta_incomplete_generalized (<a>, <b>, <z1>, <z2>)

     The basic definition of the generalized incomplete beta function is

                       z2
                     /
                     [          b - 1  a - 1
                     I   (1 - t)      t      dt
                     ]
                     /
                      z1

     Maxima simplifies `beta_incomplete_regularized' for <a> and <b> a
     positive integer.

     For realpart(a)>0 and z1=0 or z2=0, Maxima simplifies
     `beta_incomplete_generalized' to `beta_incomplete'. For
     realpart(b)>0 and z1=1 or <z2=1>, Maxima simplifies to an
     expression with `beta' and `beta_incomplete'.

     Maxima evaluates `beta_incomplete_regularized' for real and
     complex values in float and bigfloat precision.

     When `beta_expand' is `true', Maxima expands
     `beta_incomplete_generalized' for a+n and a-n, <n> a positive
     integer.

     Maxima knows the derivative of `beta_incomplete_generalized' with
     respect to the variables <a>, <b>, <z1>, and <z2> and the
     integrals with respect to the variables <z1> and <z2>.

     Examples:

     Maxima simplifies `beta_incomplete_generalized' for <a> and <b> a
     positive integer:

          (%i1) beta_incomplete_generalized(2,b,z1,z2);
                             b                      b
                     (1 - z1)  (b z1 + 1) - (1 - z2)  (b z2 + 1)
          (%o1)      -------------------------------------------
                                      b (b + 1)
          (%i2) beta_incomplete_generalized(a,2,z1,z2);
                                        a                      a
                     (a (1 - z2) + 1) z2  - (a (1 - z1) + 1) z1
          (%o2)      -------------------------------------------
                                      a (a + 1)
          (%i3) beta_incomplete_generalized(3,2,z1,z2);
                        2      2                       2      2
                (1 - z1)  (3 z1  + 2 z1 + 1) - (1 - z2)  (3 z2  + 2 z2 + 1)
          (%o3) -----------------------------------------------------------
                                            12

     Simplification for specific values z1=0, z2=0, z1=1, or z2=1:

          (%i4) assume(a > 0, b > 0)$
          (%i5) beta_incomplete_generalized(a,b,z1,0);
          (%o5)                    - beta_incomplete(a, b, z1)

          (%i6) beta_incomplete_generalized(a,b,0,z2);
          (%o6)                    - beta_incomplete(a, b, z2)

          (%i7) beta_incomplete_generalized(a,b,z1,1);
          (%o7)              beta(a, b) - beta_incomplete(a, b, z1)

          (%i8) beta_incomplete_generalized(a,b,1,z2);
          (%o8)              beta_incomplete(a, b, z2) - beta(a, b)

     Numerical evaluation for real arguments in float or bigfloat
     precision:

          (%i9) beta_incomplete_generalized(1/2,3/2,0.25,0.31);
          (%o9)                        .09638178086368676

          (%i10) fpprec:32$
          (%i10) beta_incomplete_generalized(1/2,3/2,0.25,0.31b0);
          (%o10)               9.6381780863686935309170054689964b-2

     Numerical evaluation for complex arguments in float or bigfloat
     precision:

          (%i11) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31);
          (%o11)           - .09625463003205376 %i - .003323847735353769
          (%i12) fpprec:20$
          (%i13) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31b0);
          (%o13)     - 9.6254630032054178691b-2 %i - 3.3238477353543591914b-3

     Expansion for a+n or a-n, <n> a positive integer, when
     `beta_expand' is `true':

          (%i14) beta_expand:true$

          (%i15) beta_incomplete_generalized(a+1,b,z1,z2);

                         b   a           b   a
                 (1 - z1)  z1  - (1 - z2)  z2
          (%o15) -----------------------------
                             b + a
                                a beta_incomplete_generalized(a, b, z1, z2)
                              + -------------------------------------------
                                                   b + a
          (%i16) beta_incomplete_generalized(a-1,b,z1,z2);

                 beta_incomplete_generalized(a, b, z1, z2) (- b - a + 1)
          (%o16) -------------------------------------------------------
                                          1 - a
                                              b   a - 1           b   a - 1
                                      (1 - z2)  z2      - (1 - z1)  z1
                                    - -------------------------------------
                                                      1 - a

     Derivative wrt the variable <z1> and integrals wrt <z1> and <z2>:

          (%i17) diff(beta_incomplete_generalized(a,b,z1,z2),z1);
                                         b - 1   a - 1
          (%o17)               - (1 - z1)      z1
          (%i18) integrate(beta_incomplete_generalized(a,b,z1,z2),z1);
          (%o18) beta_incomplete_generalized(a, b, z1, z2) z1
                                            + beta_incomplete(a + 1, b, z1)
          (%i19) integrate(beta_incomplete_generalized(a,b,z1,z2),z2);
          (%o19) beta_incomplete_generalized(a, b, z1, z2) z2
                                            - beta_incomplete(a + 1, b, z2)

Option variable: beta_expand

     Default value: false

     When `beta_expand' is `true', `beta(a,b)' and related functions
     are expanded for arguments like a+n or a-n, where n is an integer.

Option variable: beta_args_sum_to_integer

     Default value: false

     When `beta_args_sum_to_integer' is `true', Maxima simplifies
     `beta(a,b)', when the arguments <a> and <b> sum to an integer.

Function: psi [<n>](<x>)

     The derivative of `log (gamma (<x>))' of order `<n>+1'.  Thus,
     `psi[0](<x>)' is the first derivative, `psi[1](<x>)' is the second
     derivative, etc.

     Maxima does not know how, in general, to compute a numerical value
     of `psi', but it can compute some exact values for rational args.
     Several variables control what range of rational args `psi' will
     return an exact value, if possible.  See `maxpsiposint',
     `maxpsinegint', `maxpsifracnum', and `maxpsifracdenom'.  That is,
     <x> must lie between `maxpsinegint' and `maxpsiposint'.  If the
     absolute value of the fractional part of <x> is rational and has a
     numerator less than `maxpsifracnum' and has a denominator less
     than `maxpsifracdenom', `psi' will return an exact value.

     The function `bfpsi' in the `bffac' package can compute numerical
     values.

Option variable: maxpsiposint

     Default value: 20

     `maxpsiposint' is the largest positive value for which `psi[n](x)'
     will try to compute an exact value.

Option variable: maxpsinegint

     Default value: -10

     `maxpsinegint' is the most negative value for which `psi[n](x)'
     will try to compute an exact value.  That is if <x> is less than
     `maxnegint', `psi[n](<x>)' will not return simplified answer, even
     if it could.

Option variable: maxpsifracnum

     Default value: 6

     Let <x> be a rational number less than one of the form `p/q'.  If
     `p' is greater than `maxpsifracnum', then `psi[<n>](<x>)' will not
     try to return a simplified value.

Option variable: maxpsifracdenom

     Default value: 6

     Let <x> be a rational number less than one of the form `p/q'.  If
     `q' is greater than `maxpsifracdenom', then `psi[<n>](<x>)' will
     not try to return a simplified value.

Function: makefact (<expr>)

     Transforms instances of binomial, gamma, and beta functions in
     <expr> into factorials.

     See also `makegamma'.

Function: numfactor (<expr>)

     Returns the numerical factor multiplying the expression <expr>,
     which should be a single term.

     `content' returns the greatest common divisor (gcd) of all terms
     in a sum.

          (%i1) gamma (7/2);
                                    15 sqrt(%pi)
          (%o1)                     ------------
                                         8
          (%i2) numfactor (%);
                                         15
          (%o2)                          --
                                         8

15.5 Exponential Integrals

The Exponential Integral and related funtions are defined in Abramowitz
and Stegun, Handbook of Mathematical Functions, Chapter 5
Function: expintegral_e1 (<z>)

     The Exponential Integral E1(z) (A&S 5.1.1)

Function: expintegral_ei (<z>)

     The Exponential Integral Ei(z) (A&S 5.1.2)

Function: expintegral_li (<z>)

     The Exponential Integral Li(z)  (A&S 5.1.3)

Function: expintegral_e (<n>,<z>)

     The Exponential Integral En(z)  (A&S 5.1.4)

Function: expintegral_si (<z>)

     The Exponential Integral Si(z) (A&S 5.2.1)

Function: expintegral_ci (<z>)

     The Exponential Integral Ci(z) (A&S 5.2.2)

Function: expintegral_shi (<z>)

     The Exponential Integral Shi(z) (A&S 5.2.3)

Function: expintegral_chi (<z>)

     The Exponential Integral Chi(z) (A&S 5.2.4)

Option variable: expintrep

     Default value: false

     Change the representation of the Exponential Integral to
     gamma_incomplete, expintegral_e1, expintegral_ei,
     expintegral_li, expintegral_trig, expintegral_hyp

Option variable: expintexpand

     Default value: false

     Expand the Exponential Integral E[n](z) for half integral values
     in terms of Erfc or Erf and for positive integers in terms of Ei

15.6 Error Function

The Error function and related funtions are defined in Abramowitz and
Stegun, Handbook of Mathematical Functions, Chapter 7
Function: erf (<z>)

     The Error Function erf(z) (A&S 7.1.1)

     See also flag `erfflag'.

Function: erfc (<z>)

     The Complementary Error Function erfc(z) (A&S 7.1.2)

     `erfc(z) = 1-erf(z)'

Function: erfi (<z>)

     The Imaginary Error Function.

     `erfi(z) = -%i*erf(%i*z)'

Function: erf_generalized (<z1>,<z2>)

     Generalized Error function Erf(z1,z2)

Function: fresnel_c (<z>)

     The Fresnel Integral C(z) = integrate(cos((%pi/2)*t^2),t,0,z).
     (A&S 7.3.1)

     The simplification fresnel_c(-x) = -fresnel_c(x) is applied when
     flag `trigsign' is true.

     The simplification fresnel_c(%i*x) =  %i*fresnel_c(x) is applied
     when flag `%iargs' is true.

     See flags `erf_representation' and `hypergeometric_representation'.

Function: fresnel_s (<z>)

     The Fresnel Integral S(z) = integrate(sin((%pi/2)*t^2),t,0,z).
     (A&S 7.3.2)

     The simplification fresnel_s(-x) = -fresnel_s(x) is applied when
     flag `trigsign' is true.

     The simplification fresnel_s(%i*x) =  %i*fresnel_s(x) is applied
     when flag `%iargs' is true.

     See flags `erf_representation' and `hypergeometric_representation'.

Option variable: erf_representation

     Default value: false

     When T erfc, erfi, erf_generalized, fresnel_s and fresnel_c are
     transformed to erf.

Option variable: hypergeometric_representation

     Default value: false

     Enables transformation to a Hypergeometric representation for
     fresnel_s and fresnel_c

15.7 Struve Functions

The Struve functions are defined in Abramowitz and Stegun, Handbook of
Mathematical Functions, Chapter 12.
Function: struve_h (<v>, <z>)

     The Struve Function H of order v and argument z. (A&S 12.1.1)

Function: struve_l (<v>, <z>)

     The Modified Struve Function L of order v and argument z. (A&S
     12.2.1)

15.8 Hypergeometric Functions

The Hypergeometric Functions are defined in Abramowitz and Stegun,
Handbook of Mathematical Functions, Chapters 13 and 15.

   Maxima has very limited knowledge of these functions.  They can be
returned from function `hgfred'.
Function: %m [<k>,<u>] (<z>)

     Whittaker M function `M[k,u](z) =
     exp(-z/2)*z^(1/2+u)*M(1/2+u-k,1+2*u,z)'.  (A&S 13.1.32)

Function: %w [<k>,<u>] (<z>)

     Whittaker W function.  (A&S 13.1.33)

Function: %f [<p>,<q>] (<[a],[b],z>)

     The pFq(a1,a2,..ap;b1,b2,..bq;z) hypergeometric function, where
     `a' a list of length `p' and `b' a list of length `q'.

Function: hypergeometric ([<a1>, ..., <ap>],[<b1>, ... ,<bq>], x)

     The hypergeometric function. Unlike Maxima's `%f' hypergeometric
     function, the function `hypergeometric' is a simplifying function;
     also, `hypergeometric' supports complex double and big floating
     point evaluation. For the Gauss hypergeometric function, that is p
     = 2 and q = 1, floating point evaluation outside the unit circle
     is supported, but in general, it is not supported.

     When the option variable `expand_hypergeometric' is true (default
     is false) and one of the arguments `a1' through `ap' is a negative
     integer (a polynomial case), `hypergeometric' returns an expanded
     polynomial.

     Examples:

          (%i1)  hypergeometric([],[],x);
          (%o1) %e^x

     Polynomial cases automatically expand when `expand_hypergeometric'
     is true:

          (%i2) hypergeometric([-3],[7],x);
          (%o2) hypergeometric([-3],[7],x)

          (%i3) hypergeometric([-3],[7],x), expand_hypergeometric : true;
          (%o3) -x^3/504+3*x^2/56-3*x/7+1

     Both double float and big float evaluation is supported:

          (%i4) hypergeometric([5.1],[7.1 + %i],0.42);
          (%o4)       1.346250786375334 - 0.0559061414208204 %i
          (%i5) hypergeometric([5,6],[8], 5.7 - %i);
          (%o5)     .007375824009774946 - .001049813688578674 %i
          (%i6) hypergeometric([5,6],[8], 5.7b0 - %i), fpprec : 30;
          (%o6) 7.37582400977494674506442010824b-3
                                    - 1.04981368857867315858055393376b-3 %i

15.9 Parabolic Cylinder Functions

The Parabolic Cylinder Functions are defined in Abramowitz and Stegun,
Handbook of Mathematical Functions, Chapter 19.

   Maxima has very limited knowledge of these functions.  They can be
returned from function `hgfred'.
Function: parabolic_cylinder_d (<v>, <z>)

     The parabolic cylinder function `parabolic_cylinder_d(v,z)'. (A&s
     19.3.1)

15.10 Functions and Variables for Special Functions

Function: specint (exp(- s*<t>) * <expr>, <t>)

     Compute the Laplace transform of <expr> with respect to the
     variable <t>.  The integrand <expr> may contain special functions.

     The following special functions are handled by `specint':
     incomplete gamma function, error functions (but not the error
     function `erfi', it is easy to transform `erfi' e.g. to the error
     function `erf'), exponential integrals, bessel functions
     (including products of bessel functions), hankel functions,
     hermite and the laguerre polynomials.

     Furthermore, `specint' can handle the hypergeometric function
     `%f[p,q]([],[],z)', the whittaker function of the first kind
     `%m[u,k](z)' and of the second kind `%w[u,k](z)'.

     The result may be in terms of special functions and can include
     unsimplified hypergeomtric functions.

     When `laplace' fails to find a Laplace transform, `specint' is
     called.  Because `laplace' knows more general rules for Laplace
     transforms, it is preferable to use `laplace' and not `specint'.

     `demo(hypgeo)' displays several examples of Laplace transforms
     computed by `specint'.

     Examples:

          (%i1) assume (p > 0, a > 0)$
          (%i2) specint (t^(1/2) * exp(-a*t/4) * exp(-p*t), t);
                                     sqrt(%pi)
          (%o2)                     ------------
                                           a 3/2
                                    2 (p + -)
                                           4
          (%i3) specint (t^(1/2) * bessel_j(1, 2 * a^(1/2) * t^(1/2))
                        * exp(-p*t), t);
                                             - a/p
                                   sqrt(a) %e
          (%o3)                    ---------------
                                          2
                                         p

     Examples for exponential integrals:

          (%i4) assume(s>0,a>0,s-a>0)$
          (%i5) ratsimp(specint(%e^(a*t)
                                *(log(a)+expintegral_e1(a*t))*%e^(-s*t),t));
                                       log(s)
          (%o5)                        ------
                                       s - a
          (%i6) logarc:true$

          (%i7) gamma_expand:true$

          radcan(specint((cos(t)*expintegral_si(t)
                               -sin(t)*expintegral_ci(t))*%e^(-s*t),t));
                                       log(s)
          (%o8)                        ------
                                        2
                                       s  + 1
          ratsimp(specint((2*t*log(a)+2/a*sin(a*t)
                                -2*t*expintegral_ci(a*t))*%e^(-s*t),t));
                                         2    2
                                    log(s  + a )
          (%o9)                     ------------
                                          2
                                         s

     Results when using the expansion of `gamma_incomplete' and when
     changing the representation to `expintegral_e1':

          (%i10) assume(s>0)$
          (%i11) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
                                                      1
                                      gamma_incomplete(-, k s)
                                                      2
          (%o11)                      ------------------------
                                         sqrt(%pi) sqrt(s)

          (%i12) gamma_expand:true$
          (%i13) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
                                        erfc(sqrt(k) sqrt(s))
          (%o13)                        ---------------------
                                               sqrt(s)

          (%i14) expintrep:expintegral_e1$
          (%i15) ratsimp(specint(1/(t+a)^2*%e^(-s*t),t));
                                        a s
                                  a s %e    expintegral_e1(a s) - 1
          (%o15)                - ---------------------------------
                                                  a

Function: hgfred (<a>, <b>, <t>)

     Simplify the generalized hypergeometric function in terms of other,
     simpler, forms.  <a> is a list of numerator parameters and <b> is
     a list of the denominator parameters.

     If `hgfred' cannot simplify the hypergeometric function, it returns
     an expression of the form `%f[p,q]([a], [b], x)' where <p> is the
     number of elements in <a>, and <q> is the number of elements in
     <b>.  This is the usual `pFq' generalized hypergeometric function.

          (%i1) assume(not(equal(z,0)));
          (%o1)                          [notequal(z, 0)]
          (%i2) hgfred([v+1/2],[2*v+1],2*%i*z);

                               v/2                               %i z
                              4    bessel_j(v, z) gamma(v + 1) %e
          (%o2)               ---------------------------------------
                                                 v
                                                z
          (%i3) hgfred([1,1],[2],z);

                                             log(1 - z)
          (%o3)                            - ----------
                                                 z
          (%i4) hgfred([a,a+1/2],[3/2],z^2);

                                         1 - 2 a          1 - 2 a
                                  (z + 1)        - (1 - z)
          (%o4)                   -------------------------------
                                           2 (1 - 2 a) z

     It can be beneficial to load orthopoly too as the following example
     shows.  Note that <L> is the generalized Laguerre polynomial.

          (%i5) load(orthopoly)$
          (%i6) hgfred([-2],[a],z);

                                              (a - 1)
                                           2 L       (z)
                                              2
          (%o6)                            -------------
                                             a (a + 1)
          (%i7) ev(%);

                                            2
                                           z        2 z
          (%o7)                         --------- - --- + 1
                                        a (a + 1)    a

Function: lambert_w (<z>)

     The principal branch of Lambert's W funtion W(z), the solution of
     `z = W(z) * exp(W(z))'.

Function: nzeta (<z>)

     The Plasma Dispersion Function `nzeta(z) =
     %i*sqrt(%pi)*exp(-z^2)*(1-erf(-%i*z))'

Function: nzetar (<z>)

     Returns `realpart(nzeta(z))'.

Function: nzetai (<z>)

     Returns `imagpart(nzeta(z))'.

16 Elliptic Functions


Introduction to Elliptic Functions and Integrals
Functions and Variables for Elliptic Functions
Functions and Variables for Elliptic Integrals

16.1 Introduction to Elliptic Functions and Integrals

Maxima includes support for Jacobian elliptic functions and for
complete and incomplete elliptic integrals.  This includes symbolic
manipulation of these functions and numerical evaluation as well.
Definitions of these functions and many of their properties can by
found in Abramowitz and Stegun, Chapter 16-17.  As much as possible, we
use the definitions and relationships given there.

   In particular, all elliptic functions and integrals use the parameter
m instead of the modulus k or the modular angle \alpha.  This is one
area where we differ from Abramowitz and Stegun who use the modular
angle for the elliptic functions.  The following relationships are true:
m = k^2 and k = \sin(\alpha)

   The elliptic functions and integrals are primarily intended to
support symbolic computation.  Therefore, most of derivatives of the
functions and integrals are known.  However, if floating-point values
are given, a floating-point result is returned.

   Support for most of the other properties of elliptic functions and
integrals other than derivatives has not yet been written.

   Some examples of elliptic functions:
     (%i1) jacobi_sn (u, m);
     (%o1)                    jacobi_sn(u, m)
     (%i2) jacobi_sn (u, 1);
     (%o2)                        tanh(u)
     (%i3) jacobi_sn (u, 0);
     (%o3)                        sin(u)
     (%i4) diff (jacobi_sn (u, m), u);
     (%o4)            jacobi_cn(u, m) jacobi_dn(u, m)
     (%i5) diff (jacobi_sn (u, m), m);
     (%o5) jacobi_cn(u, m) jacobi_dn(u, m)

           elliptic_e(asin(jacobi_sn(u, m)), m)
      (u - ------------------------------------)/(2 m)
                          1 - m

                 2
        jacobi_cn (u, m) jacobi_sn(u, m)
      + --------------------------------
                   2 (1 - m)

   Some examples of elliptic integrals:
     (%i1) elliptic_f (phi, m);
     (%o1)                  elliptic_f(phi, m)
     (%i2) elliptic_f (phi, 0);
     (%o2)                          phi
     (%i3) elliptic_f (phi, 1);
                                    phi   %pi
     (%o3)                  log(tan(--- + ---))
                                     2     4
     (%i4) elliptic_e (phi, 1);
     (%o4)                       sin(phi)
     (%i5) elliptic_e (phi, 0);
     (%o5)                          phi
     (%i6) elliptic_kc (1/2);
                                          1
     (%o6)                    elliptic_kc(-)
                                          2
     (%i7) makegamma (%);
                                      2 1
                                 gamma (-)
                                        4
     (%o7)                      -----------
                                4 sqrt(%pi)
     (%i8) diff (elliptic_f (phi, m), phi);
                                     1
     (%o8)                 ---------------------
                                         2
                           sqrt(1 - m sin (phi))
     (%i9) diff (elliptic_f (phi, m), m);
            elliptic_e(phi, m) - (1 - m) elliptic_f(phi, m)
     (%o9) (-----------------------------------------------
                                   m

                                      cos(phi) sin(phi)
                                  - ---------------------)/(2 (1 - m))
                                                  2
                                    sqrt(1 - m sin (phi))

   Support for elliptic functions and integrals was written by Raymond
Toy.  It is placed under the terms of the General Public License (GPL)
that governs the distribution of Maxima.

16.2 Functions and Variables for Elliptic Functions

Function: jacobi_sn (<u>, <m>)

     The Jacobian elliptic function sn(u,m).

Function: jacobi_cn (<u>, <m>)

     The Jacobian elliptic function cn(u,m).

Function: jacobi_dn (<u>, <m>)

     The Jacobian elliptic function dn(u,m).

Function: jacobi_ns (<u>, <m>)

     The Jacobian elliptic function ns(u,m) = 1/sn(u,m).

Function: jacobi_sc (<u>, <m>)

     The Jacobian elliptic function sc(u,m) = sn(u,m)/cn(u,m).

Function: jacobi_sd (<u>, <m>)

     The Jacobian elliptic function sd(u,m) = sn(u,m)/dn(u,m).

Function: jacobi_nc (<u>, <m>)

     The Jacobian elliptic function nc(u,m) = 1/cn(u,m).

Function: jacobi_cs (<u>, <m>)

     The Jacobian elliptic function cs(u,m) = cn(u,m)/sn(u,m).

Function: jacobi_cd (<u>, <m>)

     The Jacobian elliptic function cd(u,m) = cn(u,m)/dn(u,m).

Function: jacobi_nd (<u>, <m>)

     The Jacobian elliptic function nc(u,m) = 1/cn(u,m).

Function: jacobi_ds (<u>, <m>)

     The Jacobian elliptic function ds(u,m) = dn(u,m)/sn(u,m).

Function: jacobi_dc (<u>, <m>)

     The Jacobian elliptic function dc(u,m) = dn(u,m)/cn(u,m).

Function: inverse_jacobi_sn (<u>, <m>)

     The inverse of the Jacobian elliptic function sn(u,m).

Function: inverse_jacobi_cn (<u>, <m>)

     The inverse of the Jacobian elliptic function cn(u,m).

Function: inverse_jacobi_dn (<u>, <m>)

     The inverse of the Jacobian elliptic function dn(u,m).

Function: inverse_jacobi_ns (<u>, <m>)

     The inverse of the Jacobian elliptic function ns(u,m).

Function: inverse_jacobi_sc (<u>, <m>)

     The inverse of the Jacobian elliptic function sc(u,m).

Function: inverse_jacobi_sd (<u>, <m>)

     The inverse of the Jacobian elliptic function sd(u,m).

Function: inverse_jacobi_nc (<u>, <m>)

     The inverse of the Jacobian elliptic function nc(u,m).

Function: inverse_jacobi_cs (<u>, <m>)

     The inverse of the Jacobian elliptic function cs(u,m).

Function: inverse_jacobi_cd (<u>, <m>)

     The inverse of the Jacobian elliptic function cd(u,m).

Function: inverse_jacobi_nd (<u>, <m>)

     The inverse of the Jacobian elliptic function nc(u,m).

Function: inverse_jacobi_ds (<u>, <m>)

     The inverse of the Jacobian elliptic function ds(u,m).

Function: inverse_jacobi_dc (<u>, <m>)

     The inverse of the Jacobian elliptic function dc(u,m).

16.3 Functions and Variables for Elliptic Integrals

Function: elliptic_f (<phi>, <m>)

     The incomplete elliptic integral of the first kind, defined as

     integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi)

     See also *note elliptic_e:: and *note elliptic_kc::.

Function: elliptic_e (<phi>, <m>)

     The incomplete elliptic integral of the second kind, defined as

     elliptic_e(phi, m) = integrate(sqrt(1 - m*sin(x)^2), x, 0, phi)

     See also *note elliptic_e:: and *note elliptic_ec::.

Function: elliptic_eu (<u>, <m>)

     The incomplete elliptic integral of the second kind, defined as

     integrate(dn(v,m)^2,v,0,u) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2),
     t, 0, tau)

     where tau = sn(u,m).

     This is related to elliptic_e by

     elliptic_eu(u, m) = elliptic_e(asin(sn(u,m)),m)

     See also *note elliptic_e::.

Function: elliptic_pi (<n>, <phi>, <m>)

     The incomplete elliptic integral of the third kind, defined as

     integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, 0, phi)

     Only the derivative with respect to phi is known by Maxima.

Function: elliptic_kc (<m>)

     The complete elliptic integral of the first kind, defined as

     integrate(1/sqrt(1 - m*sin(x)^2), x, 0, %pi/2)

     For certain values of m, the value of the integral is known in
     terms of Gamma functions.  Use `makegamma' to evaluate them.

Function: elliptic_ec (<m>)

     The complete elliptic integral of the second kind, defined as

     integrate(sqrt(1 - m*sin(x)^2), x, 0, %pi/2)

     For certain values of m, the value of the integral is known in
     terms of Gamma functions.  Use `makegamma' to evaluate them.

17 Limits


Functions and Variables for Limits

17.1 Functions and Variables for Limits

Option variable: lhospitallim

     Default: 4

     `lhospitallim' is the maximum number of times L'Hospital's rule is
     used in `limit'.  This prevents infinite looping in cases like
     `limit (cot(x)/csc(x), x, 0)'.

Function: limit (<expr>, <x>, <val>, <dir>)
Function: limit (<expr>, <x>, <val>)
Function: limit (<expr>)

     Computes the limit of <expr> as the real variable <x> approaches
     the value <val> from the direction <dir>.  <dir> may have the value
     `plus' for a limit from above, `minus' for a limit from below, or
     may be omitted (implying a two-sided limit is to be computed).

     `limit' uses the following special symbols: `inf' (positive
     infinity) and `minf' (negative infinity).  On output it may also
     use `und' (undefined), `ind' (indefinite but bounded) and
     `infinity' (complex infinity).

     `infinity' (complex infinity) is returned when the limit of the
     absolute value of the expression is positive infinity, but the
     limit of the expression itself is not positive infinity or
     negative infinity.  This includes cases where the limit of the
     complex argument is a constant, as in `limit(log(x), x, minf)',
     cases where the complex argument oscillates, as in `limit((-2)^x,
     x, inf)', and cases where the complex argument is different for
     either side of a two-sided limit, as in `limit(1/x, x, 0)' and
     `limit(log(x), x, 0)'.

     `lhospitallim' is the maximum number of times L'Hospital's rule is
     used in `limit'.  This prevents infinite looping in cases like
     `limit (cot(x)/csc(x), x, 0)'.

     `tlimswitch' when true will allow the `limit' command to use
     Taylor series expansion when necessary.

     `limsubst' prevents `limit' from attempting substitutions on
     unknown forms.  This is to avoid bugs like `limit (f(n)/f(n+1), n,
     inf)' giving 1.  Setting `limsubst' to `true' will allow such
     substitutions.

     `limit' with one argument is often called upon to simplify constant
     expressions, for example, `limit (inf-1)'.

     `example (limit)' displays some examples.

     For the method see Wang, P., "Evaluation of Definite Integrals by
     Symbolic Manipulation", Ph.D. thesis, MAC TR-92, October 1971.

Option variable: limsubst

     Default value: `false'

     prevents `limit' from attempting substitutions on unknown forms.
     This is to avoid bugs like `limit (f(n)/f(n+1), n, inf)' giving 1.
     Setting `limsubst' to `true' will allow such substitutions.

Function: tlimit (<expr>, <x>, <val>, <dir>)
Function: tlimit (<expr>, <x>, <val>)
Function: tlimit (<expr>)

     Take the limit of the Taylor series expansion of `expr' in `x' at
     `val' from direction `dir'.

Option variable: tlimswitch

     Default value: `true'

     When `tlimswitch' is `true', the `limit' command will use a Taylor
     series expansion if the limit of the input expression cannot be
     computed directly.  This allows evaluation of limits such as
     `limit(x/(x-1)-1/log(x),x,1,plus)'.  When `tlimswitch' is `false'
     and the limit of input expression cannot be computed directly,
     `limit' will return an unevaluated limit expression.

18 Differentiation


Functions and Variables for Differentiation

18.1 Functions and Variables for Differentiation

Function: antid (<expr>, <x>, <u(x)>)

     Returns a two-element list, such that an antiderivative of <expr>
     with respect to <x> can be constructed from the list.  The
     expression <expr> may contain an unknown function <u> and its
     derivatives.

     Let <L>, a list of two elements, be the return value of `antid'.
     Then `<L>[1] + 'integrate (<L>[2], <x>)' is an antiderivative of
     <expr> with respect to <x>.

     When `antid' succeeds entirely, the second element of the return
     value is zero.  Otherwise, the second element is nonzero, and the
     first element is nonzero or zero.  If `antid' cannot make any
     progress, the first element is zero and the second nonzero.

     `load ("antid")' loads this function.  The `antid' package also
     defines the functions `nonzeroandfreeof' and `linear'.

     `antid' is related to `antidiff' as follows.  Let <L>, a list of
     two elements, be the return value of `antid'.  Then the return
     value of `antidiff' is equal to `<L>[1] + 'integrate (<L>[2],
     <x>)' where <x> is the variable of integration.

     Examples:

          (%i1) load ("antid")$
          (%i2) expr: exp (z(x)) * diff (z(x), x) * y(x);
                                      z(x)  d
          (%o2)                y(x) %e     (-- (z(x)))
                                            dx
          (%i3) a1: antid (expr, x, z(x));
                                 z(x)      z(x)  d
          (%o3)          [y(x) %e    , - %e     (-- (y(x)))]
                                                 dx
          (%i4) a2: antidiff (expr, x, z(x));
                                      /
                               z(x)   [   z(x)  d
          (%o4)         y(x) %e     - I %e     (-- (y(x))) dx
                                      ]         dx
                                      /
          (%i5) a2 - (first (a1) + 'integrate (second (a1), x));
          (%o5)                           0
          (%i6) antid (expr, x, y(x));
                                       z(x)  d
          (%o6)             [0, y(x) %e     (-- (z(x)))]
                                             dx
          (%i7) antidiff (expr, x, y(x));
                            /
                            [        z(x)  d
          (%o7)             I y(x) %e     (-- (z(x))) dx
                            ]              dx
                            /

Function: antidiff (<expr>, <x>, <u>(<x>))

     Returns an antiderivative of <expr> with respect to <x>.  The
     expression <expr> may contain an unknown function <u> and its
     derivatives.

     When `antidiff' succeeds entirely, the resulting expression is
     free of integral signs (that is, free of the `integrate' noun).
     Otherwise, `antidiff' returns an expression which is partly or
     entirely within an integral sign.  If `antidiff' cannot make any
     progress, the return value is entirely within an integral sign.

     `load ("antid")' loads this function.  The `antid' package also
     defines the functions `nonzeroandfreeof' and `linear'.

     `antidiff' is related to `antid' as follows.  Let <L>, a list of
     two elements, be the return value of `antid'.  Then the return
     value of `antidiff' is equal to `<L>[1] + 'integrate (<L>[2],
     <x>)' where <x> is the variable of integration.

     Examples:

          (%i1) load ("antid")$
          (%i2) expr: exp (z(x)) * diff (z(x), x) * y(x);
                                      z(x)  d
          (%o2)                y(x) %e     (-- (z(x)))
                                            dx
          (%i3) a1: antid (expr, x, z(x));
                                 z(x)      z(x)  d
          (%o3)          [y(x) %e    , - %e     (-- (y(x)))]
                                                 dx
          (%i4) a2: antidiff (expr, x, z(x));
                                      /
                               z(x)   [   z(x)  d
          (%o4)         y(x) %e     - I %e     (-- (y(x))) dx
                                      ]         dx
                                      /
          (%i5) a2 - (first (a1) + 'integrate (second (a1), x));
          (%o5)                           0
          (%i6) antid (expr, x, y(x));
                                       z(x)  d
          (%o6)             [0, y(x) %e     (-- (z(x)))]
                                             dx
          (%i7) antidiff (expr, x, y(x));
                            /
                            [        z(x)  d
          (%o7)             I y(x) %e     (-- (z(x))) dx
                            ]              dx
                            /

Function: at (<expr>, [<eqn_1>, ..., <eqn_n>])
Function: at (<expr>, <eqn>)

     Evaluates the expression <expr> with the variables assuming the
     values as specified for them in the list of equations `[<eqn_1>,
     ..., <eqn_n>]' or the single equation <eqn>.

     If a subexpression depends on any of the variables for which a
     value is specified but there is no `atvalue' specified and it
     can't be otherwise evaluated, then a noun form of the `at' is
     returned which displays in a two-dimensional form.

     `at' carries out multiple substitutions in parallel.

     See also `atvalue'.  For other functions which carry out
     substitutions, see also `subst' and `ev'.

     Examples:

          (%i1) atvalue (f(x,y), [x = 0, y = 1], a^2);
                                          2
          (%o1)                          a
          (%i2) atvalue ('diff (f(x,y), x), x = 0, 1 + y);
          (%o2)                        @2 + 1
          (%i3) printprops (all, atvalue);
                                          !
                            d             !
                           --- (f(@1, @2))!       = @2 + 1
                           d@1            !
                                          !@1 = 0

                                               2
                                    f(0, 1) = a

          (%o3)                         done
          (%i4) diff (4*f(x, y)^2 - u(x, y)^2, x);
                            d                          d
          (%o4)  8 f(x, y) (-- (f(x, y))) - 2 u(x, y) (-- (u(x, y)))
                            dx                         dx
          (%i5) at (%, [x = 0, y = 1]);
                                                   !
                        2              d           !
          (%o5)     16 a  - 2 u(0, 1) (-- (u(x, y))!            )
                                       dx          !
                                                   !x = 0, y = 1

Property: atomgrad

     `atomgrad' is the atomic gradient property of an expression.  This
     property is assigned by `gradef'.

Function: atvalue (<expr>, [<x_1> = <a_1>, ..., <x_m> = <a_m>], <c>)
Function: atvalue (<expr>, <x_1> = <a_1>, <c>)

     Assigns the value <c> to <expr> at the point `<x> = <a>'.
     Typically boundary values are established by this mechanism.

     <expr> is a function evaluation, `<f>(<x_1>, ..., <x_m>)', or a
     derivative, `diff (<f>(<x_1>, ..., <x_m>), <x_1>, <n_1>, ...,
     <x_n>, <n_m>)' in which the function arguments explicitly appear.
     <n_i> is the order of differentiation with respect to <x_i>.

     The point at which the atvalue is established is given by the list
     of equations `[<x_1> = <a_1>, ..., <x_m> = <a_m>]'.  If there is a
     single variable <x_1>, the sole equation may be given without
     enclosing it in a list.

     `printprops ([<f_1>, <f_2>, ...], atvalue)' displays the atvalues
     of the functions `<f_1>, <f_2>, ...' as specified by calls to
     `atvalue'.  `printprops (<f>, atvalue)' displays the atvalues of
     one function <f>.  `printprops (all, atvalue)' displays the
     atvalues of all functions for which atvalues are defined.

     The symbols `@1', `@2', ... represent the variables <x_1>, <x_2>,
     ... when atvalues are displayed.

     `atvalue' evaluates its arguments.  `atvalue' returns <c>, the
     atvalue.

     Examples:

          (%i1) atvalue (f(x,y), [x = 0, y = 1], a^2);
                                          2
          (%o1)                          a
          (%i2) atvalue ('diff (f(x,y), x), x = 0, 1 + y);
          (%o2)                        @2 + 1
          (%i3) printprops (all, atvalue);
                                          !
                            d             !
                           --- (f(@1, @2))!       = @2 + 1
                           d@1            !
                                          !@1 = 0

                                               2
                                    f(0, 1) = a

          (%o3)                         done
          (%i4) diff (4*f(x,y)^2 - u(x,y)^2, x);
                            d                          d
          (%o4)  8 f(x, y) (-- (f(x, y))) - 2 u(x, y) (-- (u(x, y)))
                            dx                         dx
          (%i5) at (%, [x = 0, y = 1]);
                                                   !
                        2              d           !
          (%o5)     16 a  - 2 u(0, 1) (-- (u(x, y))!            )
                                       dx          !
                                                   !x = 0, y = 1

Function: cartan

     The exterior calculus of differential forms is a basic tool of
     differential geometry developed by Elie Cartan and has important
     applications in the theory of partial differential equations.  The
     `cartan' package implements the functions `ext_diff' and
     `lie_diff', along with the operators `~' (wedge product) and `|'
     (contraction of a form with a vector.)  Type `demo (tensor)' to
     see a brief description of these commands along with examples.

     `cartan' was implemented by F.B. Estabrook and H.D. Wahlquist.

Function: del (<x>)

     `del (<x>)' represents the differential of the variable x.

     `diff' returns an expression containing `del' if an independent
     variable is not specified.  In this case, the return value is the
     so-called "total differential".

     Examples:

          (%i1) diff (log (x));
                                       del(x)
          (%o1)                        ------
                                         x
          (%i2) diff (exp (x*y));
                               x y              x y
          (%o2)            x %e    del(y) + y %e    del(x)
          (%i3) diff (x*y*z);
          (%o3)         x y del(z) + x z del(y) + y z del(x)

Function: delta (<t>)

     The Dirac Delta function.

     Currently only `laplace' knows about the `delta' function.

     Example:

          (%i1) laplace (delta (t - a) * sin(b*t), t, s);
          Is  a  positive, negative, or zero?

          p;
                                             - a s
          (%o1)                   sin(a b) %e

System variable: dependencies

     Default value: `[]'

     `dependencies' is the list of atoms which have functional
     dependencies, assigned by `depends' or `gradef'.  The
     `dependencies' list is cumulative: each call to `depends' or
     `gradef' appends additional items.

     See `depends' and `gradef'.

Function: depends (<f_1>, <x_1>, ..., <f_n>, <x_n>)

     Declares functional dependencies among variables for the purpose
     of computing derivatives.  In the absence of declared dependence,
     `diff (f, x)' yields zero.  If `depends (f, x)' is declared, `diff
     (f, x)' yields a symbolic derivative (that is, a `diff' noun).

     Each argument <f_1>, <x_1>, etc., can be the name of a variable or
     array, or a list of names.  Every element of <f_i> (perhaps just a
     single element) is declared to depend on every element of <x_i>
     (perhaps just a single element).  If some <f_i> is the name of an
     array or contains the name of an array, all elements of the array
     depend on <x_i>.

     `diff' recognizes indirect dependencies established by `depends'
     and applies the chain rule in these cases.

     `remove (<f>, dependency)' removes all dependencies declared for
     <f>.

     `depends' returns a list of the dependencies established.  The
     dependencies are appended to the global variable `dependencies'.
     `depends' evaluates its arguments.

     `diff' is the only Maxima command which recognizes dependencies
     established by `depends'.  Other functions (`integrate',
     `laplace', etc.)  only recognize dependencies explicitly
     represented by their arguments.  For example, `integrate' does not
     recognize the dependence of `f' on `x' unless explicitly
     represented as `integrate (f(x), x)'.

          (%i1) depends ([f, g], x);
          (%o1)                     [f(x), g(x)]
          (%i2) depends ([r, s], [u, v, w]);
          (%o2)               [r(u, v, w), s(u, v, w)]
          (%i3) depends (u, t);
          (%o3)                        [u(t)]
          (%i4) dependencies;
          (%o4)      [f(x), g(x), r(u, v, w), s(u, v, w), u(t)]
          (%i5) diff (r.s, u);
                                   dr           ds
          (%o5)                    -- . s + r . --
                                   du           du

          (%i6) diff (r.s, t);
                                dr du           ds du
          (%o6)                 -- -- . s + r . -- --
                                du dt           du dt

          (%i7) remove (r, dependency);
          (%o7)                         done
          (%i8) diff (r.s, t);
                                          ds du
          (%o8)                       r . -- --
                                          du dt

Option variable: derivabbrev

     Default value: `false'

     When `derivabbrev' is `true', symbolic derivatives (that is,
     `diff' nouns) are displayed as subscripts.  Otherwise, derivatives
     are displayed in the Leibniz notation `dy/dx'.

Function: derivdegree (<expr>, <y>, <x>)

     Returns the highest degree of the derivative of the dependent
     variable <y> with respect to the independent variable <x> occuring
     in <expr>.

     Example:

          (%i1) 'diff (y, x, 2) + 'diff (y, z, 3) + 'diff (y, x) * x^2;
                                   3     2
                                  d y   d y    2 dy
          (%o1)                   --- + --- + x  --
                                    3     2      dx
                                  dz    dx
          (%i2) derivdegree (%, y, x);
          (%o2)                           2

Function: derivlist (<var_1>, ..., <var_k>)

     Causes only differentiations with respect to the indicated
     variables, within the `ev' command.

Option variable: derivsubst

     Default value: `false'

     When `derivsubst' is `true', a non-syntactic substitution such as
     `subst (x, 'diff (y, t), 'diff (y, t, 2))' yields `'diff (x, t)'.

Function: diff (<expr>, <x_1>, <n_1>, ..., <x_m>, <n_m>)
Function: diff (<expr>, <x>, <n>)
Function: diff (<expr>, <x>)
Function: diff (<expr>)

     Returns the derivative or differential of <expr> with respect to
     some or all variables in <expr>.

     `diff (<expr>, <x>, <n>)' returns the <n>'th derivative of <expr>
     with respect to <x>.

     `diff (<expr>, <x_1>, <n_1>, ..., <x_m>, <n_m>)' returns the mixed
     partial derivative of <expr> with respect to <x_1>, ..., <x_m>.
     It is equivalent to `diff (... (diff (<expr>, <x_m>, <n_m>) ...),
     <x_1>, <n_1>)'.

     `diff (<expr>, <x>)' returns the first derivative of <expr> with
     respect to the variable <x>.

     `diff (<expr>)' returns the total differential of <expr>, that is,
     the sum of the derivatives of <expr> with respect to each its
     variables times the differential `del' of each variable.  No
     further simplification of `del' is offered.

     The noun form of `diff' is required in some contexts, such as
     stating a differential equation.  In these cases, `diff' may be
     quoted (as `'diff') to yield the noun form instead of carrying out
     the differentiation.

     When `derivabbrev' is `true', derivatives are displayed as
     subscripts.  Otherwise, derivatives are displayed in the Leibniz
     notation, `dy/dx'.

     Examples:

          (%i1) diff (exp (f(x)), x, 2);
                               2
                        f(x)  d               f(x)  d         2
          (%o1)       %e     (--- (f(x))) + %e     (-- (f(x)))
                                2                   dx
                              dx
          (%i2) derivabbrev: true$
          (%i3) 'integrate (f(x, y), y, g(x), h(x));
                                   h(x)
                                  /
                                  [
          (%o3)                   I     f(x, y) dy
                                  ]
                                  /
                                   g(x)
          (%i4) diff (%, x);
                 h(x)
                /
                [
          (%o4) I     f(x, y)  dy + f(x, h(x)) h(x)  - f(x, g(x)) g(x)
                ]            x                     x                  x
                /
                 g(x)

     For the tensor package, the following modifications have been
     incorporated:

     (1) The derivatives of any indexed objects in <expr> will have the
     variables <x_i> appended as additional arguments.  Then all the
     derivative indices will be sorted.

     (2) The <x_i> may be integers from 1 up to the value of the
     variable `dimension' [default value: 4].  This will cause the
     differentiation to be carried out with respect to the <x_i>'th
     member of the list `coordinates' which should be set to a list of
     the names of the coordinates, e.g., `[x, y, z, t]'. If
     `coordinates' is bound to an atomic variable, then that variable
     subscripted by <x_i> will be used for the variable of
     differentiation.  This permits an array of coordinate names or
     subscripted names like `X[1]', `X[2]', ... to be used.  If
     `coordinates' has not been assigned a value, then the variables
     will be treated as in (1) above.

Special symbol: diff

     When `diff' is present as an `evflag' in call to `ev', all
     differentiations indicated in `expr' are carried out.

Function: dscalar (<f>)

     Applies the scalar d'Alembertian to the scalar function <f>.

     `load ("ctensor")' loads this function.

Function: express (<expr>)

     Expands differential operator nouns into expressions in terms of
     partial derivatives.  `express' recognizes the operators `grad',
     `div', `curl', `laplacian'.  `express' also expands the cross
     product `~'.

     Symbolic derivatives (that is, `diff' nouns) in the return value
     of express may be evaluated by including `diff' in the `ev'
     function call or command line.  In this context, `diff' acts as an
     `evfun'.

     `load ("vect")' loads this function.

     Examples:

          (%i1) load ("vect")$
          (%i2) grad (x^2 + y^2 + z^2);
                                        2    2    2
          (%o2)                  grad (z  + y  + x )
          (%i3) express (%);
                 d    2    2    2   d    2    2    2   d    2    2    2
          (%o3) [-- (z  + y  + x ), -- (z  + y  + x ), -- (z  + y  + x )]
                 dx                 dy                 dz
          (%i4) ev (%, diff);
          (%o4)                    [2 x, 2 y, 2 z]
          (%i5) div ([x^2, y^2, z^2]);
                                        2   2   2
          (%o5)                   div [x , y , z ]
          (%i6) express (%);
                             d    2    d    2    d    2
          (%o6)              -- (z ) + -- (y ) + -- (x )
                             dz        dy        dx
          (%i7) ev (%, diff);
          (%o7)                    2 z + 2 y + 2 x
          (%i8) curl ([x^2, y^2, z^2]);
                                         2   2   2
          (%o8)                   curl [x , y , z ]
          (%i9) express (%);
                 d    2    d    2   d    2    d    2   d    2    d    2
          (%o9) [-- (z ) - -- (y ), -- (x ) - -- (z ), -- (y ) - -- (x )]
                 dy        dz       dz        dx       dx        dy
          (%i10) ev (%, diff);
          (%o10)                      [0, 0, 0]
          (%i11) laplacian (x^2 * y^2 * z^2);
                                            2  2  2
          (%o11)                laplacian (x  y  z )
          (%i12) express (%);
                   2                2                2
                  d     2  2  2    d     2  2  2    d     2  2  2
          (%o12)  --- (x  y  z ) + --- (x  y  z ) + --- (x  y  z )
                    2                2                2
                  dz               dy               dx
          (%i13) ev (%, diff);
                                2  2      2  2      2  2
          (%o13)             2 y  z  + 2 x  z  + 2 x  y
          (%i14) [a, b, c] ~ [x, y, z];
          (%o14)                [a, b, c] ~ [x, y, z]
          (%i15) express (%);
          (%o15)          [b z - c y, c x - a z, a y - b x]

Function: gradef (<f>(<x_1>, ..., <x_n>), <g_1>, ..., <g_m>)
Function: gradef (<a>, <x>, <expr>)

     Defines the partial derivatives (i.e., the components of the
     gradient) of the function <f> or variable <a>.

     `gradef (<f>(<x_1>, ..., <x_n>), <g_1>, ..., <g_m>)' defines
     `d<f>/d<x_i>' as <g_i>, where <g_i> is an expression; <g_i> may be
     a function call, but not the name of a function.  The number of
     partial derivatives <m> may be less than the number of arguments
     <n>, in which case derivatives are defined with respect to <x_1>
     through <x_m> only.

     `gradef (<a>, <x>, <expr>)' defines the derivative of variable <a>
     with respect to <x> as <expr>.  This also establishes the
     dependence of <a> on <x> (via `depends (<a>, <x>)').

     The first argument `<f>(<x_1>, ..., <x_n>)' or <a> is quoted, but
     the remaining arguments <g_1>, ..., <g_m> are evaluated.  `gradef'
     returns the function or variable for which the partial derivatives
     are defined.

     `gradef' can redefine the derivatives of Maxima's built-in
     functions.  For example, `gradef (sin(x), sqrt (1 - sin(x)^2))'
     redefines the derivative of `sin'.

     `gradef' cannot define partial derivatives for a subscripted
     function.

     `printprops ([<f_1>, ..., <f_n>], gradef)' displays the partial
     derivatives of the functions <f_1>, ..., <f_n>, as defined by
     `gradef'.

     `printprops ([<a_n>, ..., <a_n>], atomgrad)' displays the partial
     derivatives of the variables <a_n>, ..., <a_n>, as defined by
     `gradef'.

     `gradefs' is the list of the functions for which partial
     derivatives have been defined by `gradef'.  `gradefs' does not
     include any variables for which partial derivatives have been
     defined by `gradef'.

     Gradients are needed when, for example, a function is not known
     explicitly but its first derivatives are and it is desired to
     obtain higher order derivatives.

System variable: gradefs

     Default value: `[]'

     `gradefs' is the list of the functions for which partial
     derivatives have been defined by `gradef'.  `gradefs' does not
     include any variables for which partial derivatives have been
     defined by `gradef'.

Function: laplace (<expr>, <t>, <s>)

     Attempts to compute the Laplace transform of <expr> with respect
     to the variable <t> and transform parameter <s>.

     `laplace' recognizes in <expr> the functions `delta', `exp',
     `log', `sin', `cos', `sinh', `cosh', and `erf', as well as
     `derivative', `integrate', `sum', and `ilt'.  If laplace fails to
     find a transform the function `specint' is called.  `specint' can
     find the laplace transform for expressions with special functions
     like the bessel functions `bessel_j', `bessel_i', ...  and can
     handle the `unit_step' function.  See also `specint'.

     If `specint' cannot find a solution too, a noun `laplace' is
     returned.

     <expr> may also be a linear, constant coefficient differential
     equation in which case `atvalue' of the dependent variable is used.
     The required atvalue may be supplied either before or after the
     transform is computed.  Since the initial conditions must be
     specified at zero, if one has boundary conditions imposed
     elsewhere he can impose these on the general solution and
     eliminate the constants by solving the general solution for them
     and substituting their values back.

     `laplace' recognizes convolution integrals of the form `integrate
     (f(x) * g(t - x), x, 0, t)'; other kinds of convolutions are not
     recognized.

     Functional relations must be explicitly represented in <expr>;
     implicit relations, established by `depends', are not recognized.
     That is, if <f> depends on <x> and <y>, `f (x, y)' must appear in
     <expr>.

     See also `ilt', the inverse Laplace transform.

     Examples:

          (%i1) laplace (exp (2*t + a) * sin(t) * t, t, s);
                                      a
                                    %e  (2 s - 4)
          (%o1)                    ---------------
                                     2           2
                                   (s  - 4 s + 5)
          (%i2) laplace ('diff (f (x), x), x, s);
          (%o2)             s laplace(f(x), x, s) - f(0)
          (%i3) diff (diff (delta (t), t), t);
                                    2
                                   d
          (%o3)                    --- (delta(t))
                                     2
                                   dt
          (%i4) laplace (%, t, s);
                                      !
                         d            !         2
          (%o4)        - -- (delta(t))!      + s  - delta(0) s
                         dt           !
                                      !t = 0
          (%i5) assume(a>0)$
          (%i6) laplace(gamma_incomplete(a,t),t,s),gamma_expand:true;
                                                        - a - 1
                                   gamma(a)   gamma(a) s
          (%o6)                    -------- - -----------------
                                      s            1     a
                                                  (- + 1)
                                                   s
          (%i7) factor(laplace(gamma_incomplete(1/2,t),t,s));
                                                        s + 1
                                sqrt(%pi) (sqrt(s) sqrt(-----) - 1)
                                                          s
          (%o7)                 -----------------------------------
                                          3/2      s + 1
                                         s    sqrt(-----)
                                                     s
          (%i8) assume(exp(%pi*s)>1)$
          (%i9) laplace(sum((-1)^n*unit_step(t-n*%pi)*sin(t),n,0,inf),t,s),
                   simpsum;
                                   %i                         %i
                        ------------------------ - ------------------------
                                        - %pi s                    - %pi s
                        (s + %i) (1 - %e       )   (s - %i) (1 - %e       )
          (%o9)         ---------------------------------------------------
                                                 2
          (%i9) factor(%);
                                                %pi s
                                              %e
          (%o9)                   -------------------------------
                                                       %pi s
                                  (s - %i) (s + %i) (%e      - 1)

19 Integration


Introduction to Integration
Functions and Variables for Integration
Introduction to QUADPACK
Functions and Variables for QUADPACK

19.1 Introduction to Integration

Maxima has several routines for handling integration.  The `integrate'
function makes use of most of them.  There is also the `antid' package,
which handles an unspecified function (and its derivatives, of course).
For numerical uses, there is a set of adaptive integrators from
QUADPACK, named `quad_qag', `quad_qags', etc., which are described
under the heading `QUADPACK'.  Hypergeometric functions are being
worked on, see `specint' for details.  Generally speaking, Maxima only
handles integrals which are integrable in terms of the "elementary
functions" (rational functions, trigonometrics, logs, exponentials,
radicals, etc.) and a few extensions (error function, dilogarithm).  It
does not handle integrals in terms of unknown functions such as `g(x)'
and `h(x)'.

19.2 Functions and Variables for Integration

Function: changevar (<expr>, <f(x,y)>, <y>, <x>)

     Makes the change of variable given by `<f(x,y)> = 0' in all
     integrals occurring in <expr> with integration with respect to <x>.
     The new variable is <y>.

          (%i1) assume(a > 0)$
          (%i2) 'integrate (%e**sqrt(a*y), y, 0, 4);
                                4
                               /
                               [    sqrt(a) sqrt(y)
          (%o2)                I  %e                dy
                               ]
                               /
                                0
          (%i3) changevar (%, y-z^2/a, z, y);
                                0
                               /
                               [                abs(z)
                             2 I            z %e       dz
                               ]
                               /
                                - 2 sqrt(a)
          (%o3)            - ----------------------------
                                          a

     An expression containing a noun form, such as the instances of
     `'integrate' above, may be evaluated by `ev' with the `nouns' flag.
     For example, the expression returned by `changevar' above may be
     evaluated by `ev (%o3, nouns)'.

     `changevar' may also be used to changes in the indices of a sum or
     product.  However, it must be realized that when a change is made
     in a sum or product, this change must be a shift, i.e., `i = j+
     ...', not a higher degree function.  E.g.,

          (%i4) sum (a[i]*x^(i-2), i, 0, inf);
                                   inf
                                   ====
                                   \         i - 2
          (%o4)                     >    a  x
                                   /      i
                                   ====
                                   i = 0
          (%i5) changevar (%, i-2-n, n, i);
                                  inf
                                  ====
                                  \               n
          (%o5)                    >      a      x
                                  /        n + 2
                                  ====
                                  n = - 2

Function: dblint (<f>, <r>, <s>, <a>, <b>)

     A double-integral routine which was written in top-level Maxima
     and then translated and compiled to machine code.  Use `load
     (dblint)' to access this package.  It uses the Simpson's rule
     method in both the x and y directions to calculate

          /b /s(x)
          |  |
          |  |    f(x,y) dy dx
          |  |
          /a /r(x)

     The function <f> must be a translated or compiled function of two
     variables, and <r> and <s> must each be a translated or compiled
     function of one variable, while <a> and <b> must be floating point
     numbers.  The routine has two global variables which determine the
     number of divisions of the x and y intervals: `dblint_x' and
     `dblint_y', both of which are initially 10, and can be changed
     independently to other integer values (there are `2*dblint_x+1'
     points computed in the x direction, and `2*dblint_y+1' in the y
     direction).  The routine subdivides the X axis and then for each
     value of X it first computes `<r>(x)' and `<s>(x)'; then the Y axis
     between `<r>(x)' and `<s>(x)' is subdivided and the integral along
     the Y axis is performed using Simpson's rule; then the integral
     along the X axis is done using Simpson's rule with the function
     values being the Y-integrals.  This procedure may be numerically
     unstable for a great variety of reasons, but is reasonably fast:
     avoid using it on highly oscillatory functions and functions with
     singularities (poles or branch points in the region).  The Y
     integrals depend on how far apart `<r>(x)' and `<s>(x)' are, so if
     the distance `<s>(x) - <r>(x)' varies rapidly with X, there may be
     substantial errors arising from truncation with different
     step-sizes in the various Y integrals.  One can increase
     `dblint_x' and `dblint_y' in an effort to improve the coverage of
     the region, at the expense of computation time.  The function
     values are not saved, so if the function is very time-consuming,
     you will have to wait for re-computation if you change anything
     (sorry).  It is required that the functions <f>, <r>, and <s> be
     either translated or compiled prior to calling `dblint'.  This
     will result in orders of magnitude speed improvement over
     interpreted code in many cases!

     `demo (dblint)' executes a demonstration of `dblint' applied to an
     example problem.

Function: defint (<expr>, <x>, <a>, <b>)

     Attempts to compute a definite integral.  `defint' is called by
     `integrate' when limits of integration are specified, i.e., when
     `integrate' is called as `integrate (<expr>, <x>, <a>, <b>)'.
     Thus from the user's point of view, it is sufficient to call
     `integrate'.

     `defint' returns a symbolic expression, either the computed
     integral or the noun form of the integral.  See `quad_qag' and
     related functions for numerical approximation of definite
     integrals.

Option variable: erfflag

     Default value: `true'

     When `erfflag' is `false', prevents `risch' from introducing the
     `erf' function in the answer if there were none in the integrand to
     begin with.

Function: ilt (<expr>, <s>, <t>)

     Computes the inverse Laplace transform of <expr> with respect to
     <s> and parameter <t>.  <expr> must be a ratio of polynomials
     whose denominator has only linear and quadratic factors.  By using
     the functions `laplace' and `ilt' together with the `solve' or
     `linsolve' functions the user can solve a single differential or
     convolution integral equation or a set of them.

          (%i1) 'integrate (sinh(a*x)*f(t-x), x, 0, t) + b*f(t) = t**2;
                        t
                       /
                       [                                    2
          (%o1)        I  f(t - x) sinh(a x) dx + b f(t) = t
                       ]
                       /
                        0
          (%i2) laplace (%, t, s);
                                         a laplace(f(t), t, s)   2
          (%o2)  b laplace(f(t), t, s) + --------------------- = --
                                                 2    2           3
                                                s  - a           s
          (%i3) linsolve ([%], ['laplace(f(t), t, s)]);
                                                  2      2
                                               2 s  - 2 a
          (%o3)     [laplace(f(t), t, s) = --------------------]
                                              5         2     3
                                           b s  + (a - a  b) s
          (%i4) ilt (rhs (first (%)), s, t);
          Is  a b (a b - 1)  positive, negative, or zero?

          pos;
                         sqrt(a b (a b - 1)) t
                  2 cosh(---------------------)       2
                                   b               a t
          (%o4) - ----------------------------- + -------
                        3  2      2               a b - 1
                       a  b  - 2 a  b + a

                                                                 2
                                                       + ------------------
                                                          3  2      2
                                                         a  b  - 2 a  b + a

Option variable: intanalysis

     Default value: `true'

     When `true', definite integration tries to find poles in the
     integrand in the interval of integration.  If there are, then the
     integral is evaluated appropriately as a principal value integral.
     If intanalysis is `false', this check is not performed and
     integration is done assuming there are no poles.

     See also `ldefint'.

     Examples:

     Maxima can solve the following integrals, when `intanalysis' is
     set to `false':

          (%i1) integrate(1/(sqrt(x)+1),x,0,1);
                                          1
                                         /
                                         [       1
          (%o1)                          I  ----------- dx
                                         ]  sqrt(x) + 1
                                         /
                                          0

          (%i2) integrate(1/(sqrt(x)+1),x,0,1),intanalysis:false;
          (%o2)                            2 - 2 log(2)

          (%i3) integrate(cos(a)/sqrt((tan(a))^2 +1),a,-%pi/2,%pi/2);
          The number 1 isn't in the domain of atanh
           -- an error. To debug this try: debugmode(true);

          (%i4) intanalysis:false$
          (%i5) integrate(cos(a)/sqrt((tan(a))^2+1),a,-%pi/2,%pi/2);
                                                %pi
          (%o5)                                 ---
                                                 2

Function: integrate (<expr>, <x>)
Function: integrate (<expr>, <x>, <a>, <b>)

     Attempts to symbolically compute the integral of <expr> with
     respect to <x>.  `integrate (<expr>, <x>)' is an indefinite
     integral, while `integrate (<expr>, <x>, <a>, <b>)' is a definite
     integral, with limits of integration <a> and <b>.  The limits
     should not contain <x>, although `integrate' does not enforce this
     restriction.  <a> need not be less than <b>.  If <b> is equal to
     <a>, `integrate' returns zero.

     See `quad_qag' and related functions for numerical approximation of
     definite integrals.  See `residue' for computation of residues
     (complex integration).  See `antid' for an alternative means of
     computing indefinite integrals.

     The integral (an expression free of `integrate') is returned if
     `integrate' succeeds.  Otherwise the return value is the noun form
     of the integral (the quoted operator `'integrate') or an
     expression containing one or more noun forms.  The noun form of
     `integrate' is displayed with an integral sign.

     In some circumstances it is useful to construct a noun form by
     hand, by quoting `integrate' with a single quote, e.g.,
     `'integrate (<expr>, <x>)'.  For example, the integral may depend
     on some parameters which are not yet computed.  The noun may be
     applied to its arguments by `ev (<i>, nouns)' where <i> is the
     noun form of interest.

     `integrate' handles definite integrals separately from indefinite,
     and employs a range of heuristics to handle each case.  Special
     cases of definite integrals include limits of integration equal to
     zero or infinity (`inf' or `minf'), trigonometric functions with
     limits of integration equal to zero and `%pi' or `2 %pi', rational
     functions, integrals related to the definitions of the `beta' and
     `psi' functions, and some logarithmic and trigonometric integrals.
     Processing rational functions may include computation of residues.
     If an applicable special case is not found, an attempt will be
     made to compute the indefinite integral and evaluate it at the
     limits of integration.  This may include taking a limit as a limit
     of integration goes to infinity or negative infinity; see also
     `ldefint'.

     Special cases of indefinite integrals include trigonometric
     functions, exponential and logarithmic functions, and rational
     functions.  `integrate' may also make use of a short table of
     elementary integrals.

     `integrate' may carry out a change of variable if the integrand
     has the form `f(g(x)) * diff(g(x), x)'.  `integrate' attempts to
     find a subexpression `g(x)' such that the derivative of `g(x)'
     divides the integrand.  This search may make use of derivatives
     defined by the `gradef' function.  See also `changevar' and
     `antid'.

     If none of the preceding heuristics find the indefinite integral,
     the Risch algorithm is executed.  The flag `risch' may be set as
     an `evflag', in a call to `ev' or on the command line, e.g., `ev
     (integrate (<expr>, <x>), risch)' or `integrate (<expr>, <x>),
     risch'.  If `risch' is present, `integrate' calls the `risch'
     function without attempting heuristics first.  See also `risch'.

     `integrate' works only with functional relations represented
     explicitly with the `f(x)' notation.  `integrate' does not respect
     implicit dependencies established by the `depends' function.

     `integrate' may need to know some property of a parameter in the
     integrand.  `integrate' will first consult the `assume' database,
     and, if the variable of interest is not there, `integrate' will
     ask the user.  Depending on the question, suitable responses are
     `yes;' or `no;', or `pos;', `zero;', or `neg;'.

     `integrate' is not, by default, declared to be linear.  See
     `declare' and `linear'.

     `integrate' attempts integration by parts only in a few special
     cases.

     Examples:

        * Elementary indefinite and definite integrals.

               (%i1) integrate (sin(x)^3, x);
                                          3
                                       cos (x)
               (%o1)                   ------- - cos(x)
                                          3
               (%i2) integrate (x/ sqrt (b^2 - x^2), x);
                                                2    2
               (%o2)                    - sqrt(b  - x )
               (%i3) integrate (cos(x)^2 * exp(x), x, 0, %pi);
                                              %pi
                                          3 %e      3
               (%o3)                      ------- - -
                                             5      5
               (%i4) integrate (x^2 * exp(-x^2), x, minf, inf);
                                           sqrt(%pi)
               (%o4)                       ---------
                                               2

        * Use of `assume' and interactive query.

               (%i1) assume (a > 1)$
               (%i2) integrate (x**a/(x+1)**(5/2), x, 0, inf);
                   2 a + 2
               Is  -------  an integer?
                      5

               no;
               Is  2 a - 3  positive, negative, or zero?

               neg;
                                                  3
               (%o2)                  beta(a + 1, - - a)
                                                  2

        * Change of variable.  There are two changes of variable in
          this example: one using a derivative established by `gradef',
          and one using the derivation `diff(r(x))' of an unspecified
          function `r(x)'.

               (%i3) gradef (q(x), sin(x**2));
               (%o3)                         q(x)
               (%i4) diff (log (q (r (x))), x);
                                     d               2
                                    (-- (r(x))) sin(r (x))
                                     dx
               (%o4)                ----------------------
                                           q(r(x))
               (%i5) integrate (%, x);
               (%o5)                     log(q(r(x)))

        * Return value contains the `'integrate' noun form.  In this
          example, Maxima can extract one factor of the denominator of
          a rational function, but cannot factor the remainder or
          otherwise find its integral.  `grind' shows the noun form
          `'integrate' in the result.  See also `integrate_use_rootsof'
          for more on integrals of rational functions.

               (%i1) expand ((x-4) * (x^3+2*x+1));
                                   4      3      2
               (%o1)              x  - 4 x  + 2 x  - 7 x - 4
               (%i2) integrate (1/%, x);
                                             /  2
                                             [ x  + 4 x + 18
                                             I ------------- dx
                                             ]  3
                                log(x - 4)   / x  + 2 x + 1
               (%o2)            ---------- - ------------------
                                    73               73
               (%i3) grind (%);
               log(x-4)/73-('integrate((x^2+4*x+18)/(x^3+2*x+1),x))/73$

        * Defining a function in terms of an integral.  The body of a
          function is not evaluated when the function is defined.  Thus
          the body of `f_1' in this example contains the noun form of
          `integrate'.  The quote-quote operator `''' causes the
          integral to be evaluated, and the result becomes the body of
          `f_2'.

               (%i1) f_1 (a) := integrate (x^3, x, 1, a);
                                                    3
               (%o1)           f_1(a) := integrate(x , x, 1, a)
               (%i2) ev (f_1 (7), nouns);
               (%o2)                          600
               (%i3) /* Note parentheses around integrate(...) here */
                     f_2 (a) := ''(integrate (x^3, x, 1, a));
                                                  4
                                                 a    1
               (%o3)                   f_2(a) := -- - -
                                                 4    4
               (%i4) f_2 (7);
               (%o4)                          600

System variable: integration_constant

     Default value: `%c'

     When a constant of integration is introduced by indefinite
     integration of an equation, the name of the constant is
     constructed by concatenating `integration_constant' and
     `integration_constant_counter'.

     `integration_constant' may be assigned any symbol.

     Examples:

          (%i1) integrate (x^2 = 1, x);
                                     3
                                    x
          (%o1)                     -- = x + %c1
                                    3
          (%i2) integration_constant : 'k;
          (%o2)                           k
          (%i3) integrate (x^2 = 1, x);
                                      3
                                     x
          (%o3)                      -- = x + k2
                                     3

System variable: integration_constant_counter

     Default value: 0

     When a constant of integration is introduced by indefinite
     integration of an equation, the name of the constant is
     constructed by concatenating `integration_constant' and
     `integration_constant_counter'.

     `integration_constant_counter' is incremented before constructing
     the next integration constant.

     Examples:

          (%i1) integrate (x^2 = 1, x);
                                     3
                                    x
          (%o1)                     -- = x + %c1
                                    3
          (%i2) integrate (x^2 = 1, x);
                                     3
                                    x
          (%o2)                     -- = x + %c2
                                    3
          (%i3) integrate (x^2 = 1, x);
                                     3
                                    x
          (%o3)                     -- = x + %c3
                                    3
          (%i4) reset (integration_constant_counter);
          (%o4)            [integration_constant_counter]
          (%i5) integrate (x^2 = 1, x);
                                     3
                                    x
          (%o5)                     -- = x + %c1
                                    3

Option variable: integrate_use_rootsof

     Default value: `false'

     When `integrate_use_rootsof' is `true' and the denominator of a
     rational function cannot be factored, `integrate' returns the
     integral in a form which is a sum over the roots (not yet known)
     of the denominator.

     For example, with `integrate_use_rootsof' set to `false',
     `integrate' returns an unsolved integral of a rational function in
     noun form:

          (%i1) integrate_use_rootsof: false$
          (%i2) integrate (1/(1+x+x^5), x);
                  /  2
                  [ x  - 4 x + 5
                  I ------------ dx                            2 x + 1
                  ]  3    2                2            5 atan(-------)
                  / x  - x  + 1       log(x  + x + 1)          sqrt(3)
          (%o2)   ----------------- - --------------- + ---------------
                          7                 14             7 sqrt(3)

     Now we set the flag to be true and the unsolved part of the
     integral will be expressed as a summation over the roots of the
     denominator of the rational function:

          (%i3) integrate_use_rootsof: true$
          (%i4) integrate (1/(1+x+x^5), x);
                ====        2
                \       (%r4  - 4 %r4 + 5) log(x - %r4)
                 >      -------------------------------
                /                    2
                ====            3 %r4  - 2 %r4
                                3    2
                %r4 in rootsof(x  - x  + 1)
          (%o4) ----------------------------------------------------------
                         7

                                                                2 x + 1
                                            2            5 atan(-------)
                                       log(x  + x + 1)          sqrt(3)
                                     - --------------- + ---------------
                                             14             7 sqrt(3)

     Alternatively the user may compute the roots of the denominator
     separately, and then express the integrand in terms of these
     roots, e.g., `1/((x - a)*(x - b)*(x - c))' or `1/((x^2 - (a+b)*x +
     a*b)*(x - c))' if the denominator is a cubic polynomial.
     Sometimes this will help Maxima obtain a more useful result.

Function: ldefint (<expr>, <x>, <a>, <b>)

     Attempts to compute the definite integral of <expr> by using
     `limit' to evaluate the indefinite integral of <expr> with respect
     to <x> at the upper limit <b> and at the lower limit <a>.  If it
     fails to compute the definite integral, `ldefint' returns an
     expression containing limits as noun forms.

     `ldefint' is not called from `integrate', so executing `ldefint
     (<expr>, <x>, <a>, <b>)' may yield a different result than
     `integrate (<expr>, <x>, <a>, <b>)'.  `ldefint' always uses the
     same method to evaluate the definite integral, while `integrate'
     may employ various heuristics and may recognize some special cases.

Function: potential (<givengradient>)

     The calculation makes use of the global variable
     `potentialzeroloc[0]' which must be `nonlist' or of the form

          [indeterminatej=expressionj, indeterminatek=expressionk, ...]

     the former being equivalent to the nonlist expression for all
     right-hand sides in the latter.  The indicated right-hand sides
     are used as the lower limit of integration.  The success of the
     integrations may depend upon their values and order.
     `potentialzeroloc' is initially set to 0.

Function: residue (<expr>, <z>, <z_0>)

     Computes the residue in the complex plane of the expression <expr>
     when the variable <z> assumes the value <z_0>.  The residue is the
     coefficient of `(<z> - <z_0>)^(-1)' in the Laurent series for
     <expr>.

          (%i1) residue (s/(s**2+a**2), s, a*%i);
                                          1
          (%o1)                           -
                                          2
          (%i2) residue (sin(a*x)/x**4, x, 0);
                                           3
                                          a
          (%o2)                         - --
                                          6

Function: risch (<expr>, <x>)

     Integrates <expr> with respect to <x> using the transcendental
     case of the Risch algorithm.  (The algebraic case of the Risch
     algorithm has not been implemented.)  This currently handles the
     cases of nested exponentials and logarithms which the main part of
     `integrate' can't do.  `integrate' will automatically apply
     `risch' if given these cases.

     `erfflag', if `false', prevents `risch' from introducing the `erf'
     function in the answer if there were none in the integrand to begin
     with.

          (%i1) risch (x^2*erf(x), x);
                                                                  2
                       3                      2                - x
                  %pi x  erf(x) + (sqrt(%pi) x  + sqrt(%pi)) %e
          (%o1)   -------------------------------------------------
                                        3 %pi
          (%i2) diff(%, x), ratsimp;
                                       2
          (%o2)                       x  erf(x)

Function: tldefint (<expr>, <x>, <a>, <b>)

     Equivalent to `ldefint' with `tlimswitch' set to `true'.

19.3 Introduction to QUADPACK

QUADPACK is a collection of functions for the numerical computation of
one-dimensional definite integrals.  It originated from a joint project
of R. Piessens (1), E. de Doncker (2), C. Ueberhuber (3), and D.
Kahaner (4).

   The QUADPACK library included in Maxima is an automatic translation
(via the program `f2cl') of the Fortran source code of QUADPACK as it
appears in the SLATEC Common Mathematical Library, Version 4.1 (5).
The SLATEC library is dated July 1993, but the QUADPACK functions were
written some years before.  There is another version of QUADPACK at
Netlib (6); it is not clear how that version differs from the SLATEC
version.

   The QUADPACK functions included in Maxima are all automatic, in the
sense that these functions attempt to compute a result to a specified
accuracy, requiring an unspecified number of function evaluations.
Maxima's Lisp translation of QUADPACK also includes some non-automatic
functions, but they are not exposed at the Maxima level.

   Further information about QUADPACK can be found in the QUADPACK book
(7).

19.4 Functions and Variables for QUADPACK

Function: quad_qag (<f(x)>, <x>, <a>, <b>, <key>, [<epsrel>, <epsabs>, <limit>])
Function: quad_qag (<f>, <x>, <a>, <b>, <key>, [<epsrel>, <epsabs>, <limit>])

     Integration of a general function over a finite interval.
     `quad_qag' implements a simple globally adaptive integrator using
     the strategy of Aind (Piessens, 1973).  The caller may choose
     among 6 pairs of Gauss-Kronrod quadrature formulae for the rule
     evaluation component.  The high-degree rules are suitable for
     strongly oscillating integrands.

     `quad_qag' computes the integral

     integrate (f(x), x, a, b)

     The function to be integrated is <f(x)>, with dependent variable
     <x>, and the function is to be integrated between the limits <a>
     and <b>.  <key> is the integrator to be used and should be an
     integer between 1 and 6, inclusive.  The value of <key> selects
     the order of the Gauss-Kronrod integration rule.  High-order rules
     are suitable for strongly oscillating integrands.

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     The numerical integration is done adaptively by subdividing the
     integration region into sub-intervals until the desired accuracy is
     achieved.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form `key=val'.  The keyword arguments
     are:

    <epsrel>
          Desired relative error of approximation.  Default is 1d-8.

    <epsabs>
          Desired absolute error of approximation.  Default is 0.

    <limit>
          Size of internal work array.  <limit> is the maximum number
          of subintervals to use.  Default is 200.

     `quad_qag' returns a list of four elements:

        * an approximation to the integral,

        * the estimated absolute error of the approximation,

        * the number integrand evaluations,

        * an error code.

     The error code (fourth element of the return value) can have the
     values:

    `0'
          if no problems were encountered;

    `1'
          if too many sub-intervals were done;

    `2'
          if excessive roundoff error is detected;

    `3'
          if extremely bad integrand behavior occurs;

    `6'
          if the input is invalid.

Function: quad_qags (<f(x)>, <x>, <a>, <b>, [<epsrel>, <epsabs>, <limit>])
Function: quad_qags (<f>, <x>, <a>, <b>, [<epsrel>, <epsabs>, <limit>])

     Integration of a general function over a finite interval.
     `quad_qags' implements globally adaptive interval subdivision with
     extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn,
     1956).

     `quad_qags' computes the integral

     integrate (f(x), x, a, b)

     The function to be integrated is <f(x)>, with dependent variable
     <x>, and the function is to be integrated between the limits <a>
     and <b>.

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form `key=val'.  The keyword arguments
     are:

    <epsrel>
          Desired relative error of approximation.  Default is 1d-8.

    <epsabs>
          Desired absolute error of approximation.  Default is 0.

    <limit>
          Size of internal work array.  <limit> is the maximum number
          of subintervals to use.  Default is 200.

     `quad_qags' returns a list of four elements:

        * an approximation to the integral,

        * the estimated absolute error of the approximation,

        * the number integrand evaluations,

        * an error code.

     The error code (fourth element of the return value) can have the
     values:

    `0'
          no problems were encountered;

    `1'
          too many sub-intervals were done;

    `2'
          excessive roundoff error is detected;

    `3'
          extremely bad integrand behavior occurs;

    `4'
          failed to converge

    `5'
          integral is probably divergent or slowly convergent

    `6'
          if the input is invalid.

     Examples:

          (%i1) quad_qags (x^(1/2)*log(1/x), x, 0, 1, 'epsrel=1d-10);
          (%o1)   [.4444444444444448, 1.11022302462516E-15, 315, 0]

     Note that `quad_qags' is more accurate and efficient than
     `quad_qag' for this integrand.

Function: quad_qagi (<f(x)>, <x>, <a>, <b>, [<epsrel>, <epsabs>, <limit>])
Function: quad_qagi (<f>, <x>, <a>, <b>, [<epsrel>, <epsabs>, <limit>])

     Integration of a general function over an infinite or
     semi-infinite interval.  The interval is mapped onto a finite
     interval and then the same strategy as in `quad_qags' is applied.

     `quad_qagi' evaluates one of the following integrals

     integrate (f(x), x, a, inf)

     integrate (f(x), x, minf, a)

     integrate (f(x), x, minf, inf)

     using the Quadpack QAGI routine.  The function to be integrated is
     <f(x)>, with dependent variable <x>, and the function is to be
     integrated over an infinite range.

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     One of the limits of integration must be infinity.  If not, then
     `quad_qagi' will just return the noun form.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form `key=val'.  The keyword arguments
     are:

    <epsrel>
          Desired relative error of approximation.  Default is 1d-8.

    <epsabs>
          Desired absolute error of approximation.  Default is 0.

    <limit>
          Size of internal work array.  <limit> is the maximum number
          of subintervals to use.  Default is 200.

     `quad_qagi' returns a list of four elements:

        * an approximation to the integral,

        * the estimated absolute error of the approximation,

        * the number integrand evaluations,

        * an error code.

     The error code (fourth element of the return value) can have the
     values:

    `0'
          no problems were encountered;

    `1'
          too many sub-intervals were done;

    `2'
          excessive roundoff error is detected;

    `3'
          extremely bad integrand behavior occurs;

    `4'
          failed to converge

    `5'
          integral is probably divergent or slowly convergent

    `6'
          if the input is invalid.

Function: quad_qawc (<f(x)>, <x>, <c>, <a>, <b>, [<epsrel>, <epsabs>, <limit>])
Function: quad_qawc (<f>, <x>, <c>, <a>, <b>, [<epsrel>, <epsabs>, <limit>])

     Computes the Cauchy principal value of f(x)/(x - c) over a finite
     interval.  The strategy is globally adaptive, and modified
     Clenshaw-Curtis integration is used on the subranges which contain
     the point x = c.

     `quad_qawc' computes the Cauchy principal value of

     integrate (f(x)/(x - c), x, a, b)

     using the Quadpack QAWC routine.  The function to be integrated is
     `<f(x)>/(<x> - <c>)', with dependent variable <x>, and the
     function is to be integrated over the interval <a> to <b>.

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form `key=val'.  The keyword arguments
     are:

    <epsrel>
          Desired relative error of approximation.  Default is 1d-8.

    <epsabs>
          Desired absolute error of approximation.  Default is 0.

    <limit>
          Size of internal work array.  <limit> is the maximum number
          of subintervals to use.  Default is 200.

     `quad_qawc' returns a list of four elements:

        * an approximation to the integral,

        * the estimated absolute error of the approximation,

        * the number integrand evaluations,

        * an error code.

     The error code (fourth element of the return value) can have the
     values:

    `0'
          no problems were encountered;

    `1'
          too many sub-intervals were done;

    `2'
          excessive roundoff error is detected;

    `3'
          extremely bad integrand behavior occurs;

    `6'
          if the input is invalid.

Function: quad_qawf (<f(x)>, <x>, <a>, <omega>, <trig>, [<epsabs>, <limit>, <maxp1>, <limlst>])
Function: quad_qawf (<f>, <x>, <a>, <omega>, <trig>, [<epsabs>, <limit>, <maxp1>, <limlst>])

     Calculates a Fourier cosine or Fourier sine transform on a
     semi-infinite interval using the Quadpack QAWF function.  The same
     approach as in `quad_qawo' is applied on successive finite
     intervals, and convergence acceleration by means of the Epsilon
     algorithm (Wynn, 1956) is applied to the series of the integral
     contributions.

     `quad_qawf' computes the integral

     integrate (f(x)*w(x), x, a, inf)

     The weight function w is selected by <trig>:

    `cos'
          w(x) = cos (omega x)

    `sin'
          w(x) = sin (omega x)

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form `key=val'.  The keyword arguments
     are:

    <epsabs>
          Desired absolute error of approximation.  Default is 1d-10.

    <limit>
          Size of internal work array.  (<limit> - <limlst>)/2 is the
          maximum number of subintervals to use.  Default is 200.

    <maxp1>
          Maximum number of Chebyshev moments.  Must be greater than 0.
          Default is 100.

    <limlst>
          Upper bound on the number of cycles.  Must be greater than or
          equal to 3.  Default is 10.

     `quad_qawf' returns a list of four elements:

        * an approximation to the integral,

        * the estimated absolute error of the approximation,

        * the number integrand evaluations,

        * an error code.

     The error code (fourth element of the return value) can have the
     values:

    `0'
          no problems were encountered;

    `1'
          too many sub-intervals were done;

    `2'
          excessive roundoff error is detected;

    `3'
          extremely bad integrand behavior occurs;

    `6'
          if the input is invalid.

Function: quad_qawo (<f(x)>, <x>, <a>, <b>, <omega>, <trig>, [<epsrel>, <epsabs>, <limit>, <maxp1>, <limlst>])
Function: quad_qawo (<f>, <x>, <a>, <b>, <omega>, <trig>, [<epsrel>, <epsabs>, <limit>, <maxp1>, <limlst>])

     Integration of cos (omega x) f(x) or sin (omega x) f(x) over a
     finite interval, where omega is a constant.  The rule evaluation
     component is based on the modified Clenshaw-Curtis technique.
     `quad_qawo' applies adaptive subdivision with extrapolation,
     similar to `quad_qags'.

     `quad_qawo' computes the integral using the Quadpack QAWO routine:

     integrate (f(x)*w(x), x, a, b)

     The weight function w is selected by <trig>:

    `cos'
          w(x) = cos (omega x)

    `sin'
          w(x) = sin (omega x)

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form `key=val'.  The keyword arguments
     are:

    <epsrel>
          Desired relative error of approximation.  Default is 1d-8.

    <epsabs>
          Desired absolute error of approximation.  Default is 0.

    <limit>
          Size of internal work array.  <limit>/2 is the maximum number
          of subintervals to use.  Default is 200.

    <maxp1>
          Maximum number of Chebyshev moments.  Must be greater than 0.
          Default is 100.

    <limlst>
          Upper bound on the number of cycles.  Must be greater than or
          equal to 3.  Default is 10.

     `quad_qawo' returns a list of four elements:

        * an approximation to the integral,

        * the estimated absolute error of the approximation,

        * the number integrand evaluations,

        * an error code.

     The error code (fourth element of the return value) can have the
     values:

    `0'
          no problems were encountered;

    `1'
          too many sub-intervals were done;

    `2'
          excessive roundoff error is detected;

    `3'
          extremely bad integrand behavior occurs;

    `6'
          if the input is invalid.

Function: quad_qaws (<f(x)>, <x>, <a>, <b>, <alpha>, <beta>, <wfun>, [<epsrel>, <epsabs>, <limit>])
Function: quad_qaws (<f>, <x>, <a>, <b>, <alpha>, <beta>, <wfun>, [<epsrel>, <epsabs>, <limit>])

     Integration of w(x) f(x) over a finite interval, where w(x) is a
     certain algebraic or logarithmic function.  A globally adaptive
     subdivision strategy is applied, with modified Clenshaw-Curtis
     integration on the subintervals which contain the endpoints of the
     interval of integration.

     `quad_qaws' computes the integral using the Quadpack QAWS routine:

     integrate (f(x)*w(x), x, a, b)

     The weight function w is selected by <wfun>:

    `1'
          w(x) = (x - a)^alpha (b - x)^beta

    `2'
          w(x) = (x - a)^alpha (b - x)^beta log(x - a)

    `3'
          w(x) = (x - a)^alpha (b - x)^beta log(b - x)

    `4'
          w(x) = (x - a)^alpha (b - x)^beta log(x - a) log(b - x)

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form `key=val'.  The keyword arguments
     are:

    <epsrel>
          Desired relative error of approximation.  Default is 1d-8.

    <epsabs>
          Desired absolute error of approximation.  Default is 0.

    <limit>
          Size of internal work array.  <limit>is the maximum number of
          subintervals to use.  Default is 200.

     `quad_qaws' returns a list of four elements:

        * an approximation to the integral,

        * the estimated absolute error of the approximation,

        * the number integrand evaluations,

        * an error code.

     The error code (fourth element of the return value) can have the
     values:

    `0'
          no problems were encountered;

    `1'
          too many sub-intervals were done;

    `2'
          excessive roundoff error is detected;

    `3'
          extremely bad integrand behavior occurs;

    `6'
          if the input is invalid.

Function: quad_qagp (<f(x)>, <x>, <a>, <b>, <points>, [<epsrel>, <epsabs>, <limit>])
Function: quad_qagp (<f>, <x>, <a>, <b>, <points>, [<epsrel>, <epsabs>, <limit>])

     Integration of a general function over a finite interval.
     `quad_qagp' implements globally adaptive interval subdivision with
     extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn,
     1956).

     `quad_qagp' computes the integral

     integrate (f(x), x, a, b)

     The function to be integrated is <f(x)>, with dependent variable
     <x>, and the function is to be integrated between the limits <a>
     and <b>.

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     To help the integrator, the user must supply a list of points where
     the integrand is singular or discontinous.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form `key=val'.  The keyword arguments
     are:

    <epsrel>
          Desired relative error of approximation.  Default is 1d-8.

    <epsabs>
          Desired absolute error of approximation.  Default is 0.

    <limit>
          Size of internal work array.  <limit> is the maximum number
          of subintervals to use.  Default is 200.

     `quad_qagp' returns a list of four elements:

        * an approximation to the integral,

        * the estimated absolute error of the approximation,

        * the number integrand evaluations,

        * an error code.

     The error code (fourth element of the return value) can have the
     values:

    `0'
          no problems were encountered;

    `1'
          too many sub-intervals were done;

    `2'
          excessive roundoff error is detected;

    `3'
          extremely bad integrand behavior occurs;

    `4'
          failed to converge

    `5'
          integral is probably divergent or slowly convergent

    `6'
          if the input is invalid.

     Examples:

          (%i1) quad_qagp(x^3*log(abs((x^2-1)*(x^2-2))),x,0,3,[1,sqrt(2)]);
          (%o1)   [52.74074838347143, 2.6247632689546663e-7, 1029, 0]
          (%i2) quad_qags(x^3*log(abs((x^2-1)*(x^2-2))), x, 0, 3);
          (%o2)   [52.74074847951494, 4.088443219529836e-7, 1869, 0]

     The integrand has singularities at `1' and `sqrt(2)' so we supply
     these points to `quad_qagp'.  We also note that `quad_qagp' is
     more accurate and more efficient that `quad_qags'.

Function: quad_control (<parameter>, [<value>])

     Control error handling for quadpack.  The parameter should be one
     of the following symbols:

    `current_error'
          The current error number

    `control'
          Controls if messages are printed or not.  If it is set to
          zero or less, messages are suppressed.

    `max_message'
          The maximum number of times any message is to be printed.

     If <value> is not given, then the current value of the <parameter>
     is returned.  If <value> is given, the value of <parameter> is set
     to the given value.

20 Equations


Functions and Variables for Equations

20.1 Functions and Variables for Equations

System variable: %rnum_list

     Default value: `[]'

     `%rnum_list' is the list of variables introduced in solutions by
     `solve'  and `algsys'.    `%r' variables are added to `%rnum_list'
     in the order they are created.  This is convenient for doing
     substitutions into the solution later on.  It's recommended to use
     this list rather than doing `concat ('%r, j)'.

          (%i1) solve ([x + y = 3], [x,y]);
          (%o1)              [[x = 3 - %r1, y = %r1]]
          (%i2) %rnum_list;
          (%o2)                       [%r1]
          (%i3) sol : solve ([x + 2*y + 3*z = 4], [x,y,z]);
          (%o3)   [[x = - 2 %r3 - 3 %r2 + 4, y = %r3, z = %r2]]
          (%i4) %rnum_list;
          (%o4)                     [%r2, %r3]
          (%i5) for i : 1 thru length (%rnum_list) do
                  sol : subst (t[i], %rnum_list[i], sol)$
          (%i6) sol;
          (%o6)     [[x = - 2 t  - 3 t  + 4, y = t , z = t ]]
                               2      1           2       1

Option variable: algepsilon

     Default value: 10^8

     `algepsilon' is used by `algsys'.

Option variable: algexact

     Default value: `false'

     `algexact' affects the behavior of `algsys'  as follows:

     If `algexact' is `true', `algsys' always calls `solve'  and then
     uses `realroots'  on `solve''s failures.

     If `algexact' is `false', `solve' is called only if the eliminant
     was not univariate, or if it was a quadratic or biquadratic.

     Thus `algexact: true' does not guarantee only exact solutions,
     just that `algsys' will first try as hard as it can to give exact
     solutions, and only yield approximations when all else fails.

Function: algsys ([<expr_1>, ..., <expr_m>], [<x_1>, ..., <x_n>])
Function: algsys ([<eqn_1>, ..., <eqn_m>], [<x_1>, ..., <x_n>])

     Solves the simultaneous polynomials <expr_1>, ..., <expr_m> or
     polynomial equations <eqn_1>, ..., <eqn_m> for the variables
     <x_1>, ..., <x_n>.  An expression <expr> is equivalent to an
     equation `<expr> = 0'.  There may be more equations than variables
     or vice versa.

     `algsys' returns a list of solutions, with each solution given as
     a list of equations stating values of the variables <x_1>, ...,
     <x_n> which satisfy the system of equations.  If `algsys' cannot
     find a solution, an empty list `[]' is returned.

     The symbols `%r1', `%r2', ..., are introduced as needed to
     represent arbitrary parameters in the solution; these variables
     are also appended to the list `%rnum_list'.

     The method is as follows:

       1. First the equations are factored and split into subsystems.

       2. For each subsystem <S_i>, an equation <E> and a variable <x>
          are selected.  The variable is chosen to have lowest nonzero
          degree.  Then the resultant of <E> and <E_j> with respect to
          <x> is computed for each of the remaining equations <E_j> in
          the subsystem <S_i>.  This yields a new subsystem <S_i'> in
          one fewer variables, as <x> has been eliminated.  The process
          now returns to (1).

       3. Eventually, a subsystem consisting of a single equation is
          obtained.  If the equation is multivariate and no
          approximations in the form of floating point numbers have
          been introduced, then `solve'  is called to find an exact
          solution.

          In some cases, `solve' is not be able to find a solution, or
          if it does the solution may be a very large expression.

          If the equation is univariate and is either linear,
          quadratic, or biquadratic, then again `solve' is called if no
          approximations have been introduced.  If approximations have
          been introduced or the equation is not univariate and neither
          linear, quadratic, or biquadratic, then if the switch
          `realonly'  is `true', the function `realroots'  is called to
          find the real-valued solutions.  If `realonly' is `false',
          then `allroots'  is called which looks for real and
          complex-valued solutions.

          If `algsys' produces a solution which has fewer significant
          digits than required, the user can change the value of
          `algepsilon'  to a higher value.

          If `algexact' is set to `true', `solve' will always be called.

       4. Finally, the solutions obtained in step (3) are substituted
          into previous levels and the solution process returns to (1).

     When `algsys' encounters a multivariate equation which contains
     floating point approximations (usually due to its failing to find
     exact solutions at an earlier stage), then it does not attempt to
     apply exact methods to such equations and instead prints the
     message: "`algsys' cannot solve - system too complicated."

     Interactions with `radcan'  can produce large or complicated
     expressions.  In that case, it may be possible to isolate parts of
     the result with `pickapart'  or `reveal'.

     Occasionally, `radcan' may introduce an imaginary unit `%i' into a
     solution which is actually real-valued.

     Examples:

          (%i1) e1: 2*x*(1 - a1) - 2*(x - 1)*a2;
          (%o1)              2 (1 - a1) x - 2 a2 (x - 1)
          (%i2) e2: a2 - a1;
          (%o2)                        a2 - a1
          (%i3) e3: a1*(-y - x^2 + 1);
                                             2
          (%o3)                   a1 (- y - x  + 1)
          (%i4) e4: a2*(y - (x - 1)^2);
                                                 2
          (%o4)                   a2 (y - (x - 1) )
          (%i5) algsys ([e1, e2, e3, e4], [x, y, a1, a2]);
          (%o5) [[x = 0, y = %r1, a1 = 0, a2 = 0],

                                            [x = 1, y = 0, a1 = 1, a2 = 1]]
          (%i6) e1: x^2 - y^2;
                                        2    2
          (%o6)                        x  - y
          (%i7) e2: -1 - y + 2*y^2 - x + x^2;
                                   2        2
          (%o7)                 2 y  - y + x  - x - 1
          (%i8) algsys ([e1, e2], [x, y]);
                           1            1
          (%o8) [[x = - -------, y = -------],
                        sqrt(3)      sqrt(3)

                  1              1             1        1
          [x = -------, y = - -------], [x = - -, y = - -], [x = 1, y = 1]]
               sqrt(3)        sqrt(3)          3        3

Function: allroots (<expr>)
Function: allroots (<eqn>)

     Computes numerical approximations of the real and complex roots of
     the polynomial <expr> or polynomial equation <eqn> of one variable.

     The flag `polyfactor'  when `true' causes `allroots' to factor the
     polynomial over the real numbers if the polynomial is real, or
     over the complex numbers, if the polynomial is complex.

     `allroots' may give inaccurate results in case of multiple roots.
     If the polynomial is real, `allroots (%i*<p>)') may yield more
     accurate approximations than `allroots (<p>)', as `allroots'
     invokes a different algorithm in that case.

     `allroots' rejects non-polynomials.  It requires that the numerator
     after `rat''ing should be a polynomial, and it requires that the
     denominator be at most a complex number.  As a result of this
     `allroots' will always return an equivalent (but factored)
     expression, if `polyfactor' is `true'.

     For complex polynomials an algorithm by Jenkins and Traub is used
     (Algorithm 419, Comm. ACM, vol. 15, (1972), p. 97).  For real
     polynomials the algorithm used is due to Jenkins (Algorithm 493,
     ACM TOMS, vol. 1, (1975), p.178).

     Examples:

          (%i1) eqn: (1 + 2*x)^3 = 13.5*(1 + x^5);
                                      3          5
          (%o1)              (2 x + 1)  = 13.5 (x  + 1)
          (%i2) soln: allroots (eqn);
          (%o2) [x = .8296749902129361, x = - 1.015755543828121,

          x = .9659625152196369 %i - .4069597231924075,

          x = - .9659625152196369 %i - .4069597231924075, x = 1.0]
          (%i3) for e in soln
                  do (e2: subst (e, eqn), disp (expand (lhs(e2) - rhs(e2))));
                                - 3.5527136788005E-15

                               - 5.32907051820075E-15

                   4.44089209850063E-15 %i - 4.88498130835069E-15

                  - 4.44089209850063E-15 %i - 4.88498130835069E-15

                                 3.5527136788005E-15

          (%o3)                         done
          (%i4) polyfactor: true$
          (%i5) allroots (eqn);
          (%o5) - 13.5 (x - 1.0) (x - .8296749902129361)

                                     2
           (x + 1.015755543828121) (x  + .8139194463848151 x

           + 1.098699797110288)

Function: bfallroots (<expr>)
Function: bfallroots (<eqn>)

     Computes numerical approximations of the real and complex roots of
     the polynomial <expr> or polynomial equation <eqn> of one variable.

     In all respects, `bfallroots' is identical to `allroots' except
     that `bfallroots' computes the roots using bigfloats.  See
     `allroots'  for more information.

Option variable: backsubst

     Default value: `true'

     When `backsubst' is `false', prevents back substitution in
     `linsolve'  after the equations have been triangularized.  This may
     be helpful in very big problems where back substitution would cause
     the generation of extremely large expressions.

          (%i1) eq1 : x + y + z = 6$
          (%i2) eq2 : x - y + z = 2$
          (%i3) eq3 : x + y - z = 0$
          (%i4) backsubst : false$
          (%i5) linsolve ([eq1, eq2, eq3], [x,y,z]);
          (%o5)             [x = z - y, y = 2, z = 3]
          (%i6) backsubst : true$
          (%i7) linsolve ([eq1, eq2, eq3], [x,y,z]);
          (%o7)               [x = 1, y = 2, z = 3]

Option variable: breakup

     Default value: `true'

     When `breakup' is `true', `solve'  expresses solutions of cubic
     and quartic equations in terms of common subexpressions, which are
     assigned to intermediate expression labels (`%t1', `%t2', etc.).
     Otherwise, common subexpressions are not identified.

     `breakup: true' has an effect only when `programmode'  is `false'.

     Examples:

          (%i1) programmode: false$
          (%i2) breakup: true$
          (%i3) solve (x^3 + x^2 - 1);

                                  sqrt(23)    25 1/3
          (%t3)                  (--------- + --)
                                  6 sqrt(3)   54
          Solution:

                                                sqrt(3) %i   1
                                                ---------- - -
                          sqrt(3) %i   1            2        2   1
          (%t4)    x = (- ---------- - -) %t3 + -------------- - -
                              2        2            9 %t3        3

                                                sqrt(3) %i   1
                                              - ---------- - -
                        sqrt(3) %i   1              2        2   1
          (%t5)    x = (---------- - -) %t3 + ---------------- - -
                            2        2             9 %t3         3

                                             1     1
          (%t6)                  x = %t3 + ----- - -
                                           9 %t3   3
          (%o6)                    [%t4, %t5, %t6]
          (%i6) breakup: false$
          (%i7) solve (x^3 + x^2 - 1);
          Solution:

                       sqrt(3) %i   1
                       ---------- - -
                           2        2        sqrt(23)    25 1/3
          (%t7) x = --------------------- + (--------- + --)
                       sqrt(23)    25 1/3    6 sqrt(3)   54
                    9 (--------- + --)
                       6 sqrt(3)   54

                                                        sqrt(3) %i   1    1
                                                     (- ---------- - -) - -
                                                            2        2    3
                     sqrt(23)    25 1/3  sqrt(3) %i   1
          (%t8) x = (--------- + --)    (---------- - -)
                     6 sqrt(3)   54          2        2

                                                      sqrt(3) %i   1
                                                    - ---------- - -
                                                          2        2      1
                                                + --------------------- - -
                                                     sqrt(23)    25 1/3   3
                                                  9 (--------- + --)
                                                     6 sqrt(3)   54
                      sqrt(23)    25 1/3             1             1
          (%t9)  x = (--------- + --)    + --------------------- - -
                      6 sqrt(3)   54          sqrt(23)    25 1/3   3
                                           9 (--------- + --)
                                              6 sqrt(3)   54
          (%o9)                    [%t7, %t8, %t9]

Function: dimension (<eqn>)
Function: dimension (<eqn_1>, ..., <eqn_n>)

     `dimen' is a package for dimensional analysis.  `load ("dimen")'
     loads this package.  `demo ("dimen")' displays a short
     demonstration.

Option variable: dispflag

     Default value: `true'

     If set to `false' within a `block' will inhibit the display of
     output generated by the solve functions called from within the
     `block'.  Termination of the `block' with a dollar sign, $, sets
     `dispflag' to `false'.

Function: funcsolve (<eqn>, <g>(<t>))

     Returns `[<g>(<t>) = ...]'  or `[]', depending on whether or not
     there exists a rational function `<g>(<t>)' satisfying <eqn>,
     which must be a first order, linear polynomial in (for this case)
     `<g>(<t>)' and `<g>(<t>+1)'

          (%i1) eqn: (n + 1)*f(n) - (n + 3)*f(n + 1)/(n + 1) =
                (n - 1)/(n + 2);
                                      (n + 3) f(n + 1)   n - 1
          (%o1)        (n + 1) f(n) - ---------------- = -----
                                           n + 1         n + 2
          (%i2) funcsolve (eqn, f(n));

          Dependent equations eliminated:  (4 3)
                                             n
          (%o2)                f(n) = ---------------
                                      (n + 1) (n + 2)

     Warning: this is a very rudimentary implementation - many safety
     checks and obvious generalizations are missing.

Option variable: globalsolve

     Default value: `false'

     When `globalsolve' is `true', solved-for variables are assigned the
     solution values found by `linsolve', and by `solve'  when solving
     two or more linear equations.

     When `globalsolve' is `false', solutions found by `linsolve'  and
     by `solve' when solving two or more linear equations are expressed
     as equations, and the solved-for variables are not assigned.

     When solving anything other than two or more linear equations,
     `solve' ignores `globalsolve'.  Other functions which solve
     equations (e.g., `algsys' ) always ignore `globalsolve'.

     Examples:

          (%i1) globalsolve: true$
          (%i2) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]);
          Solution

                                           17
          (%t2)                        x : --
                                           7

                                             1
          (%t3)                        y : - -
                                             7
          (%o3)                     [[%t2, %t3]]
          (%i3) x;
                                         17
          (%o3)                          --
                                         7
          (%i4) y;
                                           1
          (%o4)                          - -
                                           7
          (%i5) globalsolve: false$
          (%i6) kill (x, y)$
          (%i7) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]);
          Solution

                                           17
          (%t7)                        x = --
                                           7

                                             1
          (%t8)                        y = - -
                                             7
          (%o8)                     [[%t7, %t8]]
          (%i8) x;
          (%o8)                           x
          (%i9) y;
          (%o9)                           y

Function: ieqn (<ie>, <unk>, <tech>, <n>, <guess>)

     `inteqn' is a package for solving integral equations.  `load
     ("inteqn")' loads this package.

     <ie> is the integral equation; <unk> is the unknown function;
     <tech> is the technique to be tried from those given above (<tech>
     = `first' means: try the first technique which finds a solution;
     <tech> = `all' means: try all applicable techniques); <n> is the
     maximum number of terms to take for `taylor', `neumann',
     `firstkindseries', or `fredseries' (it is also the maximum depth
     of recursion for the differentiation method); <guess> is the
     initial guess for `neumann' or `firstkindseries'.

     Default values for the 2nd thru 5th parameters are:

     <unk>: `<p>(<x>)', where <p> is the first function encountered in
     an integrand which is unknown to Maxima and <x> is the variable
     which occurs as an argument to the first occurrence of <p> found
     outside of an integral in the case of `secondkind' equations, or
     is the only other variable besides the variable of integration in
     `firstkind' equations.  If the attempt to search for <x> fails,
     the user will be asked to supply the independent variable.

     tech: `first'

     n: 1

     guess: `none' which will cause `neumann' and `firstkindseries' to
     use `<f>(<x>)' as an initial guess.

Option variable: ieqnprint

     Default value: `true'

     `ieqnprint' governs the behavior of the result returned by the
     `ieqn'  command.  When `ieqnprint' is `false', the lists returned
     by the `ieqn' function are of the form

     [<solution>, <technique used>, <nterms>, <flag>]

     where <flag> is absent if the solution is exact.

     Otherwise, it is the word `approximate' or `incomplete'
     corresponding to an inexact or non-closed form solution,
     respectively.  If a series method was used, <nterms> gives the
     number of terms taken (which could be less than the n given to
     `ieqn' if an error prevented generation of further terms).

Function: lhs (<expr>)

     Returns the left-hand side (that is, the first argument) of the
     expression <expr>, when the operator of <expr> is one of the
     relational operators `< <= = # equal notequal >= >', one of the
     assignment operators `:= ::= : ::', or a user-defined binary infix
     operator, as declared by `infix'.

     When <expr> is an atom or its operator is something other than the
     ones listed above, `lhs' returns <expr>.

     See also `rhs'.

     Examples:

          (%i1) e: aa + bb = cc;
          (%o1)                     bb + aa = cc
          (%i2) lhs (e);
          (%o2)                        bb + aa
          (%i3) rhs (e);
          (%o3)                          cc
          (%i4) [lhs (aa < bb), lhs (aa <= bb), lhs (aa >= bb),
                 lhs (aa > bb)];
          (%o4)                   [aa, aa, aa, aa]
          (%i5) [lhs (aa = bb), lhs (aa # bb), lhs (equal (aa, bb)),
                 lhs (notequal (aa, bb))];
          (%o5)                   [aa, aa, aa, aa]
          (%i6) e1: '(foo(x) := 2*x);
          (%o6)                     foo(x) := 2 x
          (%i7) e2: '(bar(y) ::= 3*y);
          (%o7)                    bar(y) ::= 3 y
          (%i8) e3: '(x : y);
          (%o8)                         x : y
          (%i9) e4: '(x :: y);
          (%o9)                        x :: y
          (%i10) [lhs (e1), lhs (e2), lhs (e3), lhs (e4)];
          (%o10)               [foo(x), bar(y), x, x]
          (%i11) infix ("][");
          (%o11)                         ][
          (%i12) lhs (aa ][ bb);
          (%o12)                         aa

Function: linsolve ([<expr_1>, ..., <expr_m>], [<x_1>, ..., <x_n>])

     Solves the list of simultaneous linear equations for the list of
     variables.  The expressions must each be polynomials in the
     variables and may be equations.

     When `globalsolve'  is `true', each solved-for variable is bound to
     its value in the solution of the equations.

     When `backsubst'  is `false', `linsolve' does not carry out back
     substitution after the equations have been triangularized.  This
     may be necessary in very big problems where back substitution
     would cause the generation of extremely large expressions.

     When `linsolve_params'  is `true', `linsolve' also generates the
     `%r' symbols used to represent arbitrary parameters described in
     the manual under `algsys'.    Otherwise, `linsolve' solves an
     under-determined system of equations with some variables expressed
     in terms of others.

     When `programmode'  is `false', `linsolve' displays the solution
     with intermediate expression (`%t') labels, and returns the list
     of labels.

          (%i1) e1: x + z = y;
          (%o1)                       z + x = y
          (%i2) e2: 2*a*x - y = 2*a^2;
                                                 2
          (%o2)                   2 a x - y = 2 a
          (%i3) e3: y - 2*z = 2;
          (%o3)                      y - 2 z = 2
          (%i4) [globalsolve: false, programmode: true];
          (%o4)                     [false, true]
          (%i5) linsolve ([e1, e2, e3], [x, y, z]);
          (%o5)            [x = a + 1, y = 2 a, z = a - 1]
          (%i6) [globalsolve: false, programmode: false];
          (%o6)                    [false, false]
          (%i7) linsolve ([e1, e2, e3], [x, y, z]);
          Solution

          (%t7)                       z = a - 1

          (%t8)                        y = 2 a

          (%t9)                       x = a + 1
          (%o9)                    [%t7, %t8, %t9]
          (%i9) ''%;
          (%o9)            [z = a - 1, y = 2 a, x = a + 1]
          (%i10) [globalsolve: true, programmode: false];
          (%o10)                    [true, false]
          (%i11) linsolve ([e1, e2, e3], [x, y, z]);
          Solution

          (%t11)                      z : a - 1

          (%t12)                       y : 2 a

          (%t13)                      x : a + 1
          (%o13)                 [%t11, %t12, %t13]
          (%i13) ''%;
          (%o13)           [z : a - 1, y : 2 a, x : a + 1]
          (%i14) [x, y, z];
          (%o14)                 [a + 1, 2 a, a - 1]
          (%i15) [globalsolve: true, programmode: true];
          (%o15)                    [true, true]
          (%i16) linsolve ([e1, e2, e3], '[x, y, z]);
          (%o16)           [x : a + 1, y : 2 a, z : a - 1]
          (%i17) [x, y, z];
          (%o17)                 [a + 1, 2 a, a - 1]

Option variable: linsolvewarn

     Default value: `true'

     When `linsolvewarn' is `true', `linsolve'  prints a message
     "Dependent equations eliminated".

Option variable: linsolve_params

     Default value: `true'

     When `linsolve_params' is `true', `linsolve'  also generates the
     `%r' symbols used to represent arbitrary parameters described in
     the manual under `algsys'.    Otherwise, `linsolve' solves an
     under-determined system of equations with some variables expressed
     in terms of others.

System variable: multiplicities

     Default value: `not_set_yet'

     `multiplicities' is set to a list of the multiplicities of the
     individual solutions returned by `solve'  or `realroots'.

Function: nroots (<p>, <low>, <high>)

     Returns the number of real roots of the real univariate polynomial
     <p> in the half-open interval `(<low>, <high>]'.  The endpoints of
     the interval may be `minf' or `inf'.

     `nroots' uses the method of Sturm sequences.

          (%i1) p: x^10 - 2*x^4 + 1/2$
          (%i2) nroots (p, -6, 9.1);
          (%o2)                           4

Function: nthroot (<p>, <n>)

     where <p> is a polynomial with integer coefficients and <n> is a
     positive integer returns `q', a polynomial over the integers, such
     that `q^n = p' or prints an error message indicating that <p> is
     not a perfect nth power.  This routine is much faster than `factor'
     or even `sqfr'.

Option variable: polyfactor

     Default value: `false'

     The option variable `polyfactor' when `true' causes `allroots'
     and `bfallroots'  to factor the polynomial over the real numbers
     if the polynomial is real, or over the complex numbers, if the
     polynomial is complex.

     See `allroots' for an example.

Option variable: programmode

     Default value: `true'

     When `programmode' is `true', `solve',

     `realroots',  `allroots',  and `linsolve'  return solutions as
     elements in a list.  (Except when `backsubst'  is set to `false',
     in which case `programmode: false' is assumed.)

     When `programmode' is `false', `solve', etc. create intermediate
     expression labels `%t1', `t2', etc., and assign the solutions to
     them.

Option variable: realonly

     Default value: `false'

     When `realonly' is `true', `algsys'  returns only those solutions
     which are free of `%i'.

Function: realroots (<expr>, <bound>)
Function: realroots (<eqn>, <bound>)
Function: realroots (<expr>)
Function: realroots (<eqn>)

     Computes rational approximations of the real roots of the
     polynomial <expr> or polynomial equation <eqn> of one variable, to
     within a tolerance of <bound>.  Coefficients of <expr> or <eqn>
     must be literal numbers; symbol constants such as `%pi' are
     rejected.

     `realroots' assigns the multiplicities of the roots it finds to
     the global variable `multiplicities'.

     `realroots' constructs a Sturm sequence to bracket each root, and
     then applies bisection to refine the approximations.  All
     coefficients are converted to rational equivalents before
     searching for roots, and computations are carried out by exact
     rational arithmetic.  Even if some coefficients are floating-point
     numbers, the results are rational (unless coerced to floats by the
     `float'  or `numer'  flags).

     When <bound> is less than 1, all integer roots are found exactly.
     When <bound> is unspecified, it is assumed equal to the global
     variable `rootsepsilon'.

     When the global variable `programmode'  is `true', `realroots'
     returns a list of the form `[x = <x_1>, x = <x_2>, ...]'.  When
     `programmode' is `false', `realroots' creates intermediate
     expression labels `%t1', `%t2', ..., assigns the results to them,
     and returns the list of labels.

     Examples:

          (%i1) realroots (-1 - x + x^5, 5e-6);
                                         612003
          (%o1)                     [x = ------]
                                         524288
          (%i2) ev (%[1], float);
          (%o2)                 x = 1.167303085327148
          (%i3) ev (-1 - x + x^5, %);
          (%o3)                - 7.396496210176905E-6

          (%i1) realroots (expand ((1 - x)^5 * (2 - x)^3 * (3 - x)), 1e-20);
          (%o1)                 [x = 1, x = 2, x = 3]
          (%i2) multiplicities;
          (%o2)                       [5, 3, 1]

Function: rhs (<expr>)

     Returns the right-hand side (that is, the second argument) of the
     expression <expr>, when the operator of <expr> is one of the
     relational operators `< <= = # equal notequal >= >', one of the
     assignment operators `:= ::= : ::', or a user-defined binary infix
     operator, as declared by `infix'.

     When <expr> is an atom or its operator is something other than the
     ones listed above, `rhs' returns 0.

     See also `lhs'.

     Examples:

          (%i1) e: aa + bb = cc;
          (%o1)                     bb + aa = cc
          (%i2) lhs (e);
          (%o2)                        bb + aa
          (%i3) rhs (e);
          (%o3)                          cc
          (%i4) [rhs (aa < bb), rhs (aa <= bb), rhs (aa >= bb),
                 rhs (aa > bb)];
          (%o4)                   [bb, bb, bb, bb]
          (%i5) [rhs (aa = bb), rhs (aa # bb), rhs (equal (aa, bb)),
                 rhs (notequal (aa, bb))];
          (%o5)                   [bb, bb, bb, bb]
          (%i6) e1: '(foo(x) := 2*x);
          (%o6)                     foo(x) := 2 x
          (%i7) e2: '(bar(y) ::= 3*y);
          (%o7)                    bar(y) ::= 3 y
          (%i8) e3: '(x : y);
          (%o8)                         x : y
          (%i9) e4: '(x :: y);
          (%o9)                        x :: y
          (%i10) [rhs (e1), rhs (e2), rhs (e3), rhs (e4)];
          (%o10)                  [2 x, 3 y, y, y]
          (%i11) infix ("][");
          (%o11)                         ][
          (%i12) rhs (aa ][ bb);
          (%o12)                         bb

Option variable: rootsconmode

     Default value: `true'

     `rootsconmode' governs the behavior of the `rootscontract' command.
     See `rootscontract'  for details.

Function: rootscontract (<expr>)

     Converts products of roots into roots of products.  For example,
     `rootscontract (sqrt(x)*y^(3/2))' yields `sqrt(x*y^3)'.

     When `radexpand'  is `true' and `domain'  is `real',
     `rootscontract' converts `abs'  into `sqrt' , e.g., `rootscontract
     (abs(x)*sqrt(y))' yields `sqrt(x^2*y)'.

     There is an option `rootsconmode'  affecting `rootscontract' as
     follows:

          Problem            Value of        Result of applying
                            rootsconmode        rootscontract

          x^(1/2)*y^(3/2)      false          (x*y^3)^(1/2)
          x^(1/2)*y^(1/4)      false          x^(1/2)*y^(1/4)
          x^(1/2)*y^(1/4)      true           (x*y^(1/2))^(1/2)
          x^(1/2)*y^(1/3)      true           x^(1/2)*y^(1/3)
          x^(1/2)*y^(1/4)      all            (x^2*y)^(1/4)
          x^(1/2)*y^(1/3)      all            (x^3*y^2)^(1/6)

     When `rootsconmode' is `false', `rootscontract' contracts only
     with respect to rational number exponents whose denominators are
     the same.  The key to the `rootsconmode: true' examples is simply
     that 2 divides into 4 but not into 3.  `rootsconmode: all'
     involves taking the least common multiple of the denominators of
     the exponents.

     `rootscontract' uses `ratsimp'  in a manner similar to
     `logcontract'.

     Examples:

          (%i1) rootsconmode: false$
          (%i2) rootscontract (x^(1/2)*y^(3/2));
                                             3
          (%o2)                      sqrt(x y )
          (%i3) rootscontract (x^(1/2)*y^(1/4));
                                             1/4
          (%o3)                     sqrt(x) y
          (%i4) rootsconmode: true$
          (%i5) rootscontract (x^(1/2)*y^(1/4));
          (%o5)                    sqrt(x sqrt(y))
          (%i6) rootscontract (x^(1/2)*y^(1/3));
                                             1/3
          (%o6)                     sqrt(x) y
          (%i7) rootsconmode: all$
          (%i8) rootscontract (x^(1/2)*y^(1/4));
                                        2   1/4
          (%o8)                       (x  y)
          (%i9) rootscontract (x^(1/2)*y^(1/3));
                                       3  2 1/6
          (%o9)                      (x  y )
          (%i10) rootsconmode: false$
          (%i11) rootscontract (sqrt(sqrt(x) + sqrt(1 + x))
                              *sqrt(sqrt(1 + x) - sqrt(x)));
          (%o11)                          1
          (%i12) rootsconmode: true$
          (%i13) rootscontract (sqrt(5+sqrt(5)) - 5^(1/4)*sqrt(1+sqrt(5)));
          (%o13)                          0

Option variable: rootsepsilon

     Default value: 1.0e-7

     `rootsepsilon' is the tolerance which establishes the confidence
     interval for the roots found by the `realroots'  function.

Function: solve (<expr>, <x>)
Function: solve (<expr>)
Function: solve ([<eqn_1>, ..., <eqn_n>], [<x_1>, ..., <x_n>])

     Solves the algebraic equation <expr> for the variable <x> and
     returns a list of solution equations in <x>.  If <expr> is not an
     equation, the equation `<expr> = 0' is assumed in its place.  <x>
     may be a function (e.g. `f(x)'), or other non-atomic expression
     except a sum or product.  <x> may be omitted if <expr> contains
     only one variable.  <expr> may be a rational expression, and may
     contain trigonometric functions, exponentials, etc.

     The following method is used:

     Let <E> be the expression and <X> be the variable.  If <E> is
     linear in <X> then it is trivially solved for <X>.  Otherwise if
     <E> is of the form `A*X^N + B' then the result is `(-B/A)^1/N)'
     times the `N''th roots of unity.

     If <E> is not linear in <X> then the gcd of the exponents of <X> in
     <E> (say <N>) is divided into the exponents and the multiplicity
     of the roots is multiplied by <N>.  Then `solve' is called again
     on the result.  If <E> factors then `solve' is called on each of
     the factors.  Finally `solve' will use the quadratic, cubic, or
     quartic formulas where necessary.

     In the case where <E> is a polynomial in some function of the
     variable to be solved for, say `F(X)', then it is first solved for
     `F(X)' (call the result <C>), then the equation `F(X)=C' can be
     solved for <X> provided the inverse of the function <F> is known.

     `breakup'  if `false' will cause `solve' to express the solutions
     of cubic or quartic equations as single expressions rather than as
     made up of several common subexpressions which is the default.

     `multiplicities'  - will be set to a list of the multiplicities of
     the individual solutions returned by `solve', `realroots',  or
     `allroots'.    Try `apropos (solve)' for the switches which affect
     `solve'.  `describe'  may then by used on the individual switch
     names if their purpose is not clear.

     `solve ([<eqn_1>, ..., <eqn_n>], [<x_1>, ..., <x_n>])' solves a
     system of simultaneous (linear or non-linear) polynomial equations
     by calling `linsolve'  or `algsys'  and returns a list of the
     solution lists in the variables.  In the case of `linsolve' this
     list would contain a single list of solutions.  It takes two lists
     as arguments.  The first list represents the equations to be
     solved; the second list is a list of the unknowns to be
     determined.  If the total number of variables in the equations is
     equal to the number of equations, the second argument-list may be
     omitted.

     When `programmode'  is `false', `solve' displays solutions with
     intermediate expression (`%t') labels, and returns the list of
     labels.

     When `globalsolve'  is `true' and the problem is to solve two or
     more linear equations, each solved-for variable is bound to its
     value in the solution of the equations.

     Examples:

          (%i1) solve (asin (cos (3*x))*(f(x) - 1), x);

          solve: using arc-trig functions to get a solution.
          Some solutions will be lost.
                                      %pi
          (%o1)                  [x = ---, f(x) = 1]
                                       6
          (%i2) ev (solve (5^f(x) = 125, f(x)), solveradcan);
                                          log(125)
          (%o2)                   [f(x) = --------]
                                           log(5)
          (%i3) [4*x^2 - y^2 = 12, x*y - x = 2];
                                2    2
          (%o3)             [4 x  - y  = 12, x y - x = 2]

          (%i4) solve (%, [x, y]);
          (%o4) [[x = 2, y = 2], [x = .5202594388652008 %i
           - .1331240357358706, y = .07678378523787788
           - 3.608003221870287 %i], [x = - .5202594388652008 %i
           - .1331240357358706, y = 3.608003221870287 %i
           + .07678378523787788], [x = - 1.733751846381093,
          y = - .1535675710019696]]

          (%i5) solve (1 + a*x + x^3, x);

                                                 3
                        sqrt(3) %i   1   sqrt(4 a  + 27)   1 1/3
          (%o5) [x = (- ---------- - -) (--------------- - -)
                            2        2      6 sqrt(3)      2

                  sqrt(3) %i   1
                 (---------- - -) a
                      2        2
           - --------------------------, x =
                        3
                sqrt(4 a  + 27)   1 1/3
             3 (--------------- - -)
                   6 sqrt(3)      2

                                    3
           sqrt(3) %i   1   sqrt(4 a  + 27)   1 1/3
          (---------- - -) (--------------- - -)
               2        2      6 sqrt(3)      2

                   sqrt(3) %i   1
                (- ---------- - -) a
                       2        2
           - --------------------------, x =
                        3
                sqrt(4 a  + 27)   1 1/3
             3 (--------------- - -)
                   6 sqrt(3)      2

                   3
           sqrt(4 a  + 27)   1 1/3               a
          (--------------- - -)    - --------------------------]
              6 sqrt(3)      2                  3
                                        sqrt(4 a  + 27)   1 1/3
                                     3 (--------------- - -)
                                           6 sqrt(3)      2
          (%i6) solve (x^3 - 1);
                       sqrt(3) %i - 1        sqrt(3) %i + 1
          (%o6)   [x = --------------, x = - --------------, x = 1]
                             2                     2
          (%i7) solve (x^6 - 1);
                     sqrt(3) %i + 1      sqrt(3) %i - 1
          (%o7) [x = --------------, x = --------------, x = - 1,
                           2                   2

                               sqrt(3) %i + 1        sqrt(3) %i - 1
                         x = - --------------, x = - --------------, x = 1]
                                     2                     2
          (%i8) ev (x^6 - 1, %[1]);
                                                6
                                (sqrt(3) %i + 1)
          (%o8)                 ----------------- - 1
                                       64
          (%i9) expand (%);
          (%o9)                           0
          (%i10) x^2 - 1;
                                        2
          (%o10)                       x  - 1
          (%i11) solve (%, x);
          (%o11)                  [x = - 1, x = 1]
          (%i12) ev (%th(2), %[1]);
          (%o12)                          0

     The symbols `%r' are used to denote arbitrary constants in a
     solution.

          (%i1) solve([x+y=1,2*x+2*y=2],[x,y]);

          solve: dependent equations eliminated: (2)
          (%o1)                      [[x = 1 - %r1, y = %r1]]

     See `algsys'  and `%rnum_list'  for more information.

Option variable: solvedecomposes

     Default value: `true'

     When `solvedecomposes' is `true', `solve' calls `polydecomp'  if
     asked to solve polynomials.

Option variable: solveexplicit

     Default value: `false'

     When `solveexplicit' is `true', inhibits `solve'  from returning
     implicit solutions, that is, solutions of the form `F(x) = 0' where
     `F' is some function.

Option variable: solvefactors

     Default value: `true'

     When `solvefactors' is `false', `solve'  does not try to factor
     the expression.  The `false' setting may be desired in some cases
     where factoring is not necessary.

Option variable: solvenullwarn

     Default value: `true'

     When `solvenullwarn' is `true', `solve'  prints a warning message
     if called with either a null equation list or a null variable
     list.  For example, `solve ([], [])' would print two warning
     messages and return `[]'.

Option variable: solveradcan

     Default value: `false'

     When `solveradcan' is `true', `solve'  calls `radcan'

     which makes `solve' slower but will allow certain problems
     containing exponentials and logarithms to be solved.

Option variable: solvetrigwarn

     Default value: `true'

     When `solvetrigwarn' is `true', `solve'  may print a message
     saying that it is using inverse trigonometric functions to solve
     the equation, and thereby losing solutions.

21 Differential Equations


Introduction to Differential Equations
Functions and Variables for Differential Equations

21.1 Introduction to Differential Equations

This section describes the functions available in Maxima to obtain
analytic solutions for some specific types of first and second-order
equations. To obtain a numerical solution for a system of differential
equations, see the additional package `dynamics'. For graphical
representations in phase space, see the additional package `plotdf'.

21.2 Functions and Variables for Differential Equations

Function: bc2 (<solution>, <xval1>, <yval1>, <xval2>, <yval2>)

     Solves a boundary value problem for a second order differential
     equation.  Here: <solution> is a general solution to the equation,
     as found by `ode2'; <xval1> specifies the value of the independent
     variable in a first point, in the form `<x> = <x1>', and <yval1>
     gives the value of the dependent variable in that point, in the
     form `<y> = <y1>'. The expressions <xval2> and <yval2> give the
     values for these variables at a second point, using the same form.

     See `ode2' for an example of its usage.

Function: desolve (<eqn>, <x>)
Function: desolve ([<eqn_1>, ..., <eqn_n>], [<x_1>, ..., <x_n>])

     The function `desolve' solves systems of linear ordinary
     differential equations using Laplace transform.  Here the <eqn>'s
     are differential equations in the dependent variables <x_1>, ...,
     <x_n>.  The functional dependence of <x_1>, ..., <x_n> on an
     independent variable, for instance <x>, must be explicitly
     indicated in the variables and its derivatives. For example, this
     would not be the correct way to define two equations:

          eqn_1: 'diff(f,x,2) = sin(x) + 'diff(g,x);
          eqn_2: 'diff(f,x) + x^2 - f = 2*'diff(g,x,2);

     The correct way would be:

          eqn_1: 'diff(f(x),x,2) = sin(x) + 'diff(g(x),x);
          eqn_2: 'diff(f(x),x) + x^2 - f(x) = 2*'diff(g(x),x,2);

     The call to the function `desolve' would then be
          desolve([eqn_1, eqn_2], [f(x),g(x)]);

     If initial conditions at `x=0' are known, they can be supplied
     before calling `desolve' by using `atvalue'.

          (%i1) 'diff(f(x),x)='diff(g(x),x)+sin(x);
                           d           d
          (%o1)            -- (f(x)) = -- (g(x)) + sin(x)
                           dx          dx
          (%i2) 'diff(g(x),x,2)='diff(f(x),x)-cos(x);
                            2
                           d            d
          (%o2)            --- (g(x)) = -- (f(x)) - cos(x)
                             2          dx
                           dx
          (%i3) atvalue('diff(g(x),x),x=0,a);
          (%o3)                           a
          (%i4) atvalue(f(x),x=0,1);
          (%o4)                           1
          (%i5) desolve([%o1,%o2],[f(x),g(x)]);
                            x
          (%o5) [f(x) = a %e  - a + 1, g(x) =

                                                          x
                                             cos(x) + a %e  - a + g(0) - 1]
          (%i6) [%o1,%o2],%o5,diff;
                       x       x      x                x
          (%o6)   [a %e  = a %e , a %e  - cos(x) = a %e  - cos(x)]

     If `desolve' cannot obtain a solution, it returns `false'.

Function: ic1 (<solution>, <xval>, <yval>)

     Solves initial value problems for first order differential
     equations.  Here <solution> is a general solution to the equation,
     as found by `ode2', <xval> gives an initial value for the
     independent variable in the form `<x> = <x0>', and <yval> gives the
     initial value for the dependent variable in the form `<y> = <y0>'.

     See `ode2' for an example of its usage.

Function: ic2 (<solution>, <xval>, <yval>, <dval>)

     Solves initial value problems for second-order differential
     equations.  Here <solution> is a general solution to the equation,
     as found by `ode2', <xval> gives the initial value for the
     independent variable in the form `<x> = <x0>', <yval> gives the
     initial value of the dependent variable in the form `<y> = <y0>',
     and <dval> gives the initial value for the first derivative of the
     dependent variable with respect to independent variable, in the
     form `diff(<y>,<x>) = <dy0>' (`diff' does not have to be quoted).

     See `ode2' for an example of its usage.

Function: ode2 (<eqn>, <dvar>, <ivar>)

     The function `ode2' solves an ordinary differential equation (ODE)
     of first or second order. It takes three arguments: an ODE given by
     <eqn>, the dependent variable <dvar>, and the independent variable
     <ivar>. When successful, it returns either an explicit or implicit
     solution for the dependent variable. `%c' is used to represent the
     integration constant in the case of first-order equations, and
     `%k1' and `%k2' the constants for second-order equations. The
     dependence of the dependent variable on the independent variable
     does not have to be written explicitly, as in the case of
     `desolve', but the independent variable must always be given as the
     third argument.

     If `ode2' cannot obtain a solution for whatever reason, it returns
     `false', after perhaps printing out an error message. The methods
     implemented for first order equations in the order in which they
     are tested are: linear, separable, exact - perhaps requiring an
     integrating factor, homogeneous, Bernoulli's equation, and a
     generalized homogeneous method. The types of second-order
     equations which can be solved are: constant coefficients, exact,
     linear homogeneous with non-constant coefficients which can be
     transformed to constant coefficients, the Euler or
     equi-dimensional equation, equations solvable by the method of
     variation of parameters, and equations which are free of either the
     independent or of the dependent variable so that they can be
     reduced to two first order linear equations to be solved
     sequentially.

     In the course of solving ODE's, several variables are set purely
     for informational purposes: `method' denotes the method of solution
     used (e.g., `linear'), `intfactor' denotes any integrating factor
     used, `odeindex' denotes the index for Bernoulli's method or for
     the generalized homogeneous method, and `yp' denotes the
     particular solution for the variation of parameters technique.

     In order to solve initial value problems (IVP) functions `ic1' and
     `ic2' are available for first and second order equations, and to
     solve second-order boundary value problems (BVP) the function `bc2'
     can be used.

     Example:

          (%i1) x^2*'diff(y,x) + 3*y*x = sin(x)/x;
                                2 dy           sin(x)
          (%o1)                x  -- + 3 x y = ------
                                  dx             x
          (%i2) ode2(%,y,x);
                                       %c - cos(x)
          (%o2)                    y = -----------
                                            3
                                           x
          (%i3) ic1(%o2,x=%pi,y=0);
                                        cos(x) + 1
          (%o3)                   y = - ----------
                                             3
                                            x
          (%i4) 'diff(y,x,2) + y*'diff(y,x)^3 = 0;
                                   2
                                  d y      dy 3
          (%o4)                   --- + y (--)  = 0
                                    2      dx
                                  dx
          (%i5) ode2(%,y,x);
                                3
                               y  + 6 %k1 y
          (%o5)                ------------ = x + %k2
                                    6
          (%i6) ratsimp(ic2(%o5,x=0,y=0,'diff(y,x)=2));
                                       3
                                    2 y  - 3 y
          (%o6)                   - ---------- = x
                                        6
          (%i7) bc2(%o5,x=0,y=1,x=1,y=3);
                                   3
                                  y  - 10 y       3
          (%o7)                   --------- = x - -
                                      6           2

22 Numerical


Introduction to fast Fourier transform
Functions and Variables for fast Fourier transform
Introduction to Fourier series
Functions and Variables for Fourier series

22.1 Introduction to fast Fourier transform

The `fft' package comprises functions for the numerical (not symbolic)
computation of the fast Fourier transform.

22.2 Functions and Variables for fast Fourier transform

Function: polartorect (<r>, <t>)

     Translates complex values of the form `r %e^(%i t)' to the form `a
     + b %i', where <r> is the magnitude and <t> is the phase.  <r> and
     <t> are 1-dimensional arrays of the same size.  The array size
     need not be a power of 2.

     The original values of the input arrays are replaced by the real
     and imaginary parts, `a' and `b', on return.  The outputs are
     calculated as

          a = r cos(t)
          b = r sin(t)

     `polartorect' is the inverse function of `recttopolar'.

     `load(fft)' loads this function.  See also `fft'.

Function: recttopolar (<a>, <b>)

     Translates complex values of the form `a + b %i' to the form `r
     %e^(%i t)', where <a> is the real part and <b> is the imaginary
     part.  <a> and <b> are 1-dimensional arrays of the same size.  The
     array size need not be a power of 2.

     The original values of the input arrays are replaced by the
     magnitude and angle, `r' and `t', on return.  The outputs are
     calculated as

          r = sqrt(a^2 + b^2)
          t = atan2(b, a)

     The computed angle is in the range `-%pi' to `%pi'.

     `recttopolar' is the inverse function of `polartorect'.

     `load(fft)' loads this function.  See also `fft'.

Function: inverse_fft (<y>)

     Computes the inverse complex fast Fourier transform.  <y> is a
     list or array (named or unnamed) which contains the data to
     transform.  The number of elements must be a power of 2.  The
     elements must be literal numbers (integers, rationals, floats, or
     bigfloats) or symbolic constants, or expressions `a + b*%i' where
     `a' and `b' are literal numbers or symbolic constants.

     `inverse_fft' returns a new object of the same type as <y>, which
     is not modified.  Results are always computed as floats or
     expressions `a + b*%i' where `a' and `b' are floats.

     The inverse discrete Fourier transform is defined as follows.  Let
     `x' be the output of the inverse transform.  Then for `j' from 0
     through `n - 1',

          x[j] = sum(y[k] exp(+2 %i %pi j k / n), k, 0, n - 1)

     `load(fft)' loads this function.

     See also `fft' (forward transform), `recttopolar', and
     `polartorect'.

     Examples:

     Real data.

          (%i1) load (fft) $
          (%i2) fpprintprec : 4 $
          (%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
          (%i4) L1 : inverse_fft (L);
          (%o4) [0.0, 14.49 %i - .8284, 0.0, 2.485 %i + 4.828, 0.0,
                                 4.828 - 2.485 %i, 0.0, - 14.49 %i - .8284]
          (%i5) L2 : fft (L1);
          (%o5) [1.0, 2.0 - 2.168L-19 %i, 3.0 - 7.525L-20 %i,
          4.0 - 4.256L-19 %i, - 1.0, 2.168L-19 %i - 2.0,
          7.525L-20 %i - 3.0, 4.256L-19 %i - 4.0]
          (%i6) lmax (abs (L2 - L));
          (%o6)                       3.545L-16

     Complex data.

          (%i1) load (fft) $
          (%i2) fpprintprec : 4 $
          (%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
          (%i4) L1 : inverse_fft (L);
          (%o4) [4.0, 2.711L-19 %i + 4.0, 2.0 %i - 2.0,
          - 2.828 %i - 2.828, 0.0, 5.421L-20 %i + 4.0, - 2.0 %i - 2.0,
          2.828 %i + 2.828]
          (%i5) L2 : fft (L1);
          (%o5) [4.066E-20 %i + 1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i,
          1.55L-19 %i - 1.0, - 4.066E-20 %i - 1.0, 1.0 - 1.0 %i,
          1.0 %i + 1.0, 1.0 - 7.368L-20 %i]
          (%i6) lmax (abs (L2 - L));
          (%o6)                       6.841L-17

Function: fft (<x>)

     Computes the complex fast Fourier transform.  <x> is a list or
     array (named or unnamed) which contains the data to transform.
     The number of elements must be a power of 2.  The elements must be
     literal numbers (integers, rationals, floats, or bigfloats) or
     symbolic constants, or expressions `a + b*%i' where `a' and `b'
     are literal numbers or symbolic constants.

     `fft' returns a new object of the same type as <x>, which is not
     modified.  Results are always computed as floats or expressions `a
     + b*%i' where `a' and `b' are floats.

     The discrete Fourier transform is defined as follows.  Let `y' be
     the output of the transform.  Then for `k' from 0 through `n - 1',

          y[k] = (1/n) sum(x[j] exp(-2 %i %pi j k / n), j, 0, n - 1)

     When the data <x> are real, real coefficients `a' and `b' can be
     computed such that

          x[j] = sum(a[k]*cos(2*%pi*j*k/n)+b[k]*sin(2*%pi*j*k/n), k, 0, n/2)

     with

          a[0] = realpart (y[0])
          b[0] = 0

     and, for k from 1 through n/2 - 1,

          a[k] = realpart (y[k] + y[n - k])
          b[k] = imagpart (y[n - k] - y[k])

     and

          a[n/2] = realpart (y[n/2])
          b[n/2] = 0

     `load(fft)' loads this function.

     See also `inverse_fft' (inverse transform), `recttopolar', and
     `polartorect'.

     Examples:

     Real data.

          (%i1) load (fft) $
          (%i2) fpprintprec : 4 $
          (%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
          (%i4) L1 : fft (L);
          (%o4) [0.0, - 1.811 %i - .1036, 0.0, .6036 - .3107 %i, 0.0,
                                   .3107 %i + .6036, 0.0, 1.811 %i - .1036]
          (%i5) L2 : inverse_fft (L1);
          (%o5) [1.0, 2.168L-19 %i + 2.0, 7.525L-20 %i + 3.0,
          4.256L-19 %i + 4.0, - 1.0, - 2.168L-19 %i - 2.0,
          - 7.525L-20 %i - 3.0, - 4.256L-19 %i - 4.0]
          (%i6) lmax (abs (L2 - L));
          (%o6)                       3.545L-16

     Complex data.

          (%i1) load (fft) $
          (%i2) fpprintprec : 4 $
          (%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
          (%i4) L1 : fft (L);
          (%o4) [0.5, .3536 %i + .3536, - 0.25 %i - 0.25,
          0.5 - 6.776L-21 %i, 0.0, - .3536 %i - .3536, 0.25 %i - 0.25,
          0.5 - 3.388L-20 %i]
          (%i5) L2 : inverse_fft (L1);
          (%o5) [1.0 - 4.066E-20 %i, 1.0 %i + 1.0, 1.0 - 1.0 %i,
          - 1.008L-19 %i - 1.0, 4.066E-20 %i - 1.0, 1.0 - 1.0 %i,
          1.0 %i + 1.0, 1.947L-20 %i + 1.0]
          (%i6) lmax (abs (L2 - L));
          (%o6)                       6.83L-17

     Computation of sine and cosine coefficients.

          (%i1) load (fft) $
          (%i2) fpprintprec : 4 $
          (%i3) L : [1, 2, 3, 4, 5, 6, 7, 8] $
          (%i4) n : length (L) $
          (%i5) x : make_array (any, n) $
          (%i6) fillarray (x, L) $
          (%i7) y : fft (x) $
          (%i8) a : make_array (any, n/2 + 1) $
          (%i9) b : make_array (any, n/2 + 1) $
          (%i10) a[0] : realpart (y[0]) $
          (%i11) b[0] : 0 $
          (%i12) for k : 1 thru n/2 - 1 do
             (a[k] : realpart (y[k] + y[n - k]),
              b[k] : imagpart (y[n - k] - y[k]));
          (%o12)                        done
          (%i13) a[n/2] : y[n/2] $
          (%i14) b[n/2] : 0 $
          (%i15) listarray (a);
          (%o15)          [4.5, - 1.0, - 1.0, - 1.0, - 0.5]
          (%i16) listarray (b);
          (%o16)           [0, - 2.414, - 1.0, - .4142, 0]
          (%i17) f(j) := sum (a[k]*cos(2*%pi*j*k/n) + b[k]*sin(2*%pi*j*k/n),
                              k, 0, n/2) $
          (%i18) makelist (float (f (j)), j, 0, n - 1);
          (%o18)      [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0]

Function: horner (<expr>, <x>)
Function: horner (<expr>)

     Returns a rearranged representation of <expr> as in Horner's rule,
     using <x> as the main variable if it is specified.  `x' may be
     omitted in which case the main variable of the canonical rational
     expression form of <expr> is used.

     `horner' sometimes improves stability if `expr' is to be
     numerically evaluated.  It is also useful if Maxima is used to
     generate programs to be run in Fortran.  See also `stringout'.

          (%i1) expr: 1e-155*x^2 - 5.5*x + 5.2e155;
                                     2
          (%o1)            1.0E-155 x  - 5.5 x + 5.2E+155
          (%i2) expr2: horner (%, x), keepfloat: true;
          (%o2)            (1.0E-155 x - 5.5) x + 5.2E+155
          (%i3) ev (expr, x=1e155);
          Maxima encountered a Lisp error:

           floating point overflow

          Automatically continuing.
          To reenable the Lisp debugger set *debugger-hook* to nil.
          (%i4) ev (expr2, x=1e155);
          (%o4)                       7.0E+154

Function: find_root (<expr>, <x>, <a>, <b>, [<abserr>, <relerr>])
Function: find_root (<f>, <a>, <b>, [<abserr>, <relerr>])
Function: bf_find_root (<expr>, <x>, <a>, <b>, [<abserr>, <relerr>])
Function: bf_find_root (<f>, <a>, <b>, [<abserr>, <relerr>])
Option variable: find_root_error
Option variable: find_root_abs
Option variable: find_root_rel

     Finds a root of the expression <expr> or the function <f> over the
     closed interval [<a>, <b>].  The expression <expr> may be an
     equation, in which case `find_root' seeks a root of `lhs(<expr>) -
     rhs(<expr>)'.

     Given that Maxima can evaluate <expr> or <f> over [<a>, <b>] and
     that <expr> or <f> is continuous, `find_root' is guaranteed to
     find the root, or one of the roots if there is more than one.

     `find_root' initially applies binary search.  If the function in
     question appears to be smooth enough, `find_root' applies linear
     interpolation instead.

     `bf_find_root' is a bigfloat version of `find_root'.  The function
     is computed using bigfloat arithmetic and a bigfloat result is
     returned.  Otherwise, `bf_find_root' is identical to `find_root',
     and the following description is equally applicable to
     `bf_find_root'.

     The accuracy of `find_root' is governed by `abserr' and `relerr',
     which are optional keyword arguments to `find_root'.  These
     keyword arguments take the form `key=val'.  The keyword arguments
     are

    <abserr>
          Desired absolute error of function value at root.  Default is
          `find_root_abs'.

    <relerr>
          Desired relative error of root.  Default is `find_root_rel'.

     `find_root' stops when the function in question evaluates to
     something less than or equal to `abserr', or if successive
     approximants <x_0>, <x_1> differ by no more than `relerr *
     max(abs(x_0), abs(x_1))'.  The default values of `find_root_abs'
     and `find_root_rel' are both zero.

     `find_root' expects the function in question to have a different
     sign at the endpoints of the search interval.  When the function
     evaluates to a number at both endpoints and these numbers have the
     same sign, the behavior of `find_root' is governed by
     `find_root_error'.  When `find_root_error' is `true', `find_root'
     prints an error message.  Otherwise `find_root' returns the value
     of `find_root_error'.  The default value of `find_root_error' is
     `true'.

     If <f> evaluates to something other than a number at any step in
     the search algorithm, `find_root' returns a partially-evaluated
     `find_root' expression.

     The order of <a> and <b> is ignored; the region in which a root is
     sought is [min(<a>, <b>), max(<a>, <b>)].

     Examples:

          (%i1) f(x) := sin(x) - x/2;
                                                  x
          (%o1)                  f(x) := sin(x) - -
                                                  2
          (%i2) find_root (sin(x) - x/2, x, 0.1, %pi);
          (%o2)                   1.895494267033981
          (%i3) find_root (sin(x) = x/2, x, 0.1, %pi);
          (%o3)                   1.895494267033981
          (%i4) find_root (f(x), x, 0.1, %pi);
          (%o4)                   1.895494267033981
          (%i5) find_root (f, 0.1, %pi);
          (%o5)                   1.895494267033981
          (%i6) find_root (exp(x) = y, x, 0, 100);
                                      x
          (%o6)           find_root(%e  = y, x, 0.0, 100.0)
          (%i7) find_root (exp(x) = y, x, 0, 100), y = 10;
          (%o7)                   2.302585092994046
          (%i8) log (10.0);
          (%o8)                   2.302585092994046
          (%i9) fpprec:32;
          (%o9)                           32
          (%i10) bf_find_root (exp(x) = y, x, 0, 100), y = 10;
          (%o10)                  2.3025850929940456840179914546844b0
          (%i11) log(10b0);
          (%o11)                  2.3025850929940456840179914546844b0

Function: newton (<expr>, <x>, <x_0>, <eps>)

     Returns an approximate solution of `<expr> = 0' by Newton's method,
     considering <expr> to be a function of one variable, <x>.  The
     search begins with `<x> = <x_0>' and proceeds until `abs(<expr>) <
     <eps>' (with <expr> evaluated at the current value of <x>).

     `newton' allows undefined variables to appear in <expr>, so long
     as the termination test `abs(<expr>) < <eps>' evaluates to `true'
     or `false'.  Thus it is not necessary that <expr> evaluate to a
     number.

     `load(newton1)' loads this function.

     See also `realroots', `allroots', `find_root', and `mnewton'.

     Examples:

          (%i1) load (newton1);
          (%o1) /usr/share/maxima/5.10.0cvs/share/numeric/newton1.mac
          (%i2) newton (cos (u), u, 1, 1/100);
          (%o2)                   1.570675277161251
          (%i3) ev (cos (u), u = %);
          (%o3)                 1.2104963335033528E-4
          (%i4) assume (a > 0);
          (%o4)                        [a > 0]
          (%i5) newton (x^2 - a^2, x, a/2, a^2/100);
          (%o5)                  1.00030487804878 a
          (%i6) ev (x^2 - a^2, x = %);
                                                     2
          (%o6)                6.098490481853958E-4 a

22.3 Introduction to Fourier series

The `fourie' package comprises functions for the symbolic computation
of Fourier series.  There are functions in the `fourie' package to
calculate Fourier integral coefficients and some functions for
manipulation of expressions.

22.4 Functions and Variables for Fourier series

Function: equalp (<x>, <y>)

     Returns `true' if `equal (<x>, <y>)' otherwise `false' (doesn't
     give an error message like `equal (x, y)' would do in this case).

Function: remfun (<f>, <expr>)
Function: remfun (<f>, <expr>, <x>)

     `remfun (<f>, <expr>)' replaces all occurrences of `<f> (<arg>)'
     by <arg> in <expr>.

     `remfun (<f>, <expr>, <x>)' replaces all occurrences of `<f>
     (<arg>)' by <arg> in <expr> only if <arg> contains the variable
     <x>.

Function: funp (<f>, <expr>)
Function: funp (<f>, <expr>, <x>)

     `funp (<f>, <expr>)' returns `true' if <expr> contains the
     function <f>.

     `funp (<f>, <expr>, <x>)' returns `true' if <expr> contains the
     function <f> and the variable <x> is somewhere in the argument of
     one of the instances of <f>.

Function: absint (<f>, <x>, <halfplane>)
Function: absint (<f>, <x>)
Function: absint (<f>, <x>, <a>, <b>)

     `absint (<f>, <x>, <halfplane>)' returns the indefinite integral
     of <f> with respect to <x> in the given halfplane (`pos', `neg',
     or `both').  <f> may contain expressions of the form `abs (x)',
     `abs (sin (x))', `abs (a) * exp (-abs (b) * abs (x))'.

     `absint (<f>, <x>)' is equivalent to `absint (<f>, <x>, pos)'.

     `absint (<f>, <x>, <a>, <b>)' returns the definite integral of <f>
     with respect to <x> from <a> to <b>.  <f> may include absolute
     values.

Function: fourier (<f>, <x>, <p>)

     Returns a list of the Fourier coefficients of `<f>(<x>)' defined
     on the interval `[-p, p]'.

Function: foursimp (<l>)

     Simplifies `sin (n %pi)' to 0 if `sinnpiflag' is `true' and `cos
     (n %pi)' to `(-1)^n' if `cosnpiflag' is `true'.

Option variable: sinnpiflag

     Default value: `true'

     See `foursimp'.

Option variable: cosnpiflag

     Default value: `true'

     See `foursimp'.

Function: fourexpand (<l>, <x>, <p>, <limit>)

     Constructs and returns the Fourier series from the list of Fourier
     coefficients <l> up through <limit> terms (<limit> may be `inf').
     <x> and <p> have same meaning as in `fourier'.

Function: fourcos (<f>, <x>, <p>)

     Returns the Fourier cosine coefficients for `<f>(<x>)' defined on
     `[0, <p>]'.

Function: foursin (<f>, <x>, <p>)

     Returns the Fourier sine coefficients for `<f>(<x>)' defined on
     `[0, <p>]'.

Function: totalfourier (<f>, <x>, <p>)

     Returns `fourexpand (foursimp (fourier (<f>, <x>, <p>)), <x>, <p>,
     'inf)'.

Function: fourint (<f>, <x>)

     Constructs and returns a list of the Fourier integral coefficients
     of `<f>(<x>)' defined on `[minf, inf]'.

Function: fourintcos (<f>, <x>)

     Returns the Fourier cosine integral coefficients for `<f>(<x>)' on
     `[0, inf]'.

Function: fourintsin (<f>, <x>)

     Returns the Fourier sine integral coefficients for `<f>(<x>)' on
     `[0, inf]'.

23 Matrices and Linear Algebra


Introduction to Matrices and Linear Algebra
Functions and Variables for Matrices and Linear Algebra

23.1 Introduction to Matrices and Linear Algebra


* Dot::
* Vectors::
* eigen::

23.2 Functions and Variables for Matrices and Linear Algebra

Function: addcol (<M>, <list_1>, ..., <list_n>)

     Appends the column(s) given by the one or more lists (or matrices)
     onto the matrix <M>.

Function: addrow (<M>, <list_1>, ..., <list_n>)

     Appends the row(s) given by the one or more lists (or matrices)
     onto the matrix <M>.

Function: adjoint (<M>)

     Returns the adjoint of the matrix <M>.  The adjoint matrix is the
     transpose of the matrix of cofactors of <M>.

Function: augcoefmatrix ([<eqn_1>, ..., <eqn_m>], [<x_1>, ..., <x_n>])

     Returns the augmented coefficient matrix for the variables <x_1>,
     ..., <x_n> of the system of linear equations <eqn_1>, ...,
     <eqn_m>.  This is the coefficient matrix with a column adjoined
     for the constant terms in each equation (i.e., those terms not
     dependent upon <x_1>, ..., <x_n>).

          (%i1) m: [2*x - (a - 1)*y = 5*b, c + b*y + a*x = 0]$
          (%i2) augcoefmatrix (m, [x, y]);
                                 [ 2  1 - a  - 5 b ]
          (%o2)                  [                 ]
                                 [ a    b      c   ]

Function: cauchy_matrix ([<x_1>, <x_2>, ..., <x_m>], [<y_1>, <y_2>, ..., <y_n>])
Function: cauchy_matrix ([<x_1>, <x_2>, ..., <x_n>])

     Returns a `n' by <m> Cauchy matrix with the elements <a[i,j]> =
     1/(<x_i>+<y_i>).  The second argument of `cauchy_matrix' is
     optional.  For this case the elements of the Cauchy matrix are
     <a[i,j]> = 1/(<x_i>+<x_j>).

     Remark: In the literature the Cauchy matrix can be found defined
     in two forms.  A second definition is <a[i,j]> = 1/(<x_i>-<y_i>).

     Examples:

          (%i1) cauchy_matrix([x1,x2],[y1,y2]);
                                [    1        1    ]
                                [ -------  ------- ]
                                [ y1 + x1  y2 + x1 ]
          (%o1)                 [                  ]
                                [    1        1    ]
                                [ -------  ------- ]
                                [ y1 + x2  y2 + x2 ]

          (%i2) cauchy_matrix([x1,x2]);
                                [   1         1    ]
                                [  ----    ------- ]
                                [  2 x1    x2 + x1 ]
          (%o2)                 [                  ]
                                [    1       1     ]
                                [ -------   ----   ]
                                [ x2 + x1   2 x2   ]

Function: charpoly (<M>, <x>)

     Returns the characteristic polynomial for the matrix <M> with
     respect to variable <x>.  That is, `determinant (<M> - diagmatrix
     (length (<M>), <x>))'.

          (%i1) a: matrix ([3, 1], [2, 4]);
                                      [ 3  1 ]
          (%o1)                       [      ]
                                      [ 2  4 ]
          (%i2) expand (charpoly (a, lambda));
                                     2
          (%o2)                lambda  - 7 lambda + 10
          (%i3) (programmode: true, solve (%));
          (%o3)               [lambda = 5, lambda = 2]
          (%i4) matrix ([x1], [x2]);
                                       [ x1 ]
          (%o4)                        [    ]
                                       [ x2 ]
          (%i5) ev (a . % - lambda*%, %th(2)[1]);
                                    [ x2 - 2 x1 ]
          (%o5)                     [           ]
                                    [ 2 x1 - x2 ]
          (%i6) %[1, 1] = 0;
          (%o6)                     x2 - 2 x1 = 0
          (%i7) x2^2 + x1^2 = 1;
                                      2     2
          (%o7)                     x2  + x1  = 1
          (%i8) solve ([%th(2), %], [x1, x2]);
                            1               2
          (%o8) [[x1 = - -------, x2 = - -------],
                         sqrt(5)         sqrt(5)

                                                       1             2
                                              [x1 = -------, x2 = -------]]
                                                    sqrt(5)       sqrt(5)

Function: coefmatrix ([<eqn_1>, ..., <eqn_m>], [<x_1>, ..., <x_n>])

     Returns the coefficient matrix for the variables <x_1>, ..., <x_n>
     of the system of linear equations <eqn_1>, ..., <eqn_m>.

          (%i1) coefmatrix([2*x-(a-1)*y+5*b = 0, b*y+a*x = 3], [x,y]);
                                           [ 2  1 - a ]
          (%o1)                            [          ]
                                           [ a    b   ]

Function: col (<M>, <i>)

     Returns the <i>'th column of the matrix <M>.  The return value is
     a matrix.

Function: columnvector (<L>)
Function: covect (<L>)

     Returns a matrix of one column and `length (<L>)' rows, containing
     the elements of the list <L>.

     `covect' is a synonym for `columnvector'.

     `load ("eigen")' loads this function.

     This is useful if you want to use parts of the outputs of the
     functions in this package in matrix calculations.

     Example:

          (%i1) load ("eigen")$
          Warning - you are redefining the Macsyma function eigenvalues
          Warning - you are redefining the Macsyma function eigenvectors
          (%i2) columnvector ([aa, bb, cc, dd]);
                                       [ aa ]
                                       [    ]
                                       [ bb ]
          (%o2)                        [    ]
                                       [ cc ]
                                       [    ]
                                       [ dd ]

Function: copymatrix (<M>)

     Returns a copy of the matrix <M>.  This is the only way to make a
     copy aside from copying <M> element by element.

     Note that an assignment of one matrix to another, as in `m2: m1',
     does not copy `m1'.  An assignment `m2 [i,j]: x' or `setelmx(x, i,
     j, m2)' also modifies `m1 [i,j]'.  Creating a copy with
     `copymatrix' and then using assignment creates a separate,
     modified copy.

Function: determinant (<M>)

     Computes the determinant of <M> by a method similar to Gaussian
     elimination.

     The form of the result depends upon the setting of the switch
     `ratmx'.

     There is a special routine for computing sparse determinants which
     is called when the switches `ratmx' and `sparse'  are both `true'.

Option variable: detout

     Default value: `false'

     When `detout' is `true', the determinant of a matrix whose inverse
     is computed is factored out of the inverse.

     For this switch to have an effect `doallmxops'  and `doscmxops'
     should be `false' (see their descriptions).  Alternatively this
     switch can be given to `ev'  which causes the other two to be set
     correctly.

     Example:

          (%i1) m: matrix ([a, b], [c, d]);
                                      [ a  b ]
          (%o1)                       [      ]
                                      [ c  d ]
          (%i2) detout: true$
          (%i3) doallmxops: false$
          (%i4) doscmxops: false$
          (%i5) invert (m);
                                    [  d   - b ]
                                    [          ]
                                    [ - c   a  ]
          (%o5)                     ------------
                                     a d - b c

Function: diagmatrix (<n>, <x>)

     Returns a diagonal matrix of size <n> by <n> with the diagonal
     elements all equal to <x>.  `diagmatrix (<n>, 1)' returns an
     identity matrix (same as `ident (<n>)').

     <n> must evaluate to an integer, otherwise `diagmatrix' complains
     with an error message.

     <x> can be any kind of expression, including another matrix.  If
     <x> is a matrix, it is not copied; all diagonal elements refer to
     the same instance, <x>.

Option variable: doallmxops

     Default value: `true'

     When `doallmxops' is `true', all operations relating to matrices
     are carried out.  When it is `false' then the setting of the
     individual `dot' switches govern which operations are performed.

Option variable: domxexpt

     Default value: `true'

     When `domxexpt' is `true', a matrix exponential, `exp (<M>)' where
     <M> is a matrix, is interpreted as a matrix with element `[i,j'
     equal to `exp (m[i,j])'.  Otherwise `exp (<M>)' evaluates to `exp
     (<ev(M)>'.

     `domxexpt' affects all expressions of the form `<base>^<power>'
     where <base> is an expression assumed scalar or constant, and
     <power> is a list or matrix.

     Example:

          (%i1) m: matrix ([1, %i], [a+b, %pi]);
                                   [   1    %i  ]
          (%o1)                    [            ]
                                   [ b + a  %pi ]
          (%i2) domxexpt: false$
          (%i3) (1 - c)^m;
                                       [   1    %i  ]
                                       [            ]
                                       [ b + a  %pi ]
          (%o3)                 (1 - c)
          (%i4) domxexpt: true$
          (%i5) (1 - c)^m;
                            [                      %i  ]
                            [    1 - c      (1 - c)    ]
          (%o5)             [                          ]
                            [        b + a         %pi ]
                            [ (1 - c)       (1 - c)    ]

Option variable: domxmxops

     Default value: `true'

     When `domxmxops' is `true', all matrix-matrix or matrix-list
     operations are carried out (but not scalar-matrix operations); if
     this switch is `false' such operations are not carried out.

Option variable: domxnctimes

     Default value: `false'

     When `domxnctimes' is `true', non-commutative products of matrices
     are carried out.

Option variable: dontfactor

     Default value: `[]'

     `dontfactor' may be set to a list of variables with respect to
     which factoring is not to occur.  (The list is initially empty.)
     Factoring also will not take place with respect to any variables
     which are less important, according the variable ordering assumed
     for canonical rational expression (CRE) form, than those on the
     `dontfactor' list.

Option variable: doscmxops

     Default value: `false'

     When `doscmxops' is `true', scalar-matrix operations are carried
     out.

Option variable: doscmxplus

     Default value: `false'

     When `doscmxplus' is `true', scalar-matrix operations yield a
     matrix result.  This switch is not subsumed under `doallmxops'.

Option variable: dot0nscsimp

     Default value: `true'

     When `dot0nscsimp' is `true', a non-commutative product of zero
     and a nonscalar term is simplified to a commutative product.

Option variable: dot0simp

     Default value: `true'

     When `dot0simp' is `true', a non-commutative product of zero and a
     scalar term is simplified to a commutative product.

Option variable: dot1simp

     Default value: `true'

     When `dot1simp' is `true', a non-commutative product of one and
     another term is simplified to a commutative product.

Option variable: dotassoc

     Default value: `true'

     When `dotassoc' is `true', an expression `(A.B).C' simplifies to
     `A.(B.C)'.

Option variable: dotconstrules

     Default value: `true'

     When `dotconstrules' is `true', a non-commutative product of a
     constant and another term is simplified to a commutative product.
     Turning on this flag effectively turns on `dot0simp',

     `dot0nscsimp',  and `dot1simp'  as well.

Option variable: dotdistrib

     Default value: `false'

     When `dotdistrib' is `true', an expression `A.(B + C)' simplifies
     to `A.B + A.C'.

Option variable: dotexptsimp

     Default value: `true'

     When `dotexptsimp' is `true', an expression `A.A' simplifies to
     `A^^2'.

Option variable: dotident

     Default value: 1

     `dotident' is the value returned by `X^^0'.

Option variable: dotscrules

     Default value: `false'

     When `dotscrules' is `true', an expression `A.SC' or `SC.A'
     simplifies to `SC*A' and `A.(SC*B)' simplifies to `SC*(A.B)'.

Function: echelon (<M>)

     Returns the echelon form of the matrix <M>, as produced by
     Gaussian elimination.  The echelon form is computed from <M> by
     elementary row operations such that the first non-zero element in
     each row in the resulting matrix is one and the column elements
     under the first one in each row are all zero.

     `triangularize'  also carries out Gaussian elimination, but it
     does not normalize the leading non-zero element in each row.

     `lu_factor'  and `cholesky'  are other functions which yield
     triangularized matrices.

          (%i1) M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]);
                                 [  3   7  aa  bb ]
                                 [                ]
          (%o1)                  [ - 1  8  5   2  ]
                                 [                ]
                                 [  9   2  11  4  ]
          (%i2) echelon (M);
                            [ 1  - 8  - 5      - 2     ]
                            [                          ]
                            [         28       11      ]
                            [ 0   1   --       --      ]
          (%o2)             [         37       37      ]
                            [                          ]
                            [              37 bb - 119 ]
                            [ 0   0    1   ----------- ]
                            [              37 aa - 313 ]

Function: eigenvalues (<M>)
Function: eivals (<M>)

     Returns a list of two lists containing the eigenvalues of the
     matrix <M>.  The first sublist of the return value is the list of
     eigenvalues of the matrix, and the second sublist is the list of
     the multiplicities of the eigenvalues in the corresponding order.

     `eivals' is a synonym for `eigenvalues'.

     `eigenvalues' calls the function `solve'  to find the roots of the
     characteristic polynomial of the matrix.  Sometimes `solve' may
     not be able to find the roots of the polynomial; in that case some
     other functions in this package (except `innerproduct',
     `unitvector',

     `columnvector'  and `gramschmidt' ) will not work.

     In some cases the eigenvalues found by `solve' may be complicated
     expressions.  (This may happen when `solve' returns a
     not-so-obviously real expression for an eigenvalue which is known
     to be real.)  It may be possible to simplify the eigenvalues using
     some other functions.

     The package `eigen.mac' is loaded automatically when `eigenvalues'
     or `eigenvectors'  is referenced.  If `eigen.mac' is not already
     loaded, `load ("eigen")' loads it.  After loading, all functions
     and variables in the package are available.

Function: eigenvectors (<M>)
Function: eivects (<M>)

     Computes eigenvectors of the matrix <M>.  The return value is a
     list of two elements.  The first is a list of the eigenvalues of
     <M> and a list of the multiplicities of the eigenvalues.  The
     second is a list of lists of eigenvectors.  There is one list of
     eigenvectors for each eigenvalue.  There may be one or more
     eigenvectors in each list.

     `eivects' is a synonym for `eigenvectors'.

     The package `eigen.mac' is loaded automatically when `eigenvalues'
     or `eigenvectors' is referenced.  If `eigen.mac' is not already
     loaded, `load ("eigen")' loads it.  After loading, all functions
     and variables in the package are available.

     The flags that affect this function are:

     `nondiagonalizable' is set to `true' or `false' depending on
     whether the matrix is nondiagonalizable or diagonalizable after
     `eigenvectors' returns.

     `hermitianmatrix' when `true', causes the degenerate eigenvectors
     of the Hermitian matrix to be orthogonalized using the
     Gram-Schmidt algorithm.

     `knowneigvals' when `true' causes the `eigen' package to assume
     the eigenvalues of the matrix are known to the user and stored
     under the global name `listeigvals'.  `listeigvals' should be set
     to a list similar to the output `eigenvalues'.

     The function `algsys'  is used here to solve for the eigenvectors.
     Sometimes if the eigenvalues are messy, `algsys' may not be able
     to find a solution.  In some cases, it may be possible to simplify
     the eigenvalues by first finding them using `eigenvalues' command
     and then using other functions to reduce them to something
     simpler.  Following simplification, `eigenvectors' can be called
     again with the `knowneigvals' flag set to `true'.

     See also `eigenvalues'.

     Examples:

     A matrix which has just one eigenvector per eigenvalue.

          (%i1) M1 : matrix ([11, -1], [1, 7]);
                                     [ 11  - 1 ]
          (%o1)                      [         ]
                                     [ 1    7  ]
          (%i2) [vals, vecs] : eigenvectors (M1);
          (%o2) [[[9 - sqrt(3), sqrt(3) + 9], [1, 1]],
                                  [[[1, sqrt(3) + 2]], [[1, 2 - sqrt(3)]]]]
          (%i3) for i thru length (vals[1]) do disp (val[i] = vals[1][i],
            mult[i] = vals[2][i], vec[i] = vecs[i]);
                                 val  = 9 - sqrt(3)
                                    1

                                      mult  = 1
                                          1

                              vec  = [[1, sqrt(3) + 2]]
                                 1

                                 val  = sqrt(3) + 9
                                    2

                                      mult  = 1
                                          2

                              vec  = [[1, 2 - sqrt(3)]]
                                 2

          (%o3)                         done

     A matrix which has two eigenvectors for one eigenvalue (namely 2).

          (%i1) M1 : matrix ([0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 2, 0],
                             [0, 0, 0, 2]);
                                   [ 0  1  0  0 ]
                                   [            ]
                                   [ 0  0  0  0 ]
          (%o1)                    [            ]
                                   [ 0  0  2  0 ]
                                   [            ]
                                   [ 0  0  0  2 ]
          (%i2) [vals, vecs] : eigenvectors (M1);
          (%o2) [[[0, 2], [2, 2]], [[[1, 0, 0, 0]],
                                             [[0, 0, 1, 0], [0, 0, 0, 1]]]]
          (%i3) for i thru length (vals[1]) do disp (val[i] = vals[1][i],
            mult[i] = vals[2][i], vec[i] = vecs[i]);
                                      val  = 0
                                         1

                                      mult  = 2
                                          1

                                vec  = [[1, 0, 0, 0]]
                                   1

                                      val  = 2
                                         2

                                      mult  = 2
                                          2

                         vec  = [[0, 0, 1, 0], [0, 0, 0, 1]]
                            2

          (%o3)                         done

Function: ematrix (<m>, <n>, <x>, <i>, <j>)

     Returns an <m> by <n> matrix, all elements of which are zero
     except for the `[<i>, <j>]' element which is <x>.

Function: entermatrix (<m>, <n>)

     Returns an <m> by <n> matrix, reading the elements interactively.

     If <n> is equal to <m>, Maxima prompts for the type of the matrix
     (diagonal, symmetric, antisymmetric, or general) and for each
     element.  Each response is terminated by a semicolon `;' or dollar
     sign `$'.

     If <n> is not equal to <m>, Maxima prompts for each element.

     The elements may be any expressions, which are evaluated.
     `entermatrix' evaluates its arguments.

          (%i1) n: 3$
          (%i2) m: entermatrix (n, n)$

          Is the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric
          4. General
          Answer 1, 2, 3 or 4 :
          1$
          Row 1 Column 1:
          (a+b)^n$
          Row 2 Column 2:
          (a+b)^(n+1)$
          Row 3 Column 3:
          (a+b)^(n+2)$

          Matrix entered.
          (%i3) m;
                          [        3                     ]
                          [ (b + a)      0         0     ]
                          [                              ]
          (%o3)           [                  4           ]
                          [    0      (b + a)      0     ]
                          [                              ]
                          [                            5 ]
                          [    0         0      (b + a)  ]

Function: genmatrix (<a>, <i_2>, <j_2>, <i_1>, <j_1>)
Function: genmatrix (<a>, <i_2>, <j_2>, <i_1>)
Function: genmatrix (<a>, <i_2>, <j_2>)

     Returns a matrix generated from <a>, taking element `<a>[<i_1>,
     <j_1>]' as the upper-left element and `<a>[<i_2>, <j_2>]' as the
     lower-right element of the matrix.  Here <a> is a declared array
     (created by `array' but not by `make_array' ) or an undeclared
     array, or an array function, or a lambda expression of two
     arguments.  (An array function is created like other functions
     with `:='  or `define',  but arguments are enclosed in square
     brackets instead of parentheses.)

     If <j_1> is omitted, it is assumed equal to <i_1>.  If both <j_1>
     and <i_1> are omitted, both are assumed equal to 1.

     If a selected element `i,j' of the array is undefined, the matrix
     will contain a symbolic element `<a>[i,j]'.

     Examples:

          (%i1) h [i, j] := 1 / (i + j - 1);
                                              1
          (%o1)                  h     := ---------
                                  i, j    i + j - 1
          (%i2) genmatrix (h, 3, 3);
                                     [    1  1 ]
                                     [ 1  -  - ]
                                     [    2  3 ]
                                     [         ]
                                     [ 1  1  1 ]
          (%o2)                      [ -  -  - ]
                                     [ 2  3  4 ]
                                     [         ]
                                     [ 1  1  1 ]
                                     [ -  -  - ]
                                     [ 3  4  5 ]
          (%i3) array (a, fixnum, 2, 2);
          (%o3)                           a
          (%i4) a [1, 1] : %e;
          (%o4)                          %e
          (%i5) a [2, 2] : %pi;
          (%o5)                          %pi
          (%i6) genmatrix (a, 2, 2);
                                     [ %e   0  ]
          (%o6)                      [         ]
                                     [ 0   %pi ]
          (%i7) genmatrix (lambda ([i, j], j - i), 3, 3);
                                   [  0    1   2 ]
                                   [             ]
          (%o7)                    [ - 1   0   1 ]
                                   [             ]
                                   [ - 2  - 1  0 ]
          (%i8) genmatrix (B, 2, 2);
                                  [ B      B     ]
                                  [  1, 1   1, 2 ]
          (%o8)                   [              ]
                                  [ B      B     ]
                                  [  2, 1   2, 2 ]

Function: gramschmidt (<x>)
Function: gramschmidt (<x>, <F>)

     Carries out the Gram-Schmidt orthogonalization algorithm on <x>,
     which is either a matrix or a list of lists.  <x> is not modified
     by `gramschmidt'.  The inner product employed by `gramschmidt' is
     <F>, if present, otherwise the inner product is the function
     `innerproduct'.

     If <x> is a matrix, the algorithm is applied to the rows of <x>.
     If <x> is a list of lists, the algorithm is applied to the
     sublists, which must have equal numbers of elements.  In either
     case, the return value is a list of lists, the sublists of which
     are orthogonal and span the same space as <x>.  If the dimension
     of the span of <x> is less than the number of rows or sublists,
     some sublists of the return value are zero.

     `factor'  is called at each stage of the algorithm to simplify
     intermediate results.  As a consequence, the return value may
     contain factored integers.

     `load(eigen)' loads this function.

     Example:

     Gram-Schmidt algorithm using default inner product function.

          (%i1) load (eigen)$
          (%i2) x: matrix ([1, 2, 3], [9, 18, 30], [12, 48, 60]);
                                   [ 1   2   3  ]
                                   [            ]
          (%o2)                    [ 9   18  30 ]
                                   [            ]
                                   [ 12  48  60 ]
          (%i3) y: gramschmidt (x);
                                 2      2            4     3
                                3      3   3 5      2  3  2  3
          (%o3)  [[1, 2, 3], [- ---, - --, ---], [- ----, ----, 0]]
                                2 7    7   2 7       5     5
          (%i4) map (innerproduct, [y[1], y[2], y[3]], [y[2], y[3], y[1]]);
          (%o4)                       [0, 0, 0]

     Gram-Schmidt algorithm using a specified inner product function.

          (%i1) load (eigen)$
          (%i2) ip (f, g) := integrate (f * g, u, a, b);
          (%o2)          ip(f, g) := integrate(f g, u, a, b)
          (%i3) y : gramschmidt([1, sin(u), cos(u)], ip), a= -%pi/2, b=%pi/2;
                                         %pi cos(u) - 2
          (%o3)              [1, sin(u), --------------]
                                              %pi
          (%i4) map (ip, [y[1], y[2], y[3]], [y[2], y[3], y[1]]),
                                                          a= -%pi/2, b=%pi/2;
          (%o4)                       [0, 0, 0]

Function: ident (<n>)

     Returns an <n> by <n> identity matrix.

Function: innerproduct (<x>, <y>)
Function: inprod (<x>, <y>)

     Returns the inner product (also called the scalar product or dot
     product) of <x> and <y>, which are lists of equal length, or both
     1-column or 1-row matrices of equal length.  The return value is
     `conjugate (x) . y', where `.' is the noncommutative
     multiplication operator.

     `load ("eigen")' loads this function.

     `inprod' is a synonym for `innerproduct'.

Function: invert (<M>)

     Returns the inverse of the matrix <M>.  The inverse is computed by
     the adjoint method.

     This allows a user to compute the inverse of a matrix with bfloat
     entries or polynomials with floating point coefficients without
     converting to cre-form.

     Cofactors are computed by the `determinant'  function, so if
     `ratmx'  is `false' the inverse is computed without changing the
     representation of the elements.

     The current implementation is inefficient for matrices of high
     order.

     When `detout'  is `true', the determinant is factored out of the
     inverse.

     The elements of the inverse are not automatically expanded.  If
     <M> has polynomial elements, better appearing output can be
     generated by `expand (invert (m)), detout'.  If it is desirable to
     then divide through by the determinant this can be accomplished by
     `xthru (%)' or alternatively from scratch by

          expand (adjoint (m)) / expand (determinant (m))
          invert (m) := adjoint (m) / determinant (m)

     See `^^' (noncommutative exponent) for another method of inverting
     a matrix.

Function: list_matrix_entries (<M>)

     Returns a list containing the elements of the matrix <M>.

     Example:

          (%i1) list_matrix_entries(matrix([a,b],[c,d]));
          (%o1)                     [a, b, c, d]

Option variable: lmxchar

     Default value: `['

     `lmxchar' is the character displayed as the left delimiter of a
     matrix.  See also `rmxchar' .

     Example:

          (%i1) lmxchar: "|"$
          (%i2) matrix ([a, b, c], [d, e, f], [g, h, i]);
                                     | a  b  c ]
                                     |         ]
          (%o2)                      | d  e  f ]
                                     |         ]
                                     | g  h  i ]

Function: matrix (<row_1>, ..., <row_n>)

     Returns a rectangular matrix which has the rows <row_1>, ...,
     <row_n>.  Each row is a list of expressions.  All rows must be the
     same length.

     The operations `+' (addition), `-' (subtraction), `*'
     (multiplication), and `/' (division), are carried out element by
     element when the operands are two matrices, a scalar and a matrix,
     or a matrix and a scalar.  The operation `^' (exponentiation,
     equivalently `**') is carried out element by element if the
     operands are a scalar and a matrix or a matrix and a scalar, but
     not if the operands are two matrices.  All operations are normally
     carried out in full, including `.' (noncommutative multiplication).

     Matrix multiplication is represented by the noncommutative
     multiplication operator `.'.  The corresponding noncommutative
     exponentiation operator is `^^'.  For a matrix `<A>', `<A>.<A> =
     <A>^^2' and `<A>^^-1' is the inverse of <A>, if it exists.

     There are switches for controlling simplification of expressions
     involving dot and matrix-list operations.  These are `doallmxops',
     `domxexpt',  `domxmxops',

     `doscmxops',  and `doscmxplus'.

     There are additional options which are related to matrices.  These
     are: `lmxchar',  `rmxchar',  `ratmx',

     `listarith',  `detout',  `scalarmatrix' and `sparse'.

     There are a number of functions which take matrices as arguments
     or yield matrices as return values.  See `eigenvalues',
     `eigenvectors',  `determinant',

     `charpoly',  `genmatrix',  `addcol',

     `addrow',  `copymatrix',  `transpose',

     `echelon',  and `rank'.

     Examples:

        * Construction of matrices from lists.

          (%i1) x: matrix ([17, 3], [-8, 11]);
                                     [ 17   3  ]
          (%o1)                      [         ]
                                     [ - 8  11 ]
          (%i2) y: matrix ([%pi, %e], [a, b]);
                                     [ %pi  %e ]
          (%o2)                      [         ]
                                     [  a   b  ]

        * Addition, element by element.

          (%i3) x + y;
                                [ %pi + 17  %e + 3 ]
          (%o3)                 [                  ]
                                [  a - 8    b + 11 ]

        * Subtraction, element by element.

          (%i4) x - y;
                                [ 17 - %pi  3 - %e ]
          (%o4)                 [                  ]
                                [ - a - 8   11 - b ]

        * Multiplication, element by element.

          (%i5) x * y;
                                  [ 17 %pi  3 %e ]
          (%o5)                   [              ]
                                  [ - 8 a   11 b ]

        * Division, element by element.

          (%i6) x / y;
                                  [ 17       - 1 ]
                                  [ ---  3 %e    ]
                                  [ %pi          ]
          (%o6)                   [              ]
                                  [   8    11    ]
                                  [ - -    --    ]
                                  [   a    b     ]

        * Matrix to a scalar exponent, element by element.

          (%i7) x ^ 3;
                                   [ 4913    27  ]
          (%o7)                    [             ]
                                   [ - 512  1331 ]

        * Scalar base to a matrix exponent, element by element.

          (%i8) exp(y);
                                   [   %pi    %e ]
                                   [ %e     %e   ]
          (%o8)                    [             ]
                                   [    a     b  ]
                                   [  %e    %e   ]

        * Matrix base to a matrix exponent.  This is not carried out
          element by element.

          (%i9) x ^ y;
                                          [ %pi  %e ]
                                          [         ]
                                          [  a   b  ]
                               [ 17   3  ]
          (%o9)                [         ]
                               [ - 8  11 ]

        * Noncommutative matrix multiplication.

          (%i10) x . y;
                            [ 3 a + 17 %pi  3 b + 17 %e ]
          (%o10)            [                           ]
                            [ 11 a - 8 %pi  11 b - 8 %e ]
          (%i11) y . x;
                          [ 17 %pi - 8 %e  3 %pi + 11 %e ]
          (%o11)          [                              ]
                          [  17 a - 8 b     11 b + 3 a   ]

        * Noncommutative matrix exponentiation.  A scalar base <b> to a
          matrix power <M> is carried out element by element and so
          `b^^m' is the same as `b^m'.

          (%i12) x ^^ 3;
                                  [  3833   1719 ]
          (%o12)                  [              ]
                                  [ - 4584  395  ]
          (%i13) %e ^^ y;
                                   [   %pi    %e ]
                                   [ %e     %e   ]
          (%o13)                   [             ]
                                   [    a     b  ]
                                   [  %e    %e   ]

        * A matrix raised to a -1 exponent with noncommutative
          exponentiation is the matrix inverse, if it exists.

          (%i14) x ^^ -1;
                                   [ 11      3  ]
                                   [ ---  - --- ]
                                   [ 211    211 ]
          (%o14)                   [            ]
                                   [  8    17   ]
                                   [ ---   ---  ]
                                   [ 211   211  ]
          (%i15) x . (x ^^ -1);
                                      [ 1  0 ]
          (%o15)                      [      ]
                                      [ 0  1 ]

Function: matrixmap (<f>, <M>)

     Returns a matrix with element `i,j' equal to `<f>(<M>[i,j])'.

     See also `map',  `fullmap',  `fullmapl',  and `apply'.

Function: matrixp (<expr>)

     Returns `true' if <expr> is a matrix, otherwise `false'.

Option variable: matrix_element_add

     Default value: `+'

     `matrix_element_add' is the operation invoked in place of addition
     in a matrix multiplication.  `matrix_element_add' can be assigned
     any n-ary operator (that is, a function which handles any number
     of arguments).  The assigned value may be the name of an operator
     enclosed in quote marks, the name of a function, or a lambda
     expression.

     See also `matrix_element_mult'  and `matrix_element_transpose'.

     Example:

          (%i1) matrix_element_add: "*"$
          (%i2) matrix_element_mult: "^"$
          (%i3) aa: matrix ([a, b, c], [d, e, f]);
                                     [ a  b  c ]
          (%o3)                      [         ]
                                     [ d  e  f ]
          (%i4) bb: matrix ([u, v, w], [x, y, z]);
                                     [ u  v  w ]
          (%o4)                      [         ]
                                     [ x  y  z ]
          (%i5) aa . transpose (bb);
                               [  u  v  w   x  y  z ]
                               [ a  b  c   a  b  c  ]
          (%o5)                [                    ]
                               [  u  v  w   x  y  z ]
                               [ d  e  f   d  e  f  ]

Option variable: matrix_element_mult

     Default value: `*'

     `matrix_element_mult' is the operation invoked in place of
     multiplication in a matrix multiplication.  `matrix_element_mult'
     can be assigned any binary operator.  The assigned value may be
     the name of an operator enclosed in quote marks, the name of a
     function, or a lambda expression.

     The dot operator `.' is a useful choice in some contexts.

     See also `matrix_element_add'  and `matrix_element_transpose'.

     Example:

          (%i1) matrix_element_add: lambda ([[x]], sqrt (apply ("+", x)))$
          (%i2) matrix_element_mult: lambda ([x, y], (x - y)^2)$
          (%i3) [a, b, c] . [x, y, z];
                                    2          2          2
          (%o3)         sqrt((c - z)  + (b - y)  + (a - x) )
          (%i4) aa: matrix ([a, b, c], [d, e, f]);
                                     [ a  b  c ]
          (%o4)                      [         ]
                                     [ d  e  f ]
          (%i5) bb: matrix ([u, v, w], [x, y, z]);
                                     [ u  v  w ]
          (%o5)                      [         ]
                                     [ x  y  z ]
          (%i6) aa . transpose (bb);
                         [             2          2          2  ]
                         [ sqrt((c - w)  + (b - v)  + (a - u) ) ]
          (%o6)  Col 1 = [                                      ]
                         [             2          2          2  ]
                         [ sqrt((f - w)  + (e - v)  + (d - u) ) ]

                                   [             2          2          2  ]
                                   [ sqrt((c - z)  + (b - y)  + (a - x) ) ]
                           Col 2 = [                                      ]
                                   [             2          2          2  ]
                                   [ sqrt((f - z)  + (e - y)  + (d - x) ) ]

Option variable: matrix_element_transpose

     Default value: `false'

     `matrix_element_transpose' is the operation applied to each
     element of a matrix when it is transposed.  `matrix_element_mult'
     can be assigned any unary operator.  The assigned value may be the
     name of an operator enclosed in quote marks, the name of a
     function, or a lambda expression.

     When `matrix_element_transpose' equals `transpose',

     the `transpose' function is applied to every element.  When
     `matrix_element_transpose' equals `nonscalars', the `transpose'
     function is applied to every nonscalar element.  If some element
     is an atom, the `nonscalars' option applies `transpose' only if
     the atom is declared nonscalar, while the `transpose' option
     always applies `transpose'.

     The default value, `false', means no operation is applied.

     See also `matrix_element_add'  and `matrix_element_mult'.

     Examples:

          (%i1) declare (a, nonscalar)$
          (%i2) transpose ([a, b]);
                                  [ transpose(a) ]
          (%o2)                   [              ]
                                  [      b       ]
          (%i3) matrix_element_transpose: nonscalars$
          (%i4) transpose ([a, b]);
                                  [ transpose(a) ]
          (%o4)                   [              ]
                                  [      b       ]
          (%i5) matrix_element_transpose: transpose$
          (%i6) transpose ([a, b]);
                                  [ transpose(a) ]
          (%o6)                   [              ]
                                  [ transpose(b) ]
          (%i7) matrix_element_transpose: lambda ([x], realpart(x)
                - %i*imagpart(x))$
          (%i8) m: matrix ([1 + 5*%i, 3 - 2*%i], [7*%i, 11]);
                               [ 5 %i + 1  3 - 2 %i ]
          (%o8)                [                    ]
                               [   7 %i       11    ]
          (%i9) transpose (m);
                                [ 1 - 5 %i  - 7 %i ]
          (%o9)                 [                  ]
                                [ 2 %i + 3    11   ]

Function: mattrace (<M>)

     Returns the trace (that is, the sum of the elements on the main
     diagonal) of the square matrix <M>.

     `mattrace' is called by `ncharpoly',  an alternative to Maxima's
     `charpoly'.

     `load ("nchrpl")' loads this function.

Function: minor (<M>, <i>, <j>)

     Returns the <i>, <j> minor of the matrix <M>.  That is, <M> with
     row <i> and column <j> removed.

Function: ncharpoly (<M>, <x>)

     Returns the characteristic polynomial of the matrix <M> with
     respect to <x>.  This is an alternative to Maxima's `charpoly'.

     `ncharpoly' works by computing traces of powers of the given
     matrix, which are known to be equal to sums of powers of the roots
     of the characteristic polynomial.  From these quantities the
     symmetric functions of the roots can be calculated, which are
     nothing more than the coefficients of the characteristic
     polynomial.  `charpoly' works by forming the determinant of `<x> *
     ident [n] - a'.  Thus `ncharpoly' wins, for example, in the case
     of large dense matrices filled with integers, since it avoids
     polynomial arithmetic altogether.

     `load ("nchrpl")' loads this file.

Function: newdet (<M>)

     Computes the determinant of the matrix <M> by the
     Johnson-Gentleman tree minor algorithm.  `newdet' returns the
     result in CRE form.

Function: permanent (<M>)

     Computes the permanent of the matrix <M> by the Johnson-Gentleman
     tree minor algorithm.  A permanent is like a determinant but with
     no sign changes.  `permanent' returns the result in CRE form.

     See also `newdet'.

Function: rank (<M>)

     Computes the rank of the matrix <M>.  That is, the order of the
     largest non-singular subdeterminant of <M>.

     <rank> may return the wrong answer if it cannot determine that a
     matrix element that is equivalent to zero is indeed so.

Option variable: ratmx

     Default value: `false'

     When `ratmx' is `false', determinant and matrix addition,
     subtraction, and multiplication are performed in the
     representation of the matrix elements and cause the result of
     matrix inversion to be left in general representation.

     When `ratmx' is `true', the 4 operations mentioned above are
     performed in CRE form and the result of matrix inverse is in CRE
     form.  Note that this may cause the elements to be expanded
     (depending on the setting of `ratfac' ) which might not always be
     desired.

Function: row (<M>, <i>)

     Returns the <i>'th row of the matrix <M>.  The return value is a
     matrix.

Option variable: rmxchar

     Default value: `]'

     `rmxchar' is the character drawn on the right-hand side of a
     matrix.

     See also `lmxchar'.

Option variable: scalarmatrixp

     Default value: `true'

     When `scalarmatrixp' is `true', then whenever a 1 x 1 matrix is
     produced as a result of computing the dot product of matrices it
     is simplified to a scalar, namely the sole element of the matrix.

     When `scalarmatrixp' is `all', then all 1 x 1 matrices are
     simplified to scalars.

     When `scalarmatrixp' is `false', 1 x 1 matrices are not simplified
     to scalars.

Function: scalefactors (<coordinatetransform>)

     Here the argument <coordinatetransform> evaluates to the form
     `[[expression1, expression2, ...], indeterminate1, indeterminat2,
     ...]', where the variables <indeterminate1>, <indeterminate2>,
     etc. are the curvilinear coordinate variables and where a set of
     rectangular Cartesian components is given in terms of the
     curvilinear coordinates by `[expression1, expression2, ...]'.
     `coordinates' is set to the vector `[indeterminate1,
     indeterminate2,...]', and `dimension' is set to the length of this
     vector.  SF[1], SF[2], ..., SF[DIMENSION] are set to the
     coordinate scale factors, and `sfprod' is set to the product of
     these scale factors.  Initially, `coordinates' is `[X, Y, Z]',
     `dimension' is 3, and SF[1]=SF[2]=SF[3]=SFPROD=1, corresponding to
     3-dimensional rectangular Cartesian coordinates.  To expand an
     expression into physical components in the current coordinate
     system, there is a function with usage of the form

Function: setelmx (<x>, <i>, <j>, <M>)

     Assigns <x> to the (<i>, <j>)'th element of the matrix <M>, and
     returns the altered matrix.

     `<M> [<i>, <j>]: <x>' has the same effect, but returns <x> instead
     of <M>.

Function: similaritytransform (<M>)
Function: simtran (<M>)

     `similaritytransform' computes a similarity transform of the matrix
     `M'.  It returns a list which is the output of the
     `uniteigenvectors' command.  In addition if the flag
     `nondiagonalizable' is `false' two global matrices `leftmatrix'
     and `rightmatrix' are computed.  These matrices have the property
     that `leftmatrix . <M> . rightmatrix' is a diagonal matrix with
     the eigenvalues of <M> on the diagonal.  If `nondiagonalizable' is
     `true' the left and right matrices are not computed.

     If the flag `hermitianmatrix' is `true' then `leftmatrix' is the
     complex conjugate of the transpose of `rightmatrix'.  Otherwise
     `leftmatrix' is the inverse of `rightmatrix'.

     `rightmatrix' is the matrix the columns of which are the unit
     eigenvectors of <M>.  The other flags (see `eigenvalues' and
     `eigenvectors') have the same effects since `similaritytransform'
     calls the other functions in the package in order to be able to
     form `rightmatrix'.

     `load ("eigen")' loads this function.

     `simtran' is a synonym for `similaritytransform'.

Option variable: sparse

     Default value: `false'

     When `sparse' is `true', and if `ratmx' is `true', then
     `determinant' will use special routines for computing sparse
     determinants.

Function: submatrix (<i_1>, ..., <i_m>, <M>, <j_1>, ..., <j_n>)
Function: submatrix (<i_1>, ..., <i_m>, <M>)
Function: submatrix (<M>, <j_1>, ..., <j_n>)

     Returns a new matrix composed of the matrix <M> with rows <i_1>,
     ..., <i_m> deleted, and columns <j_1>, ..., <j_n> deleted.

Function: transpose (<M>)

     Returns the transpose of <M>.

     If <M> is a matrix, the return value is another matrix <N> such
     that `N[i,j] = M[j,i]'.

     If <M> is a list, the return value is a matrix <N> of `length (m)'
     rows and 1 column, such that `N[i,1] = M[i]'.

     Otherwise <M> is a symbol, and the return value is a noun
     expression `'transpose (<M>)'.

Function: triangularize (<M>)

     Returns the upper triangular form of the matrix `M', as produced
     by Gaussian elimination.  The return value is the same as
     `echelon', except that the leading nonzero coefficient in each row
     is not normalized to 1.

     `lu_factor' and `cholesky' are other functions which yield
     triangularized matrices.

          (%i1) M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]);
                                 [  3   7  aa  bb ]
                                 [                ]
          (%o1)                  [ - 1  8  5   2  ]
                                 [                ]
                                 [  9   2  11  4  ]
          (%i2) triangularize (M);
                       [ - 1   8         5            2      ]
                       [                                     ]
          (%o2)        [  0   - 74     - 56         - 22     ]
                       [                                     ]
                       [  0    0    626 - 74 aa  238 - 74 bb ]

Function: uniteigenvectors (<M>)
Function: ueivects (<M>)

     Computes unit eigenvectors of the matrix <M>.  The return value is
     a list of lists, the first sublist of which is the output of the
     `eigenvalues' command, and the other sublists of which are the
     unit eigenvectors of the matrix corresponding to those eigenvalues
     respectively.

     The flags mentioned in the description of the `eigenvectors'
     command have the same effects in this one as well.

     When `knowneigvects' is `true', the `eigen' package assumes that
     the eigenvectors of the matrix are known to the user and are
     stored under the global name `listeigvects'.  `listeigvects' should
     be set to a list similar to the output of the `eigenvectors'
     command.

     If `knowneigvects' is set to `true' and the list of eigenvectors is
     given the setting of the flag `nondiagonalizable' may not be
     correct.  If that is the case please set it to the correct value.
     The author assumes that the user knows what he is doing and will
     not try to diagonalize a matrix the eigenvectors of which do not
     span the vector space of the appropriate dimension.

     `load ("eigen")' loads this function.

     `ueivects' is a synonym for `uniteigenvectors'.

Function: unitvector (<x>)
Function: uvect (<x>)

     Returns <x>/norm(<x>); this is a unit vector in the same direction
     as <x>.

     `load ("eigen")' loads this function.

     `uvect' is a synonym for `unitvector'.

Function: vectorpotential (<givencurl>)

     Returns the vector potential of a given curl vector, in the
     current coordinate system.  `potentialzeroloc' has a similar role
     as for `potential', but the order of the left-hand sides of the
     equations must be a cyclic permutation of the coordinate variables.

Function: vectorsimp (<expr>)

     Applies simplifications and expansions according to the following
     global flags:

     `expandall', `expanddot', `expanddotplus', `expandcross', `expandcrossplus',
     `expandcrosscross', `expandgrad', `expandgradplus', `expandgradprod',
     `expanddiv', `expanddivplus', `expanddivprod', `expandcurl', `expandcurlplus',
     `expandcurlcurl', `expandlaplacian', `expandlaplacianplus',
     and `expandlaplacianprod'.

     All these flags have default value `false'.  The `plus' suffix
     refers to employing additivity or distributivity.  The `prod'
     suffix refers to the expansion for an operand that is any kind of
     product.

    `expandcrosscross'
          Simplifies p ~ (q ~ r) to (p . r)*q - (p . q)*r.

    `expandcurlcurl'
          Simplifies curl curl p to grad div p + div grad p.

    `expandlaplaciantodivgrad'
          Simplifies laplacian p to div grad p.

    `expandcross'
          Enables `expandcrossplus' and `expandcrosscross'.

    `expandplus'
          Enables `expanddotplus', `expandcrossplus', `expandgradplus',
          `expanddivplus', `expandcurlplus', and `expandlaplacianplus'.

    `expandprod'
          Enables `expandgradprod', `expanddivprod', and
          `expandlaplacianprod'.

     These flags have all been declared `evflag'.

Option variable: vect_cross

     Default value: `false'

     When `vect_cross' is `true', it allows DIFF(X~Y,T) to work where ~
     is defined in SHARE;VECT (where VECT_CROSS is set to `true',
     anyway.)

Function: zeromatrix (<m>, <n>)

     Returns an <m> by <n> matrix, all elements of which are zero.

24 Affine


Introduction to Affine
Functions and Variables for Affine

24.1 Introduction to Affine

`affine' is a package to work with groups of polynomials.

24.2 Functions and Variables for Affine

Function: fast_linsolve ([<expr_1>, ..., <expr_m>], [<x_1>, ..., <x_n>])

     Solves the simultaneous linear equations <expr_1>, ..., <expr_m>
     for the variables <x_1>, ..., <x_n>.  Each <expr_i> may be an
     equation or a general expression; if given as a general
     expression, it is treated as an equation of the form `<expr_i> =
     0'.

     The return value is a list of equations of the form `[<x_1> =
     <a_1>, ..., <x_n> = <a_n>]' where <a_1>, ..., <a_n> are all free
     of <x_1>, ..., <x_n>.

     `fast_linsolve' is faster than `linsolve' for system of equations
     which are sparse.

     `load(affine)' loads this function.

Function: grobner_basis ([<expr_1>, ..., <expr_m>])

     Returns a Groebner basis for the equations <expr_1>, ..., <expr_m>.
     The function `polysimp' can then be used to simplify other
     functions relative to the equations.

          grobner_basis ([3*x^2+1, y*x])$

          polysimp (y^2*x + x^3*9 + 2) ==> -3*x + 2

     `polysimp(f)' yields 0 if and only if <f> is in the ideal
     generated by <expr_1>, ..., <expr_m>, that is, if and only if <f>
     is a polynomial combination of the elements of <expr_1>, ...,
     <expr_m>.

     `load(affine)' loads this function.

Function: set_up_dot_simplifications (<eqns>, <check_through_degree>)
Function: set_up_dot_simplifications (<eqns>)

     The <eqns> are polynomial equations in non commutative variables.
     The value of `current_variables' is the list of variables used for
     computing degrees.  The equations must be homogeneous, in order
     for the procedure to terminate.

     If you have checked overlapping simplifications in
     `dot_simplifications' above the degree of <f>, then the following
     is true: `dotsimp (<f>)' yields 0 if and only if <f> is in the
     ideal generated by the equations, i.e., if and only if <f> is a
     polynomial combination of the elements of the equations.

     The degree is that returned by `nc_degree'.   This in turn is
     influenced by the weights of individual variables.

     `load(affine)' loads this function.

Function: declare_weights (<x_1>, <w_1>, ..., <x_n>, <w_n>)

     Assigns weights <w_1>, ..., <w_n> to <x_1>, ..., <x_n>,
     respectively.  These are the weights used in computing `nc_degree'.

     `load(affine)' loads this function.

Function: nc_degree (<p>)

     Returns the degree of a noncommutative polynomial <p>.  See
     `declare_weights'.

     `load(affine)' loads this function.

Function: dotsimp (<f>)

     Returns 0 if and only if <f> is in the ideal generated by the
     equations, i.e., if and only if <f> is a polynomial combination of
     the elements of the equations.

     `load(affine)' loads this function.

Function: fast_central_elements ([<x_1>, ..., <x_n>], <n>)

     If `set_up_dot_simplifications' has been previously done, finds
     the central polynomials in the variables <x_1>, ..., <x_n> in the
     given degree, <n>.

     For example:
          set_up_dot_simplifications ([y.x + x.y], 3);
          fast_central_elements ([x, y], 2);
          [y.y, x.x];

     `load(affine)' loads this function.

Function: check_overlaps (<n>, <add_to_simps>)

     Checks the overlaps thru degree <n>, making sure that you have
     sufficient simplification rules in each degree, for `dotsimp' to
     work correctly.  This process can be speeded up if you know before
     hand what the dimension of the space of monomials is.  If it is of
     finite global dimension, then `hilbert' should be used.  If you
     don't know the monomial dimensions, do not specify a
     `rank_function'.  An optional third argument `reset', `false' says
     don't bother to query about resetting things.

     `load(affine)' loads this function.

Function: mono ([<x_1>, ..., <x_n>], <n>)

     Returns the list of independent monomials relative to the current
     dot simplifications of degree <n> in the variables <x_1>, ...,
     <x_n>.

     `load(affine)' loads this function.

Function: monomial_dimensions (<n>)

     Compute the Hilbert series through degree <n> for the current
     algebra.

     `load(affine)' loads this function.

Function: extract_linear_equations ([<p_1>, ..., <p_n>], [<m_1>, ..., <m_n>])

     Makes a list of the coefficients of the noncommutative polynomials
     <p_1>, ..., <p_n> of the noncommutative monomials <m_1>, ...,
     <m_n>.  The coefficients should be scalars.   Use
     `list_nc_monomials' to build the list of monomials.

     `load(affine)' loads this function.

Function: list_nc_monomials ([<p_1>, ..., <p_n>])
Function: list_nc_monomials (<p>)

     Returns a list of the non commutative monomials occurring in a
     polynomial <p> or a list of polynomials <p_1>, ..., <p_n>.

     `load(affine)' loads this function.

Option variable: all_dotsimp_denoms

     Default value: `false'

     When `all_dotsimp_denoms' is a list, the denominators encountered
     by `dotsimp' are appended to the list.  `all_dotsimp_denoms' may
     be initialized to an empty list `[]' before calling `dotsimp'.

     By default, denominators are not collected by `dotsimp'.

25 itensor


Introduction to itensor
Functions and Variables for itensor

25.1 Introduction to itensor

Maxima implements symbolic tensor manipulation of two distinct types:
component tensor manipulation (`ctensor' package) and indicial tensor
manipulation (`itensor' package).

   Nota bene: Please see the note on 'new tensor notation' below.

   Component tensor manipulation means that geometrical tensor objects
are represented as arrays or matrices. Tensor operations such as
contraction or covariant differentiation are carried out by actually
summing over repeated (dummy) indices with `do' statements.  That is,
one explicitly performs operations on the appropriate tensor components
stored in an array or matrix.

   Indicial tensor manipulation is implemented by representing tensors
as functions of their covariant, contravariant and derivative indices.
Tensor operations such as contraction or covariant differentiation are
performed by manipulating the indices themselves rather than the
components to which they correspond.

   These two approaches to the treatment of differential, algebraic and
analytic processes in the context of Riemannian geometry have various
advantages and disadvantages which reveal themselves only through the
particular nature and difficulty of the user's problem.  However, one
should keep in mind the following characteristics of the two
implementations:

   The representation of tensors and tensor operations explicitly in
terms of their components makes `ctensor' easy to use. Specification of
the metric and the computation of the induced tensors and invariants is
straightforward. Although all of Maxima's powerful simplification
capacity is at hand, a complex metric with intricate functional and
coordinate dependencies can easily lead to expressions whose size is
excessive and whose structure is hidden. In addition, many calculations
involve intermediate expressions which swell causing programs to
terminate before completion. Through experience, a user can avoid many
of these difficulties.

   Because of the special way in which tensors and tensor operations
are represented in terms of symbolic operations on their indices,
expressions which in the component representation would be unmanageable
can sometimes be greatly simplified by using the special routines for
symmetrical objects in `itensor'. In this way the structure of a large
expression may be more transparent. On the other hand, because of the
the special indicial representation in `itensor', in some cases the
user may find difficulty with the specification of the metric, function
definition, and the evaluation of differentiated "indexed" objects.

   The `itensor' package can carry out differentiation with respect to
an indexed variable, which allows one to use the package when dealing
with Lagrangian and Hamiltonian formalisms. As it is possible to
differentiate a field Lagrangian with respect to an (indexed) field
variable, one can use Maxima to derive the corresponding Euler-Lagrange
equations in indicial form. These equations can be translated into
component tensor (`ctensor') programs using the `ic_convert' function,
allowing us to solve the field equations in a particular coordinate
representation, or to recast the equations of motion in Hamiltonian
form. See `einhil.dem' and `bradic.dem' for two comprehensive examples.
The first, `einhil.dem', uses the Einstein-Hilbert action to derive the
Einstein field tensor in the homogeneous and isotropic case (Friedmann
equations) and the spherically symmetric, static case (Schwarzschild
solution.) The second, `bradic.dem', demonstrates how to compute the
Friedmann equations from the action of Brans-Dicke gravity theory, and
also derives the Hamiltonian associated with the theory's scalar field.

25.2 Functions and Variables for itensor

Function: dispcon (<tensor_1>, <tensor_2>, ...)
Function: dispcon (all)

     Displays the contraction properties of its arguments as were given
     to `defcon'.  `dispcon (all)' displays all the contraction
     properties which were defined.

Function: entertensor (<name>)

     is a function which, by prompting, allows one to create an indexed
     object called <name> with any number of tensorial and derivative
     indices. Either a single index or a list of indices (which may be
     null) is acceptable input (see the example under `covdiff').

Function: changename (<old>, <new>, <expr>)

     will change the name of all indexed objects called <old> to <new>
     in <expr>. <old> may be either a symbol or a list of the form
     `[<name>, <m>, <n>]' in which case only those indexed objects
     called <name> with <m> covariant and <n> contravariant indices
     will be renamed to <new>.

Function: listoftens

     Lists all tensors in a tensorial expression, complete with their
     indices. E.g.,

Function: ishow (<expr>)

     displays <expr> with the indexed objects in it shown having their
     covariant indices as subscripts and contravariant indices as
     superscripts. The derivative indices are displayed as subscripts,
     separated from the covariant indices by a comma (see the examples
     throughout this document).

Function: indices (<expr>)

     Returns a list of two elements.  The first is a list of the free
     indices in <expr> (those that occur only once). The second is the
     list of the dummy indices in <expr> (those that occur exactly
     twice) as the following example demonstrates.

Function: rename (<expr>)
Function: rename (<expr>, <count>)

     Returns an expression equivalent to <expr> but with the dummy
     indices in each term chosen from the set `[%1, %2,...]', if the
     optional second argument is omitted. Otherwise, the dummy indices
     are indexed beginning at the value of <count>.  Each dummy index
     in a product will be different. For a sum, `rename' will operate
     upon each term in the sum resetting the counter with each term. In
     this way `rename' can serve as a tensorial simplifier. In
     addition, the indices will be sorted alphanumerically (if `allsym'
     is `true') with respect to covariant or contravariant indices
     depending upon the value of `flipflag'.  If `flipflag' is `false'
     then the indices will be renamed according to the order of the
     contravariant indices. If `flipflag' is `true' the renaming will
     occur according to the order of the covariant indices. It often
     happens that the combined effect of the two renamings will reduce
     an expression more than either one by itself.

Function: show (<expr>)

     Displays `expr' with the indexed objects in it shown having
     covariant indices as subscripts, contravariant indices as
     superscripts.  The derivative indices are displayed as subscripts,
     separated from the covariant indices by a comma.

Option variable: flipflag

     Default: `false'. If `false' then the indices will be renamed
     according to the order of the contravariant indices, otherwise
     according to the order of the covariant indices.

     If `flipflag' is `false' then `rename' forms a list of the
     contravariant indices as they are encountered from left to right
     (if `true' then of the covariant indices). The first dummy index
     in the list is renamed to `%1', the next to `%2', etc.  Then
     sorting occurs after the `rename'-ing (see the example under
     `rename').

Function: defcon (<tensor_1>)
Function: defcon (<tensor_1>, <tensor_2>, <tensor_3>)

     gives <tensor_1> the property that the contraction of a product of
     <tensor_1> and <tensor_2> results in <tensor_3> with the
     appropriate indices.  If only one argument, <tensor_1>, is given,
     then the contraction of the product of <tensor_1> with any indexed
     object having the appropriate indices (say `my_tensor') will yield
     an indexed object with that name, i.e. `my_tensor', and with a new
     set of indices reflecting the contractions performed.      For
     example, if `imetric:g', then `defcon(g)' will implement the
     raising and lowering of indices through contraction with the metric
     tensor.      More than one `defcon' can be given for the same
     indexed object; the latest one given which applies in a particular
     contraction will be used.  `contractions' is a list of those
     indexed objects which have been given contraction properties with
     `defcon'.

Function: remcon (<tensor_1>, ..., <tensor_n>)
Function: remcon (all)

     removes all the contraction properties from the <tensor_1>, ...,
     <tensor_n>). `remcon(all)' removes all contraction properties from
     all indexed objects.

Function: contract (<expr>)

     Carries out the tensorial contractions in <expr> which may be any
     combination of sums and products. This function uses the
     information given to the `defcon' function. For best results,
     `expr' should be fully expanded. `ratexpand' is the fastest way to
     expand products and powers of sums if there are no variables in
     the denominators of the terms. The `gcd' switch should be `false'
     if GCD cancellations are unnecessary.

Function: indexed_tensor (<tensor>)

     Must be executed before assigning components to a <tensor> for
     which a built in value already exists as with `ichr1', `ichr2',
     `icurvature'. See the example under `icurvature'.

Function: components (<tensor>, <expr>)

     permits one to assign an indicial value to an expression <expr>
     giving the values of the components of <tensor>. These are
     automatically substituted for the tensor whenever it occurs with
     all of its indices. The tensor must be of the form `t([...],[...])'
     where either list may be empty. <expr> can be any indexed
     expression involving other objects with the same free indices as
     <tensor>. When used to assign values to the metric tensor wherein
     the components contain dummy indices one must be careful to define
     these indices to avoid the generation of multiple dummy indices.
     Removal of this assignment is given to the function `remcomps'.

     It is important to keep in mind that `components' cares only about
     the valence of a tensor, not about any particular index ordering.
     Thus assigning components to, say, `x([i,-j],[])', `x([-j,i],[])',
     or `x([i],[j])' all produce the same result, namely components
     being assigned to a tensor named `x' with valence `(1,1)'.

     Components can be assigned to an indexed expression in four ways,
     two of which involve the use of the `components' command:

     1) As an indexed expression. For instance:

          (%i2) components(g([],[i,j]),e([],[i])*p([],[j]))$
          (%i3) ishow(g([],[i,j]))$
                                                i  j
          (%t3)                                e  p

     2) As a matrix:

Function: remcomps (<tensor>)

     Unbinds all values from <tensor> which were assigned with the
     `components' function.

Function: showcomps (<tensor>)

     Shows component assignments of a tensor, as made using the
     `components' command. This function can be particularly useful
     when a matrix is assigned to an indicial tensor using
     `components', as demonstrated by the following example:

Function: idummy ()

     Increments `icounter' and returns as its value an index of the form
     `%n' where n is a positive integer.  This guarantees that dummy
     indices which are needed in forming expressions will not conflict
     with indices already in use (see the example under `indices').

Option variable: idummyx

     Default value: `%'

     Is the prefix for dummy indices (see the example under `indices').

Option variable: icounter

     Default value: `1'

     Determines the numerical suffix to be used in generating the next
     dummy index in the tensor package.  The prefix is determined by
     the option `idummy' (default: `%').

Function: kdelta (<L1>, <L2>)

     is the generalized Kronecker delta function defined in the
     `itensor' package with <L1> the list of covariant indices and <L2>
     the list of contravariant indices.  `kdelta([i],[j])' returns the
     ordinary Kronecker delta.  The command `ev(<expr>,kdelta)' causes
     the evaluation of an expression containing `kdelta([],[])' to the
     dimension of the manifold.

     In what amounts to an abuse of this notation, `itensor' also allows
     `kdelta' to have 2 covariant and no contravariant, or 2
     contravariant and no covariant indices, in effect providing a
     co(ntra)variant "unit matrix" capability. This is strictly
     considered a programming aid and not meant to imply that
     `kdelta([i,j],[])' is a valid tensorial object.

Function: kdels (<L1>, <L2>)

     Symmetrized Kronecker delta, used in some calculations. For
     instance:

Function: levi_civita (<L>)

     is the permutation (or Levi-Civita) tensor which yields 1 if the
     list <L> consists of an even permutation of integers, -1 if it
     consists of an odd permutation, and 0 if some indices in <L> are
     repeated.

Function: lc2kdt (<expr>)

     Simplifies expressions containing the Levi-Civita symbol,
     converting these to Kronecker-delta expressions when possible. The
     main difference between this function and simply evaluating the
     Levi-Civita symbol is that direct evaluation often results in
     Kronecker expressions containing numerical indices. This is often
     undesirable as it prevents further simplification.  The `lc2kdt'
     function avoids this problem, yielding expressions that are more
     easily simplified with `rename' or `contract'.

Function: lc_l

     Simplification rule used for expressions containing the
     unevaluated Levi-Civita symbol (`levi_civita'). Along with `lc_u',
     it can be used to simplify many expressions more efficiently than
     the evaluation of `levi_civita'.  For example:

Function: lc_u

     Simplification rule used for expressions containing the
     unevaluated Levi-Civita symbol (`levi_civita'). Along with `lc_u',
     it can be used to simplify many expressions more efficiently than
     the evaluation of `levi_civita'.  For details, see `lc_l'.

Function: canten (<expr>)

     Simplifies <expr> by renaming (see `rename') and permuting dummy
     indices. `rename' is restricted to sums of tensor products in
     which no derivatives are present. As such it is limited and should
     only be used if `canform' is not capable of carrying out the
     required simplification.

     The `canten' function returns a mathematically correct result only
     if its argument is an expression that is fully symmetric in its
     indices.  For this reason, `canten' returns an error if `allsym'
     is not set to `true'.

Function: concan (<expr>)

     Similar to `canten' but also performs index contraction.

Option variable: allsym

     Default: `false'. if `true' then all indexed objects are assumed
     symmetric in all of their covariant and contravariant indices. If
     `false' then no symmetries of any kind are assumed in these
     indices. Derivative indices are always taken to be symmetric
     unless `iframe_flag' is set to `true'.

Function: decsym (<tensor>, <m>, <n>, [<cov_1>, <cov_2>, ...], [<contr_1>, <contr_2>, ...])

     Declares symmetry properties for <tensor> of <m> covariant and <n>
     contravariant indices. The <cov_i> and <contr_i> are
     pseudofunctions expressing symmetry relations among the covariant
     and contravariant indices respectively.  These are of the form
     `symoper(<index_1>, <index_2>,...)' where `symoper' is one of
     `sym', `anti' or `cyc' and the <index_i> are integers indicating
     the position of the index in the <tensor>.  This will declare
     <tensor> to be symmetric, antisymmetric or cyclic respectively in
     the <index_i>. `symoper(all)' is also an allowable form which
     indicates all indices obey the symmetry condition. For example,
     given an object `b' with 5 covariant indices,
     `decsym(b,5,3,[sym(1,2),anti(3,4)],[cyc(all)])' declares `b'
     symmetric in its first and second and antisymmetric in its third
     and fourth covariant indices, and cyclic in all of its
     contravariant indices.  Either list of symmetry declarations may
     be null.  The function which performs the simplifications is
     `canform' as the example below illustrates.

Function: remsym (<tensor>, <m>, <n>)

     Removes all symmetry properties from <tensor> which has <m>
     covariant indices and <n> contravariant indices.

Function: canform (<expr>)
Function: canform (<expr>, <rename>)

     Simplifies <expr> by renaming dummy indices and reordering all
     indices as dictated by symmetry conditions imposed on them. If
     `allsym' is `true' then all indices are assumed symmetric,
     otherwise symmetry information provided by `decsym' declarations
     will be used. The dummy indices are renamed in the same manner as
     in the `rename' function. When `canform' is applied to a large
     expression the calculation may take a considerable amount of time.
     This time can be shortened by calling `rename' on the expression
     first.  Also see the example under `decsym'. Note: `canform' may
     not be able to reduce an expression completely to its simplest
     form although it will always return a mathematically correct
     result.

     The optional second parameter <rename>, if set to `false',
     suppresses renaming.

Function: diff (<expr>, <v_1>, [<n_1>, [<v_2>, <n_2>] ...])

     is the usual Maxima differentiation function which has been
     expanded in its abilities for `itensor'. It takes the derivative
     of <expr> with respect to <v_1> <n_1> times, with respect to <v_2>
     <n_2> times, etc. For the tensor package, the function has been
     modified so that the <v_i> may be integers from 1 up to the value
     of the variable `dim'.  This will cause the differentiation to be
     carried out with respect to the <v_i>th member of the list
     `vect_coords'.  If `vect_coords' is bound to an atomic variable,
     then that variable subscripted by <v_i> will be used for the
     variable of differentiation.  This permits an array of coordinate
     names or subscripted names like `x[1]', `x[2]', ...  to be used.

     A further extension adds the ability to `diff' to compute
     derivatives with respect to an indexed variable. In particular,
     the tensor package knows how to differentiate expressions
     containing combinations of the metric tensor and its derivatives
     with respect to the metric tensor and its first and second
     derivatives. This capability is particularly useful when
     considering Lagrangian formulations of a gravitational theory,
     allowing one to derive the Einstein tensor and field equations
     from the action principle.

Function: idiff (<expr>, <v_1>, [<n_1>, [<v_2>, <n_2>] ...])

     Indicial differentiation. Unlike `diff', which differentiates with
     respect to an independent variable, `idiff)' can be used to
     differentiate with respect to a coordinate. For an indexed object,
     this amounts to appending the <v_i> as derivative indices.
     Subsequently, derivative indices will be sorted, unless
     `iframe_flag' is set to `true'.

     `idiff' can also differentiate the determinant of the metric
     tensor. Thus, if `imetric' has been bound to `G' then
     `idiff(determinant(g),k)' will return `2 * determinant(g) *
     ichr2([%i,k],[%i])' where the dummy index `%i' is chosen
     appropriately.

Function: liediff (<v>, <ten>)

     Computes the Lie-derivative of the tensorial expression <ten> with
     respect to the vector field <v>. <ten> should be any indexed
     tensor expression; <v> should be the name (without indices) of a
     vector field. For example:

Function: rediff (<ten>)

     Evaluates all occurrences of the `idiff' command in the tensorial
     expression <ten>.

Function: undiff (<expr>)

     Returns an expression equivalent to <expr> but with all derivatives
     of indexed objects replaced by the noun form of the `idiff'
     function. Its arguments would yield that indexed object if the
     differentiation were carried out.  This is useful when it is
     desired to replace a differentiated indexed object with some
     function definition resulting in <expr> and then carry out the
     differentiation by saying `ev(<expr>, idiff)'.

Function: evundiff (<expr>)

     Equivalent to the execution of `undiff', followed by `ev' and
     `rediff'.

     The point of this operation is to easily evalute expressions that
     cannot be directly evaluated in derivative form. For instance, the
     following causes an error:

          (%i1) load(itensor);
          (%o1)      /share/tensor/itensor.lisp
          (%i2) icurvature([i,j,k],[l],m);
          Maxima encountered a Lisp error:

           Error in $ICURVATURE [or a callee]:
           $ICURVATURE [or a callee] requires less than three arguments.

          Automatically continuing.
          To reenable the Lisp debugger set *debugger-hook* to nil.

     However, if `icurvature' is entered in noun form, it can be
     evaluated using `evundiff':

          (%i3) ishow('icurvature([i,j,k],[l],m))$
                                                   l
          (%t3)                          icurvature
                                                   i j k,m
          (%i4) ishow(evundiff(%))$
                       l              l         %1           l           %1
          (%t4) - ichr2        - ichr2     ichr2      - ichr2       ichr2
                       i k,j m        %1 j      i k,m        %1 j,m      i k

                       l              l         %1           l           %1
                + ichr2        + ichr2     ichr2      + ichr2       ichr2
                       i j,k m        %1 k      i j,m        %1 k,m      i j

     Note: In earlier versions of Maxima, derivative forms of the
     Christoffel-symbols also could not be evaluated. This has been
     fixed now, so `evundiff' is no longer necessary for expressions
     like this:

          (%i5) imetric(g);
          (%o5)                                done
          (%i6) ishow(ichr2([i,j],[k],l))$
                 k %3
                g     (g         - g         + g        )
                        j %3,i l    i j,%3 l    i %3,j l
          (%t6) -----------------------------------------
                                    2

                                   k %3
                                  g     (g       - g       + g      )
                                   ,l     j %3,i    i j,%3    i %3,j
                                + -----------------------------------
                                                   2

Function: flush (<expr>, <tensor_1>, <tensor_2>, ...)

     Set to zero, in <expr>, all occurrences of the <tensor_i> that
     have no derivative indices.

Function: flushd (<expr>, <tensor_1>, <tensor_2>, ...)

     Set to zero, in <expr>, all occurrences of the <tensor_i> that
     have derivative indices.

Function: flushnd (<expr>, <tensor>, <n>)

     Set to zero, in <expr>, all occurrences of the differentiated
     object <tensor> that have <n> or more derivative indices as the
     following example demonstrates.

          (%i1) load(itensor);
          (%o1)      /share/tensor/itensor.lisp
          (%i2) ishow(a([i],[J,r],k,r)+a([i],[j,r,s],k,r,s))$
                                          J r      j r s
          (%t2)                          a      + a
                                          i,k r    i,k r s
          (%i3) ishow(flushnd(%,a,3))$
                                               J r
          (%t3)                               a
                                               i,k r

Function: coord (<tensor_1>, <tensor_2>, ...)

     Gives <tensor_i> the coordinate differentiation property that the
     derivative of contravariant vector whose name is one of the
     <tensor_i> yields a Kronecker delta. For example, if `coord(x)' has
     been done then `idiff(x([],[i]),j)' gives `kdelta([i],[j])'.
     `coord' is a list of all indexed objects having this property.

Function: remcoord (<tensor_1>, <tensor_2>, ...)
Function: remcoord (all)

     Removes the coordinate differentiation property from the `tensor_i'
     that was established by the function `coord'.  `remcoord(all)'
     removes this property from all indexed objects.

Function: makebox (<expr>)

     Display <expr> in the same manner as `show'; however, any tensor
     d'Alembertian occurring in <expr> will be indicated using the
     symbol `[]'.  For example, `[]p([m],[n])' represents
     `g([],[i,j])*p([m],[n],i,j)'.

Function: conmetderiv (<expr>, <tensor>)

     Simplifies expressions containing ordinary derivatives of both
     covariant and contravariant forms of the metric tensor (the
     current restriction).  For example, `conmetderiv' can relate the
     derivative of the contravariant metric tensor with the Christoffel
     symbols as seen from the following:

Function: simpmetderiv (<expr>)
Function: simpmetderiv (<expr>[, <stop>])

     Simplifies expressions containing products of the derivatives of
     the metric tensor. Specifically, `simpmetderiv' recognizes two
     identities:

Function: flush1deriv (<expr>, <tensor>)

     Set to zero, in `expr', all occurrences of `tensor' that have
     exactly one derivative index.

Function: imetric (<g>)
System variable: imetric

     Specifies the metric by assigning the variable `imetric:<g>' in
     addition, the contraction properties of the metric <g> are set up
     by executing the commands `defcon(<g>), defcon(<g>, <g>, kdelta)'.
     The variable `imetric' (unbound by default), is bound to the
     metric, assigned by the `imetric(<g>)' command.

Function: idim (<n>)

     Sets the dimensions of the metric. Also initializes the
     antisymmetry properties of the Levi-Civita symbols for the given
     dimension.

Function: ichr1 ([<i>, <j>, <k>])

     Yields the Christoffel symbol of the first kind via the definition
                 (g      + g      - g     )/2 .
                   ik,j     jk,i     ij,k
     To evaluate the Christoffel symbols for a particular metric, the
     variable `imetric' must be assigned a name as in the example under
     `chr2'.

Function: ichr2 ([<i>, <j>], [<k>])

     Yields the Christoffel symbol of the second kind defined by the
     relation
                                 ks
             ichr2([i,j],[k]) = g    (g      + g      - g     )/2
                                       is,j     js,i     ij,s

Function: icurvature ([<i>, <j>, <k>], [<h>])

     Yields the Riemann curvature tensor in terms of the Christoffel
     symbols of the second kind (`ichr2').  The following notation is
     used:
                      h             h            h         %1         h
            icurvature     = - ichr2      - ichr2     ichr2    + ichr2
                      i j k         i k,j        %1 j      i k        i j,k
                                      h          %1
                               + ichr2      ichr2
                                      %1 k       i j

Function: covdiff (<expr>, <v_1>, <v_2>, ...)

     Yields the covariant derivative of <expr> with respect to the
     variables <v_i> in terms of the Christoffel symbols of the second
     kind (`ichr2').  In order to evaluate these, one should use
     `ev(<expr>,ichr2)'.

Function: lorentz_gauge (<expr>)

     Imposes the Lorentz condition by substituting 0 for all indexed
     objects in <expr> that have a derivative index identical to a
     contravariant index.

Function: igeodesic_coords (<expr>, <name>)

     Causes undifferentiated Christoffel symbols and first derivatives
     of the metric tensor vanish in <expr>. The <name> in the
     `igeodesic_coords' function refers to the metric <name> (if it
     appears in <expr>) while the connection coefficients must be
     called with the names `ichr1' and/or `ichr2'. The following example
     demonstrates the verification of the cyclic identity satisfied by
     the Riemann curvature tensor using the `igeodesic_coords' function.

Function: iframes ()

     Since in this version of Maxima, contraction identities for `ifr'
     and `ifri' are always defined, as is the frame bracket (`ifb'),
     this function does nothing.

Variable: ifb

     The frame bracket. The contribution of the frame metric to the
     connection coefficients is expressed using the frame bracket:

Variable: icc1

     Connection coefficients of the first kind. In `itensor', defined as

Variable: icc2

     Connection coefficients of the second kind. In `itensor', defined
     as

Variable: ifc1

     Frame coefficient of the first kind (also known as Ricci-rotation
     coefficients.) This tensor represents the contribution of the
     frame metric to the connection coefficient of the first kind.
     Defined as:

Variable: ifc2

     Frame coefficient of the second kind. This tensor represents the
     contribution of the frame metric to the connection coefficient of
     the second kind. Defined as a permutation of the frame bracket
     (`ifb') with the appropriate indices raised and lowered as
     necessary:

Variable: ifr

     The frame field. Contracts with the inverse frame field (`ifri') to
     form the frame metric (`ifg').

Variable: ifri

     The inverse frame field. Specifies the frame base (dual basis
     vectors). Along with the frame metric, it forms the basis of all
     calculations based on frames.

Variable: ifg

     The frame metric. Defaults to `kdelta', but can be changed using
     `components'.

Variable: ifgi

     The inverse frame metric. Contracts with the frame metric (`ifg')
     to `kdelta'.

Option variable: iframe_bracket_form

     Default value: `true'

     Specifies how the frame bracket (`ifb') is computed.

Variable: inm

     The nonmetricity vector. Conformal nonmetricity is defined through
     the covariant derivative of the metric tensor. Normally zero, the
     metric tensor's covariant derivative will evaluate to the
     following when `inonmet_flag' is set to `true':

Variable: inmc1

     Covariant permutation of the nonmetricity vector components.
     Defined as

Variable: inmc2

     Contravariant permutation of the nonmetricity vector components.
     Used in the connection coefficients if `inonmet_flag' is `true'.
     Defined as:

Variable: ikt1

     Covariant permutation of the torsion tensor (also known as
     contorsion).  Defined as:

Variable: ikt2

     Contravariant permutation of the torsion tensor (also known as
     contorsion).  Defined as:

Variable: itr

     The torsion tensor. For a metric with torsion, repeated covariant
     differentiation on a scalar function will not commute, as
     demonstrated by the following example:

Operator: ~

     The wedge product operator is denoted by the tilde `~'. This is a
     binary operator. Its arguments should be expressions involving
     scalars, covariant tensors of rank one, or covariant tensors of
     rank `l' that have been declared antisymmetric in all covariant
     indices.

     The behavior of the wedge product operator is controlled by the
     `igeowedge_flag' flag, as in the following example:

          (%i1) load(itensor);
          (%o1)      /share/tensor/itensor.lisp
          (%i2) ishow(a([i])~b([j]))$
                                           a  b  - b  a
                                            i  j    i  j
          (%t2)                            -------------
                                                 2
          (%i3) decsym(a,2,0,[anti(all)],[]);
          (%o3)                                done
          (%i4) ishow(a([i,j])~b([k]))$
                                    a    b  + b  a    - a    b
                                     i j  k    i  j k    i k  j
          (%t4)                     ---------------------------
                                                 3
          (%i5) igeowedge_flag:true;
          (%o5)                                true
          (%i6) ishow(a([i])~b([j]))$
          (%t6)                            a  b  - b  a
                                            i  j    i  j
          (%i7) ishow(a([i,j])~b([k]))$
          (%t7)                     a    b  + b  a    - a    b
                                     i j  k    i  j k    i k  j

Operator: |

     The vertical bar `|' denotes the "contraction with a vector" binary
     operation. When a totally antisymmetric covariant tensor is
     contracted with a contravariant vector, the result is the same
     regardless which index was used for the contraction. Thus, it is
     possible to define the contraction operation in an index-free
     manner.

     In the `itensor' package, contraction with a vector is always
     carried out with respect to the first index in the literal sorting
     order. This ensures better simplification of expressions involving
     the `|' operator. For instance:

          (%i1) load(itensor);
          (%o1)      /share/tensor/itensor.lisp
          (%i2) decsym(a,2,0,[anti(all)],[]);
          (%o2)                                done
          (%i3) ishow(a([i,j],[])|v)$
                                              %1
          (%t3)                              v   a
                                                  %1 j
          (%i4) ishow(a([j,i],[])|v)$
                                               %1
          (%t4)                             - v   a
                                                   %1 j

     Note that it is essential that the tensors used with the `|'
     operator be declared totally antisymmetric in their covariant
     indices. Otherwise, the results will be incorrect.

Function: extdiff (<expr>, <i>)

     Computes the exterior derivative of <expr> with respect to the
     index <i>. The exterior derivative is formally defined as the wedge
     product of the partial derivative operator and a differential
     form. As such, this operation is also controlled by the setting of
     `igeowedge_flag'.  For instance:

          (%i1) load(itensor);
          (%o1)      /share/tensor/itensor.lisp
          (%i2) ishow(extdiff(v([i]),j))$
                                            v    - v
                                             j,i    i,j
          (%t2)                             -----------
                                                 2
          (%i3) decsym(a,2,0,[anti(all)],[]);
          (%o3)                                done
          (%i4) ishow(extdiff(a([i,j]),k))$
                                     a      - a      + a
                                      j k,i    i k,j    i j,k
          (%t4)                      ------------------------
                                                3
          (%i5) igeowedge_flag:true;
          (%o5)                                true
          (%i6) ishow(extdiff(v([i]),j))$
          (%t6)                             v    - v
                                             j,i    i,j
          (%i7) ishow(extdiff(a([i,j]),k))$
          (%t7)                    - (a      - a      + a     )
                                       k j,i    k i,j    j i,k

Function: hodge (<expr>)

     Compute the Hodge-dual of <expr>. For instance:

Option variable: igeowedge_flag

     Default value: `false'

     Controls the behavior of the wedge product and exterior
     derivative. When set to `false' (the default), the notion of
     differential forms will correspond with that of a totally
     antisymmetric covariant tensor field.  When set to `true',
     differential forms will agree with the notion of the volume
     element.

Function: tentex (<expr>)

     To use the `tentex' function, you must first load `tentex', as in
     the following example:

          (%i1) load(itensor);
          (%o1)      /share/tensor/itensor.lisp
          (%i2) load(tentex);
          (%o2)       /share/tensor/tentex.lisp
          (%i3) idummyx:m;
          (%o3)                                  m
          (%i4) ishow(icurvature([j,k,l],[i]))$
                      m1       i           m1       i           i
          (%t4)  ichr2    ichr2     - ichr2    ichr2     - ichr2
                      j k      m1 l        j l      m1 k        j l,k

                                                                i
                                                         + ichr2
                                                                j k,l
          (%i5) tentex(%)$
          $$\Gamma_{j\,k}^{m_1}\,\Gamma_{l\,m_1}^{i}-\Gamma_{j\,l}^{m_1}\,
           \Gamma_{k\,m_1}^{i}-\Gamma_{j\,l,k}^{i}+\Gamma_{j\,k,l}^{i}$$

     Note the use of the `idummyx' assignment, to avoid the appearance
     of the percent sign in the TeX expression, which may lead to
     compile errors.

     NB: This version of the `tentex' function is somewhat experimental.

Function: ic_convert (<eqn>)

     Converts the `itensor' equation <eqn> to a `ctensor' assignment
     statement.  Implied sums over dummy indices are made explicit
     while indexed objects are transformed into arrays (the array
     subscripts are in the order of covariant followed by contravariant
     indices of the indexed objects). The derivative of an indexed
     object will be replaced by the noun form of `diff' taken with
     respect to `ct_coords' subscripted by the derivative index. The
     Christoffel symbols `ichr1' and `ichr2' will be translated to
     `lcs' and `mcs', respectively and if `metricconvert' is `true'
     then all occurrences of the metric with two covariant
     (contravariant) indices will be renamed to `lg' (`ug'). In
     addition, `do' loops will be introduced summing over all free
     indices so that the transformed assignment statement can be
     evaluated by just doing `ev'. The following examples demonstrate
     the features of this function.

          (%i1) load(itensor);
          (%o1)      /share/tensor/itensor.lisp
          (%i2) eqn:ishow(t([i,j],[k])=f([],[])*g([l,m],[])*a([],[m],j)
                *b([i],[l,k]))$
                                       k        m   l k
          (%t2)                       t    = f a   b    g
                                       i j      ,j  i    l m
          (%i3) ic_convert(eqn);
          (%o3) for i thru dim do (for j thru dim do (
                 for k thru dim do
                  t        : f sum(sum(diff(a , ct_coords ) b
                   i, j, k                   m           j   i, l, k

           g    , l, 1, dim), m, 1, dim)))
            l, m
          (%i4) imetric(g);
          (%o4)                                done
          (%i5) metricconvert:true;
          (%o5)                                true
          (%i6) ic_convert(eqn);
          (%o6) for i thru dim do (for j thru dim do (
                 for k thru dim do
                  t        : f sum(sum(diff(a , ct_coords ) b
                   i, j, k                   m           j   i, l, k

           lg    , l, 1, dim), m, 1, dim)))
             l, m

26 ctensor


Introduction to ctensor
Functions and Variables for ctensor

26.1 Introduction to ctensor

`ctensor' is a component tensor manipulation package.  To use the
`ctensor' package, type `load(ctensor)'.  To begin an interactive
session with `ctensor', type `csetup()'.  You are first asked to
specify the dimension of the manifold. If the dimension is 2, 3 or 4
then the list of coordinates defaults to `[x,y]', `[x,y,z]' or
`[x,y,z,t]' respectively.  These names may be changed by assigning a
new list of coordinates to the variable `ct_coords' (described below)
and the user is queried about this. Care must be taken to avoid the
coordinate names conflicting with other object definitions.

   Next, the user enters the metric either directly or from a file by
specifying its ordinal position.  The metric is stored in the matrix
`lg'. Finally, the metric inverse is computed and stored in the matrix
`ug'. One has the option of carrying out all calculations in a power
series.

   A sample protocol is begun below for the static, spherically
symmetric metric (standard coordinates) which will be applied to the
problem of deriving Einstein's vacuum equations (which lead to the
Schwarzschild solution) as an example. Many of the functions in
`ctensor' will be displayed for the standard metric as examples.

     (%i1) load(ctensor);
     (%o1)      /share/tensor/ctensor.mac
     (%i2) csetup();
     Enter the dimension of the coordinate system:
     4;
     Do you wish to change the coordinate names?
     n;
     Do you want to
     1. Enter a new metric?

     2. Enter a metric from a file?

     3. Approximate a metric with a Taylor series?
     1;

     Is the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric  4. General
     Answer 1, 2, 3 or 4
     1;
     Row 1 Column 1:
     a;
     Row 2 Column 2:
     x^2;
     Row 3 Column 3:
     x^2*sin(y)^2;
     Row 4 Column 4:
     -d;

     Matrix entered.
     Enter functional dependencies with the DEPENDS function or 'N' if none
     depends([a,d],x);
     Do you wish to see the metric?
     y;
                               [ a  0       0        0  ]
                               [                        ]
                               [     2                  ]
                               [ 0  x       0        0  ]
                               [                        ]
                               [         2    2         ]
                               [ 0  0   x  sin (y)   0  ]
                               [                        ]
                               [ 0  0       0       - d ]
     (%o2)                                done
     (%i3) christof(mcs);
                                                 a
                                                  x
     (%t3)                          mcs        = ---
                                       1, 1, 1   2 a

                                                  1
     (%t4)                           mcs        = -
                                        1, 2, 2   x

                                                  1
     (%t5)                           mcs        = -
                                        1, 3, 3   x

                                                 d
                                                  x
     (%t6)                          mcs        = ---
                                       1, 4, 4   2 d

                                                   x
     (%t7)                          mcs        = - -
                                       2, 2, 1     a

                                                cos(y)
     (%t8)                         mcs        = ------
                                      2, 3, 3   sin(y)

                                                    2
                                               x sin (y)
     (%t9)                      mcs        = - ---------
                                   3, 3, 1         a

     (%t10)                   mcs        = - cos(y) sin(y)
                                 3, 3, 2

                                                 d
                                                  x
     (%t11)                         mcs        = ---
                                       4, 4, 1   2 a
     (%o11)                               done

26.2 Functions and Variables for ctensor

Function: csetup ()

     A function in the `ctensor' (component tensor) package which
     initializes the package and allows the user to enter a metric
     interactively. See `ctensor' for more details.

Function: cmetric (<dis>)
Function: cmetric ()

     A function in the `ctensor' (component tensor) package that
     computes the metric inverse and sets up the package for further
     calculations.

     If `cframe_flag' is `false', the function computes the inverse
     metric `ug' from the (user-defined) matrix `lg'. The metric
     determinant is also computed and stored in the variable `gdet'.
     Furthermore, the package determines if the metric is diagonal and
     sets the value of `diagmetric' accordingly. If the optional
     argument <dis> is present and not equal to `false', the user is
     prompted to see the metric inverse.

     If `cframe_flag' is `true', the function expects that the values of
     `fri' (the inverse frame matrix) and `lfg' (the frame metric) are
     defined. From these, the frame matrix `fr' and the inverse frame
     metric `ufg' are computed.

Function: ct_coordsys (<coordinate_system>, <extra_arg>)
Function: ct_coordsys (<coordinate_system>)

     Sets up a predefined coordinate system and metric. The argument
     <coordinate_system> can be one of the following symbols:

Function: init_ctensor ()

     Initializes the `ctensor' package.

     The `init_ctensor' function reinitializes the `ctensor' package.
     It removes all arrays and matrices used by `ctensor', resets all
     flags, resets `dim' to 4, and resets the frame metric to the
     Lorentz-frame.

Function: christof (<dis>)

     A function in the `ctensor' (component tensor) package.  It
     computes the Christoffel symbols of both kinds.  The argument
     <dis> determines which results are to be immediately displayed.
     The Christoffel symbols of the first and second kinds are stored
     in the arrays `lcs[i,j,k]' and `mcs[i,j,k]' respectively and
     defined to be symmetric in the first two indices. If the argument
     to `christof' is `lcs' or `mcs' then the unique non-zero values of
     `lcs[i,j,k]' or `mcs[i,j,k]', respectively, will be displayed. If
     the argument is `all' then the unique non-zero values of
     `lcs[i,j,k]' and `mcs[i,j,k]' will be displayed.  If the argument
     is `false' then the display of the elements will not occur. The
     array elements `mcs[i,j,k]' are defined in such a manner that the
     final index is contravariant.

Function: ricci (<dis>)

     A function in the `ctensor' (component tensor) package. `ricci'
     computes the covariant (symmetric) components `ric[i,j]' of the
     Ricci tensor.  If the argument <dis> is `true', then the non-zero
     components are displayed.

Function: uricci (<dis>)

     This function first computes the covariant components `ric[i,j]'
     of the Ricci tensor.  Then the mixed Ricci tensor is computed
     using the contravariant metric tensor.  If the value of the
     argument <dis> is `true', then these mixed components, `uric[i,j]'
     (the index `i' is covariant and the index `j' is contravariant),
     will be displayed directly.  Otherwise, `ricci(false)' will simply
     compute the entries of the array `uric[i,j]' without displaying
     the results.

Function: scurvature ()

     Returns the scalar curvature (obtained by contracting the Ricci
     tensor) of the Riemannian manifold with the given metric.

Function: einstein (<dis>)

     A function in the `ctensor' (component tensor) package.
     `einstein' computes the mixed Einstein tensor after the
     Christoffel symbols and Ricci tensor have been obtained (with the
     functions `christof' and `ricci').  If the argument <dis> is
     `true', then the non-zero values of the mixed Einstein tensor
     `ein[i,j]' will be displayed where `j' is the contravariant index.
     The variable `rateinstein' will cause the rational simplification
     on these components. If `ratfac' is `true' then the components will
     also be factored.

Function: leinstein (<dis>)

     Covariant Einstein-tensor. `leinstein' stores the values of the
     covariant Einstein tensor in the array `lein'. The covariant
     Einstein-tensor is computed from the mixed Einstein tensor `ein'
     by multiplying it with the metric tensor. If the argument <dis> is
     `true', then the non-zero values of the covariant Einstein tensor
     are displayed.

Function: riemann (<dis>)

     A function in the `ctensor' (component tensor) package.  `riemann'
     computes the Riemann curvature tensor from the given metric and
     the corresponding Christoffel symbols. The following index
     conventions are used:

                          l      _l       _l       _l   _m    _l   _m
           R[i,j,k,l] =  R    = |      - |      + |    |   - |    |
                          ijk     ij,k     ik,j     mk   ij    mj   ik

     This notation is consistent with the notation used by the `itensor'
     package and its `icurvature' function.  If the optional argument
     <dis> is `true', the unique non-zero components `riem[i,j,k,l]'
     will be displayed.  As with the Einstein tensor, various switches
     set by the user control the simplification of the components of
     the Riemann tensor.  If `ratriemann' is `true', then rational
     simplification will be done. If `ratfac' is `true' then each of
     the components will also be factored.

     If the variable `cframe_flag' is `false', the Riemann tensor is
     computed directly from the Christoffel-symbols. If `cframe_flag' is
     `true', the covariant Riemann-tensor is computed first from the
     frame field coefficients.

Function: lriemann (<dis>)

     Covariant Riemann-tensor (`lriem[]').

     Computes the covariant Riemann-tensor as the array `lriem'. If the
     argument <dis> is `true', unique non-zero values are displayed.

     If the variable `cframe_flag' is `true', the covariant Riemann
     tensor is computed directly from the frame field coefficients.
     Otherwise, the (3,1) Riemann tensor is computed first.

     For information on index ordering, see `riemann'.

Function: uriemann (<dis>)

     Computes the contravariant components of the Riemann curvature
     tensor as array elements `uriem[i,j,k,l]'.  These are displayed if
     <dis> is `true'.

Function: rinvariant ()

     Forms the Kretchmann-invariant (`kinvariant') obtained by
     contracting the tensors

          lriem[i,j,k,l]*uriem[i,j,k,l].

     This object is not automatically simplified since it can be very
     large.

Function: weyl (<dis>)

     Computes the Weyl conformal tensor.  If the argument <dis> is
     `true', the non-zero components `weyl[i,j,k,l]' will be displayed
     to the user.  Otherwise, these components will simply be computed
     and stored.  If the switch `ratweyl' is set to `true', then the
     components will be rationally simplified; if `ratfac' is `true'
     then the results will be factored as well.

Function: ctaylor ()

     The `ctaylor' function truncates its argument by converting it to
     a Taylor-series using `taylor', and then calling `ratdisrep'. This
     has the combined effect of dropping terms higher order in the
     expansion variable `ctayvar'. The order of terms that should be
     dropped is defined by `ctaypov'; the point around which the series
     expansion is carried out is specified in `ctaypt'.

     As an example, consider a simple metric that is a perturbation of
     the Minkowski metric. Without further restrictions, even a diagonal
     metric produces expressions for the Einstein tensor that are far
     too complex:

Function: frame_bracket (<fr>, <fri>, <diagframe>)

     The frame bracket (`fb[]').

     Computes the frame bracket according to the following definition:

             c          c         c        d     e
          ifb   = ( ifri    - ifri    ) ifr   ifr
             ab         d,e       e,d      a     b

Function: nptetrad ()

     Computes a Newman-Penrose null tetrad (`np') and its raised-index
     counterpart (`npi'). See `petrov' for an example.

     The null tetrad is constructed on the assumption that a
     four-dimensional orthonormal frame metric with metric signature
     (-,+,+,+) is being used.  The components of the null tetrad are
     related to the inverse frame matrix as follows:

Function: psi (<dis>)

     Computes the five Newman-Penrose coefficients `psi[0]'...`psi[4]'.
     If `dis' is set to `true', the coefficients are displayed.  See
     `petrov' for an example.

     These coefficients are computed from the Weyl-tensor in a
     coordinate base.  If a frame base is used, the Weyl-tensor is
     first converted to a coordinate base, which can be a
     computationally expensive procedure. For this reason, in some
     cases it may be more advantageous to use a coordinate base in the
     first place before the Weyl tensor is computed. Note however, that
     constructing a Newman-Penrose null tetrad requires a frame base.
     Therefore, a meaningful computation sequence may begin with a
     frame base, which is then used to compute `lg' (computed
     automatically by `cmetric' and then `ug'. At this point, you can
     switch back to a coordinate base by setting `cframe_flag' to false
     before beginning to compute the Christoffel symbols. Changing to a
     frame base at a later stage could yield inconsistent results, as
     you may end up with a mixed bag of tensors, some computed in a
     frame base, some in a coordinate base, with no means to
     distinguish between the two.

Function: petrov ()

     Computes the Petrov classification of the metric characterized by
     `psi[0]'...`psi[4]'.

     For example, the following demonstrates how to obtain the
     Petrov-classification of the Kerr metric:

          (%i1) load(ctensor);
          (%o1)       /share/tensor/ctensor.mac
          (%i2) (cframe_flag:true,gcd:spmod,ctrgsimp:true,ratfac:true);
          (%o2)                                true
          (%i3) ct_coordsys(exteriorschwarzschild,all);
          (%o3)                                done
          (%i4) ug:invert(lg)$
          (%i5) weyl(false);
          (%o5)                                done
          (%i6) nptetrad(true);
          (%t6) np =

          [ sqrt(r - 2 m)           sqrt(r)                                 ]
          [---------------   ---------------------    0            0        ]
          [sqrt(2) sqrt(r)   sqrt(2) sqrt(r - 2 m)                          ]
          [                                                                 ]
          [ sqrt(r - 2 m)            sqrt(r)                                ]
          [---------------  - ---------------------   0            0        ]
          [sqrt(2) sqrt(r)    sqrt(2) sqrt(r - 2 m)                         ]
          [                                                                 ]
          [                                          r      %i r sin(theta) ]
          [       0                    0          -------   --------------- ]
          [                                       sqrt(2)       sqrt(2)     ]
          [                                                                 ]
          [                                          r       %i r sin(theta)]
          [       0                    0          -------  - ---------------]
          [                                       sqrt(2)        sqrt(2)    ]

                                       sqrt(r)         sqrt(r - 2 m)
          (%t7) npi = matrix([- ---------------------,---------------, 0, 0],
                                sqrt(2) sqrt(r - 2 m) sqrt(2) sqrt(r)

                    sqrt(r)            sqrt(r - 2 m)
          [- ---------------------, - ---------------, 0, 0],
             sqrt(2) sqrt(r - 2 m)    sqrt(2) sqrt(r)

                     1               %i
          [0, 0, ---------, --------------------],
                 sqrt(2) r  sqrt(2) r sin(theta)

                     1                 %i
          [0, 0, ---------, - --------------------])
                 sqrt(2) r    sqrt(2) r sin(theta)

          (%o7)                                done
          (%i7) psi(true);
          (%t8)                              psi  = 0
                                                0

          (%t9)                              psi  = 0
                                                1

                                                    m
          (%t10)                             psi  = --
                                                2    3
                                                    r

          (%t11)                             psi  = 0
                                                3

          (%t12)                             psi  = 0
                                                4
          (%o12)                               done
          (%i12) petrov();
          (%o12)                                 D

     The Petrov classification function is based on the algorithm
     published in "Classifying geometries in general relativity: III
     Classification in practice" by Pollney, Skea, and d'Inverno,
     Class. Quant. Grav. 17 2885-2902 (2000).  Except for some simple
     test cases, the implementation is untested as of December 19,
     2004, and is likely to contain errors.

Function: contortion (<tr>)

     Computes the (2,1) contortion coefficients from the torsion tensor
     <tr>.

Function: nonmetricity (<nm>)

     Computes the (2,1) nonmetricity coefficients from the nonmetricity
     vector <nm>.

Function: ctransform (<M>)

     A function in the `ctensor' (component tensor) package which will
     perform a coordinate transformation upon an arbitrary square
     symmetric matrix <M>. The user must input the functions which
     define the transformation.  (Formerly called `transform'.)

Function: findde (<A>, <n>)

     returns a list of the unique differential equations (expressions)
     corresponding to the elements of the <n> dimensional square array
     <A>. Presently, <n> may be 2 or 3. `deindex' is a global list
     containing the indices of <A> corresponding to these unique
     differential equations. For the Einstein tensor (`ein'), which is
     a two dimensional array, if computed for the metric in the example
     below, `findde' gives the following independent differential
     equations:

          (%i1) load(ctensor);
          (%o1)       /share/tensor/ctensor.mac
          (%i2) derivabbrev:true;
          (%o2)                                true
          (%i3) dim:4;
          (%o3)                                  4
          (%i4) lg:matrix([a, 0, 0, 0], [ 0, x^2, 0, 0],
                                        [0, 0, x^2*sin(y)^2, 0], [0,0,0,-d]);
                                    [ a  0       0        0  ]
                                    [                        ]
                                    [     2                  ]
                                    [ 0  x       0        0  ]
          (%o4)                     [                        ]
                                    [         2    2         ]
                                    [ 0  0   x  sin (y)   0  ]
                                    [                        ]
                                    [ 0  0       0       - d ]
          (%i5) depends([a,d],x);
          (%o5)                            [a(x), d(x)]
          (%i6) ct_coords:[x,y,z,t];
          (%o6)                            [x, y, z, t]
          (%i7) cmetric();
          (%o7)                                done
          (%i8) einstein(false);
          (%o8)                                done
          (%i9) findde(ein,2);
                                                      2
          (%o9) [d  x - a d + d, 2 a d d    x - a (d )  x - a  d d  x
                  x                     x x         x        x    x

                                                        2          2
                                    + 2 a d d   - 2 a  d , a  x + a  - a]
                                             x       x      x
          (%i10) deindex;
          (%o10)                     [[1, 1], [2, 2], [4, 4]]

Function: cograd ()

     Computes the covariant gradient of a scalar function allowing the
     user to choose the corresponding vector name as the example under
     `contragrad' illustrates.

Function: contragrad ()

     Computes the contravariant gradient of a scalar function allowing
     the user to choose the corresponding vector name as the example
     below for the Schwarzschild metric illustrates:

          (%i1) load(ctensor);
          (%o1)       /share/tensor/ctensor.mac
          (%i2) derivabbrev:true;
          (%o2)                                true
          (%i3) ct_coordsys(exteriorschwarzschild,all);
          (%o3)                                done
          (%i4) depends(f,r);
          (%o4)                               [f(r)]
          (%i5) cograd(f,g1);
          (%o5)                                done
          (%i6) listarray(g1);
          (%o6)                            [0, f , 0, 0]
                                                r
          (%i7) contragrad(f,g2);
          (%o7)                                done
          (%i8) listarray(g2);
                                         f  r - 2 f  m
                                          r        r
          (%o8)                      [0, -------------, 0, 0]
                                               r

Function: dscalar ()

     computes the tensor d'Alembertian of the scalar function once
     dependencies have been declared upon the function. For example:

          (%i1) load(ctensor);
          (%o1)       /share/tensor/ctensor.mac
          (%i2) derivabbrev:true;
          (%o2)                                true
          (%i3) ct_coordsys(exteriorschwarzschild,all);
          (%o3)                                done
          (%i4) depends(p,r);
          (%o4)                               [p(r)]
          (%i5) factor(dscalar(p));
                                    2
                              p    r  - 2 m p    r + 2 p  r - 2 m p
                               r r           r r        r          r
          (%o5)               --------------------------------------
                                                 2
                                                r

Function: checkdiv ()

     computes the covariant divergence of the mixed second rank tensor
     (whose first index must be covariant) by printing the
     corresponding n components of the vector field (the divergence)
     where n = `dim'. If the argument to the function is `g' then the
     divergence of the Einstein tensor will be formed and must be zero.
     In addition, the divergence (vector) is given the array name `div'.

Function: cgeodesic (<dis>)

     A function in the `ctensor' (component tensor) package.
     `cgeodesic' computes the geodesic equations of motion for a given
     metric.  They are stored in the array `geod[i]'.  If the argument
     <dis> is `true' then these equations are displayed.

Function: bdvac (<f>)

     generates the covariant components of the vacuum field equations of
     the Brans- Dicke gravitational theory. The scalar field is
     specified by the argument <f>, which should be a (quoted) function
     name with functional dependencies, e.g., `'p(x)'.

     The components of the second rank covariant field tensor are
     represented by the array `bd'.

Function: invariant1 ()

     generates the mixed Euler- Lagrange tensor (field equations) for
     the invariant density of R^2. The field equations are the
     components of an array named `inv1'.

Function: invariant2 ()

     *** NOT YET IMPLEMENTED ***

     generates the mixed Euler- Lagrange tensor (field equations) for
     the invariant density of `ric[i,j]*uriem[i,j]'. The field
     equations are the components of an array named `inv2'.

Function: bimetric ()

     *** NOT YET IMPLEMENTED ***

     generates the field equations of Rosen's bimetric theory. The field
     equations are the components of an array named `rosen'.

Function: diagmatrixp (<M>)

     Returns `true' if <M> is a diagonal matrix or (2D) array.

Function: symmetricp (<M>)

     Returns `true' if <M> is a symmetric matrix or (2D) array.

Function: ntermst (<f>)

     gives the user a quick picture of the "size" of the doubly
     subscripted tensor (array) <f>.  It prints two element lists where
     the second element corresponds to NTERMS of the components
     specified by the first elements.  In this way, it is possible to
     quickly find the non-zero expressions and attempt simplification.

Function: cdisplay (<ten>)

     displays all the elements of the tensor <ten>, as represented by a
     multidimensional array. Tensors of rank 0 and 1, as well as other
     types of variables, are displayed as with `ldisplay'. Tensors of
     rank 2 are displayed as 2-dimensional matrices, while tensors of
     higher rank are displayed as a list of 2-dimensional matrices. For
     instance, the Riemann-tensor of the Schwarzschild metric can be
     viewed as:

          (%i1) load(ctensor);
          (%o1)       /share/tensor/ctensor.mac
          (%i2) ratfac:true;
          (%o2)                                true
          (%i3) ct_coordsys(exteriorschwarzschild,all);
          (%o3)                                done
          (%i4) riemann(false);
          (%o4)                                done
          (%i5) cdisplay(riem);
                    [ 0               0                   0           0     ]
                    [                                                       ]
                    [                              2                        ]
                    [      3 m (r - 2 m)   m    2 m                         ]
                    [ 0  - ------------- + -- - ----      0           0     ]
                    [            4          3     4                         ]
                    [           r          r     r                          ]
                    [                                                       ]
          riem    = [                                m (r - 2 m)            ]
              1, 1  [ 0               0              -----------      0     ]
                    [                                     4                 ]
                    [                                    r                  ]
                    [                                                       ]
                    [                                           m (r - 2 m) ]
                    [ 0               0                   0     ----------- ]
                    [                                                4      ]
                    [                                               r       ]

                                          [    2 m (r - 2 m)       ]
                                          [ 0  -------------  0  0 ]
                                          [          4             ]
                                          [         r              ]
                               riem     = [                        ]
                                   1, 2   [ 0        0        0  0 ]
                                          [                        ]
                                          [ 0        0        0  0 ]
                                          [                        ]
                                          [ 0        0        0  0 ]

                                          [         m (r - 2 m)    ]
                                          [ 0  0  - -----------  0 ]
                                          [              4         ]
                                          [             r          ]
                               riem     = [                        ]
                                   1, 3   [ 0  0        0        0 ]
                                          [                        ]
                                          [ 0  0        0        0 ]
                                          [                        ]
                                          [ 0  0        0        0 ]

                                          [            m (r - 2 m) ]
                                          [ 0  0  0  - ----------- ]
                                          [                 4      ]
                                          [                r       ]
                               riem     = [                        ]
                                   1, 4   [ 0  0  0        0       ]
                                          [                        ]
                                          [ 0  0  0        0       ]
                                          [                        ]
                                          [ 0  0  0        0       ]

                                         [       0         0  0  0 ]
                                         [                         ]
                                         [       2 m               ]
                                         [ - ------------  0  0  0 ]
                              riem     = [    2                    ]
                                  2, 1   [   r  (r - 2 m)          ]
                                         [                         ]
                                         [       0         0  0  0 ]
                                         [                         ]
                                         [       0         0  0  0 ]

                       [     2 m                                         ]
                       [ ------------  0        0               0        ]
                       [  2                                              ]
                       [ r  (r - 2 m)                                    ]
                       [                                                 ]
                       [      0        0        0               0        ]
                       [                                                 ]
            riem     = [                         m                       ]
                2, 2   [      0        0  - ------------        0        ]
                       [                     2                           ]
                       [                    r  (r - 2 m)                 ]
                       [                                                 ]
                       [                                         m       ]
                       [      0        0        0         - ------------ ]
                       [                                     2           ]
                       [                                    r  (r - 2 m) ]

                                          [ 0  0       0        0 ]
                                          [                       ]
                                          [            m          ]
                                          [ 0  0  ------------  0 ]
                               riem     = [        2              ]
                                   2, 3   [       r  (r - 2 m)    ]
                                          [                       ]
                                          [ 0  0       0        0 ]
                                          [                       ]
                                          [ 0  0       0        0 ]

                                          [ 0  0  0       0       ]
                                          [                       ]
                                          [               m       ]
                                          [ 0  0  0  ------------ ]
                               riem     = [           2           ]
                                   2, 4   [          r  (r - 2 m) ]
                                          [                       ]
                                          [ 0  0  0       0       ]
                                          [                       ]
                                          [ 0  0  0       0       ]

                                                [ 0  0  0  0 ]
                                                [            ]
                                                [ 0  0  0  0 ]
                                                [            ]
                                     riem     = [ m          ]
                                         3, 1   [ -  0  0  0 ]
                                                [ r          ]
                                                [            ]
                                                [ 0  0  0  0 ]

                                                [ 0  0  0  0 ]
                                                [            ]
                                                [ 0  0  0  0 ]
                                                [            ]
                                     riem     = [    m       ]
                                         3, 2   [ 0  -  0  0 ]
                                                [    r       ]
                                                [            ]
                                                [ 0  0  0  0 ]

                                         [   m                      ]
                                         [ - -   0   0       0      ]
                                         [   r                      ]
                                         [                          ]
                                         [        m                 ]
                                         [  0   - -  0       0      ]
                              riem     = [        r                 ]
                                  3, 3   [                          ]
                                         [  0    0   0       0      ]
                                         [                          ]
                                         [              2 m - r     ]
                                         [  0    0   0  ------- + 1 ]
                                         [                 r        ]

                                              [ 0  0  0    0   ]
                                              [                ]
                                              [ 0  0  0    0   ]
                                              [                ]
                                   riem     = [            2 m ]
                                       3, 4   [ 0  0  0  - --- ]
                                              [             r  ]
                                              [                ]
                                              [ 0  0  0    0   ]

                                          [       0        0  0  0 ]
                                          [                        ]
                                          [       0        0  0  0 ]
                                          [                        ]
                               riem     = [       0        0  0  0 ]
                                   4, 1   [                        ]
                                          [      2                 ]
                                          [ m sin (theta)          ]
                                          [ -------------  0  0  0 ]
                                          [       r                ]

                                          [ 0        0        0  0 ]
                                          [                        ]
                                          [ 0        0        0  0 ]
                                          [                        ]
                               riem     = [ 0        0        0  0 ]
                                   4, 2   [                        ]
                                          [         2              ]
                                          [    m sin (theta)       ]
                                          [ 0  -------------  0  0 ]
                                          [          r             ]

                                        [ 0  0          0          0 ]
                                        [                            ]
                                        [ 0  0          0          0 ]
                                        [                            ]
                             riem     = [ 0  0          0          0 ]
                                 4, 3   [                            ]
                                        [                2           ]
                                        [         2 m sin (theta)    ]
                                        [ 0  0  - ---------------  0 ]
                                        [                r           ]

                     [        2                                             ]
                     [   m sin (theta)                                      ]
                     [ - -------------         0                0         0 ]
                     [         r                                            ]
                     [                                                      ]
                     [                         2                            ]
                     [                    m sin (theta)                     ]
          riem     = [        0         - -------------         0         0 ]
              4, 4   [                          r                           ]
                     [                                                      ]
                     [                                          2           ]
                     [                                   2 m sin (theta)    ]
                     [        0                0         ---------------  0 ]
                     [                                          r           ]
                     [                                                      ]
                     [        0                0                0         0 ]

          (%o5)                                done

Function: deleten (<L>, <n>)

     Returns a new list consisting of <L> with the <n>'th element
     deleted.

Option variable: dim

     Default value: 4

     An option in the `ctensor' (component tensor) package.  `dim' is
     the dimension of the manifold with the default 4. The command
     `dim: n' will reset the dimension to any other value `n'.

Option variable: diagmetric

     Default value: `false'

     An option in the `ctensor' (component tensor) package.  If
     `diagmetric' is `true' special routines compute all geometrical
     objects (which contain the metric tensor explicitly) by taking
     into consideration the diagonality of the metric. Reduced run
     times will, of course, result. Note: this option is set
     automatically by `csetup' if a diagonal metric is specified.

Option variable: ctrgsimp

     Causes trigonometric simplifications to be used when tensors are
     computed. Presently, `ctrgsimp' affects only computations
     involving a moving frame.

Option variable: cframe_flag

     Causes computations to be performed relative to a moving frame as
     opposed to a holonomic metric. The frame is defined by the inverse
     frame array `fri' and the frame metric `lfg'. For computations
     using a Cartesian frame, `lfg' should be the unit matrix of the
     appropriate dimension; for computations in a Lorentz frame, `lfg'
     should have the appropriate signature.

Option variable: ctorsion_flag

     Causes the contortion tensor to be included in the computation of
     the connection coefficients. The contortion tensor itself is
     computed by `contortion' from the user-supplied tensor `tr'.

Option variable: cnonmet_flag

     Causes the nonmetricity coefficients to be included in the
     computation of the connection coefficients. The nonmetricity
     coefficients are computed from the user-supplied nonmetricity
     vector `nm' by the function `nonmetricity'.

Option variable: ctayswitch

     If set to `true', causes some `ctensor' computations to be carried
     out using Taylor-series expansions. Presently, `christof', `ricci',
     `uricci', `einstein', and `weyl' take into account this setting.

Option variable: ctayvar

     Variable used for Taylor-series expansion if `ctayswitch' is set to
     `true'.

Option variable: ctaypov

     Maximum power used in Taylor-series expansion when `ctayswitch' is
     set to `true'.

Option variable: ctaypt

     Point around which Taylor-series expansion is carried out when
     `ctayswitch' is set to `true'.

System variable: gdet

     The determinant of the metric tensor `lg'. Computed by `cmetric'
     when `cframe_flag' is set to `false'.

Option variable: ratchristof

     Causes rational simplification to be applied by `christof'.

Option variable: rateinstein

     Default value: `true'

     If `true' rational simplification will be performed on the
     non-zero components of Einstein tensors; if `ratfac' is `true'
     then the components will also be factored.

Option variable: ratriemann

     Default value: `true'

     One of the switches which controls simplification of Riemann
     tensors; if `true', then rational simplification will be done; if
     `ratfac' is `true' then each of the components will also be
     factored.

Option variable: ratweyl

     Default value: `true'

     If `true', this switch causes the `weyl' function to apply
     rational simplification to the values of the Weyl tensor. If
     `ratfac' is `true', then the components will also be factored.

Variable: lfg

     The covariant frame metric. By default, it is initialized to the
     4-dimensional Lorentz frame with signature (+,+,+,-). Used when
     `cframe_flag' is `true'.

Variable: ufg

     The inverse frame metric. Computed from `lfg' when `cmetric' is
     called while `cframe_flag' is set to `true'.

Variable: riem

     The (3,1) Riemann tensor. Computed when the function `riemann' is
     invoked. For information about index ordering, see the description
     of `riemann'.

     If `cframe_flag' is `true', `riem' is computed from the covariant
     Riemann-tensor `lriem'.

Variable: lriem

     The covariant Riemann tensor. Computed by `lriemann'.

Variable: uriem

     The contravariant Riemann tensor. Computed by `uriemann'.

Variable: ric

     The mixed Ricci-tensor. Computed by `ricci'.

Variable: uric

     The contravariant Ricci-tensor. Computed by `uricci'.

Variable: lg

     The metric tensor. This tensor must be specified (as a `dim' by
     `dim' matrix) before other computations can be performed.

Variable: ug

     The inverse of the metric tensor. Computed by `cmetric'.

Variable: weyl

     The Weyl tensor. Computed by `weyl'.

Variable: fb

     Frame bracket coefficients, as computed by `frame_bracket'.

Variable: kinvariant

     The Kretchmann invariant. Computed by `rinvariant'.

Variable: np

     A Newman-Penrose null tetrad. Computed by `nptetrad'.

Variable: npi

     The raised-index Newman-Penrose null tetrad. Computed by
     `nptetrad'.  Defined as `ug.np'. The product `np.transpose(npi)'
     is constant:

          (%i39) trigsimp(np.transpose(npi));
                                        [  0   - 1  0  0 ]
                                        [                ]
                                        [ - 1   0   0  0 ]
          (%o39)                        [                ]
                                        [  0    0   0  1 ]
                                        [                ]
                                        [  0    0   1  0 ]

Variable: tr

     User-supplied rank-3 tensor representing torsion. Used by
     `contortion'.

Variable: kt

     The contortion tensor, computed from `tr' by `contortion'.

Variable: nm

     User-supplied nonmetricity vector. Used by `nonmetricity'.

Variable: nmc

     The nonmetricity coefficients, computed from `nm' by
     `nonmetricity'.

System variable: tensorkill

     Variable indicating if the tensor package has been initialized.
     Set and used by `csetup', reset by `init_ctensor'.

Option variable: ct_coords

     Default value: `[]'

     An option in the `ctensor' (component tensor) package.
     `ct_coords' contains a list of coordinates.  While normally
     defined when the function `csetup' is called, one may redefine the
     coordinates with the assignment `ct_coords: [j1, j2, ..., jn]'
     where the j's are the new coordinate names.  See also `csetup'.

27 atensor


Introduction to atensor
Functions and Variables for atensor

27.1 Introduction to atensor

`atensor' is an algebraic tensor manipulation package. To use `atensor',
type `load(atensor)', followed by a call to the `init_atensor' function.

   The essence of `atensor' is a set of simplification rules for the
noncommutative (dot) product operator ("`.'"). `atensor' recognizes
several algebra types; the corresponding simplification rules are put
into effect when the `init_atensor' function is called.

   The capabilities of `atensor' can be demonstrated by defining the
algebra of quaternions as a Clifford-algebra Cl(0,2) with two basis
vectors. The three quaternionic imaginary units are then the two basis
vectors and their product, i.e.:

         i = v     j = v     k = v  . v
              1         2         1    2

   Although the `atensor' package has a built-in definition for the
quaternion algebra, it is not used in this example, in which we
endeavour to build the quaternion multiplication table as a matrix:

     (%i1) load(atensor);
     (%o1)       /share/tensor/atensor.mac
     (%i2) init_atensor(clifford,0,0,2);
     (%o2)                                done
     (%i3) atensimp(v[1].v[1]);
     (%o3)                                 - 1
     (%i4) atensimp((v[1].v[2]).(v[1].v[2]));
     (%o4)                                 - 1
     (%i5) q:zeromatrix(4,4);
                                     [ 0  0  0  0 ]
                                     [            ]
                                     [ 0  0  0  0 ]
     (%o5)                           [            ]
                                     [ 0  0  0  0 ]
                                     [            ]
                                     [ 0  0  0  0 ]
     (%i6) q[1,1]:1;
     (%o6)                                  1
     (%i7) for i thru adim do q[1,i+1]:q[i+1,1]:v[i];
     (%o7)                                done
     (%i8) q[1,4]:q[4,1]:v[1].v[2];
     (%o8)                               v  . v
                                          1    2
     (%i9) for i from 2 thru 4 do for j from 2 thru 4 do
           q[i,j]:atensimp(q[i,1].q[1,j]);
     (%o9)                                done
     (%i10) q;
                        [    1        v         v      v  . v  ]
                        [              1         2      1    2 ]
                        [                                      ]
                        [   v         - 1     v  . v    - v    ]
                        [    1                 1    2      2   ]
     (%o10)             [                                      ]
                        [   v      - v  . v     - 1      v     ]
                        [    2        1    2              1    ]
                        [                                      ]
                        [ v  . v      v        - v       - 1   ]
                        [  1    2      2          1            ]

   `atensor' recognizes as base vectors indexed symbols, where the
symbol is that stored in `asymbol' and the index runs between 1 and
`adim'.  For indexed symbols, and indexed symbols only, the bilinear
forms `sf', `af', and `av' are evaluated. The evaluation substitutes
the value of `aform[i,j]' in place of `fun(v[i],v[j])' where `v'
represents the value of `asymbol' and `fun' is either `af' or `sf'; or,
it substitutes `v[aform[i,j]]' in place of `av(v[i],v[j])'.

   Needless to say, the functions `sf', `af' and `av' can be redefined.

   When the `atensor' package is loaded, the following flags are set:

     dotscrules:true;
     dotdistrib:true;
     dotexptsimp:false;

   If you wish to experiment with a nonassociative algebra, you may also
consider setting `dotassoc' to `false'. In this case, however,
`atensimp' will not always be able to obtain the desired
simplifications.

27.2 Functions and Variables for atensor

Function: init_atensor (<alg_type>, <opt_dims>)
Function: init_atensor (<alg_type>)

     Initializes the `atensor' package with the specified algebra type.
     <alg_type> can be one of the following:

     `universal': The universal algebra has no commutation rules.

     `grassmann': The Grassman algebra is defined by the commutation
     relation `u.v+v.u=0'.

     `clifford': The Clifford algebra is defined by the commutation
     relation `u.v+v.u=-2*sf(u,v)' where `sf' is a symmetric
     scalar-valued function. For this algebra, <opt_dims> can be up to
     three nonnegative integers, representing the number of positive,
     degenerate, and negative dimensions of the algebra, respectively.
     If any <opt_dims> values are supplied, `atensor' will configure the
     values of `adim' and `aform' appropriately. Otherwise, `adim' will
     default to 0 and `aform' will not be defined.

     `symmetric': The symmetric algebra is defined by the commutation
     relation `u.v-v.u=0'.

     `symplectic': The symplectic algebra is defined by the commutation
     relation `u.v-v.u=2*af(u,v)' where `af' is an antisymmetric
     scalar-valued function. For the symplectic algebra, <opt_dims> can
     be up to two nonnegative integers, representing the nondegenerate
     and degenerate dimensions, respectively. If any <opt_dims> values
     are supplied, `atensor' will configure the values of `adim' and
     `aform' appropriately. Otherwise, `adim' will default to 0 and
     `aform' will not be defined.

     `lie_envelop': The algebra of the Lie envelope is defined by the
     commutation relation `u.v-v.u=2*av(u,v)' where `av' is an
     antisymmetric function.

     The `init_atensor' function also recognizes several predefined
     algebra types:

     `complex' implements the algebra of complex numbers as the
     Clifford algebra Cl(0,1). The call `init_atensor(complex)' is
     equivalent to `init_atensor(clifford,0,0,1)'.

     `quaternion' implements the algebra of quaternions. The call
     `init_atensor (quaternion)' is equivalent to `init_atensor
     (clifford,0,0,2)'.

     `pauli' implements the algebra of Pauli-spinors as the
     Clifford-algebra Cl(3,0). A call to `init_atensor(pauli)' is
     equivalent to `init_atensor(clifford,3)'.

     `dirac' implements the algebra of Dirac-spinors as the
     Clifford-algebra Cl(3,1). A call to `init_atensor(dirac)' is
     equivalent to `init_atensor(clifford,3,0,1)'.

Function: atensimp (<expr>)

     Simplifies an algebraic tensor expression <expr> according to the
     rules configured by a call to `init_atensor'. Simplification
     includes recursive application of commutation relations and
     resolving calls to `sf', `af', and `av' where applicable. A
     safeguard is used to ensure that the function always terminates,
     even for complex expressions.

Function: alg_type

     The algebra type. Valid values are `universal', `grassmann',
     `clifford', `symmetric', `symplectic' and `lie_envelop'.

Variable: adim

     Default value: 0

     The dimensionality of the algebra. `atensor' uses the value of
     `adim' to determine if an indexed object is a valid base vector.
     See `abasep'.

Variable: aform

     Default value: `ident(3)'

     Default values for the bilinear forms `sf', `af', and `av'. The
     default is the identity matrix `ident(3)'.

Variable: asymbol

     Default value: `v'

     The symbol for base vectors.

Function: sf (<u>, <v>)

     A symmetric scalar function that is used in commutation relations.
     The default implementation checks if both arguments are base
     vectors using `abasep' and if that is the case, substitutes the
     corresponding value from the matrix `aform'.

Function: af (<u>, <v>)

     An antisymmetric scalar function that is used in commutation
     relations.  The default implementation checks if both arguments
     are base vectors using `abasep' and if that is the case,
     substitutes the corresponding value from the matrix `aform'.

Function: av (<u>, <v>)

     An antisymmetric function that is used in commutation relations.
     The default implementation checks if both arguments are base
     vectors using `abasep' and if that is the case, substitutes the
     corresponding value from the matrix `aform'.

     For instance:

          (%i1) load(atensor);
          (%o1)       /share/tensor/atensor.mac
          (%i2) adim:3;
          (%o2)                                  3
          (%i3) aform:matrix([0,3,-2],[-3,0,1],[2,-1,0]);
                                         [  0    3   - 2 ]
                                         [               ]
          (%o3)                          [ - 3   0    1  ]
                                         [               ]
                                         [  2   - 1   0  ]
          (%i4) asymbol:x;
          (%o4)                                  x
          (%i5) av(x[1],x[2]);
          (%o5)                                 x
                                                 3

Function: abasep (<v>)

     Checks if its argument is an `atensor' base vector. That is, if it
     is an indexed symbol, with the symbol being the same as the value
     of `asymbol', and the index having a numeric value between 1 and
     `adim'.

28 Sums, Products, and Series


Sums and Products
Introduction to Series
Functions and Variables for Series
Poisson series

28.1 Sums and Products

Function: bashindices (<expr>)

     Transforms the expression <expr> by giving each summation and
     product a unique index.  This gives `changevar' greater precision
     when it is working with summations or products.  The form of the
     unique index is `j<number>'. The quantity <number> is determined
     by referring to `gensumnum', which can be changed by the user.
     For example, `gensumnum:0$' resets it.

Function: lsum (<expr>, <x>, <L>)

     Represents the sum of <expr> for each element <x> in <L>.  A noun
     form `'lsum' is returned if the argument <L> does not evaluate to
     a list.

     Examples:

          (%i1) lsum (x^i, i, [1, 2, 7]);
                                      7    2
          (%o1)                      x  + x  + x
          (%i2) lsum (i^2, i, rootsof (x^3 - 1));
                               ====
                               \      2
          (%o2)                 >    i
                               /
                               ====
                                             3
                               i in rootsof(x  - 1)

Function: intosum (<expr>)

     Moves multiplicative factors outside a summation to inside.  If
     the index is used in the outside expression, then the function
     tries to find a reasonable index, the same as it does for
     `sumcontract'.  This is essentially the reverse idea of the
     `outative' property of summations, but note that it does not
     remove this property, it only bypasses it.

     In some cases, a `scanmap (multthru, <expr>)' may be necessary
     before the `intosum'.

Function: product (<expr>, <i>, <i_0>, <i_1>)

     Represents a product of the values of <expr> as the index <i>
     varies from <i_0> to <i_1>.  The noun form `'product' is displayed
     as an uppercase letter pi.

     `product' evaluates <expr> and lower and upper limits <i_0> and
     <i_1>, `product' quotes (does not evaluate) the index <i>.

     If the upper and lower limits differ by an integer, <expr> is
     evaluated for each value of the index <i>, and the result is an
     explicit product.

     Otherwise, the range of the index is indefinite.  Some rules are
     applied to simplify the product.  When the global variable
     `simpproduct' is `true', additional rules are applied.  In some
     cases, simplification yields a result which is not a product;
     otherwise, the result is a noun form `'product'.

     See also `nouns' and `evflag'.

     Examples:

          (%i1) product (x + i*(i+1)/2, i, 1, 4);
          (%o1)           (x + 1) (x + 3) (x + 6) (x + 10)
          (%i2) product (i^2, i, 1, 7);
          (%o2)                       25401600
          (%i3) product (a[i], i, 1, 7);
          (%o3)                 a  a  a  a  a  a  a
                                 1  2  3  4  5  6  7
          (%i4) product (a(i), i, 1, 7);
          (%o4)          a(1) a(2) a(3) a(4) a(5) a(6) a(7)
          (%i5) product (a(i), i, 1, n);
                                       n
                                     /===\
                                      ! !
          (%o5)                       ! !  a(i)
                                      ! !
                                     i = 1
          (%i6) product (k, k, 1, n);
                                         n
                                       /===\
                                        ! !
          (%o6)                         ! !  k
                                        ! !
                                       k = 1
          (%i7) product (k, k, 1, n), simpproduct;
          (%o7)                          n!
          (%i8) product (integrate (x^k, x, 0, 1), k, 1, n);
                                       n
                                     /===\
                                      ! !    1
          (%o8)                       ! !  -----
                                      ! !  k + 1
                                     k = 1
          (%i9) product (if k <= 5 then a^k else b^k, k, 1, 10);
                                        15  40
          (%o9)                        a   b

Option variable: simpsum

     Default value: `false'

     When `simpsum' is `true', the result of a `sum' is simplified.
     This simplification may sometimes be able to produce a closed
     form.  If `simpsum' is `false' or if the quoted form `'sum' is
     used, the value is a sum noun form which is a representation of
     the sigma notation used in mathematics.

Function: sum (<expr>, <i>, <i_0>, <i_1>)

     Represents a summation of the values of <expr> as the index <i>
     varies from <i_0> to <i_1>.  The noun form `'sum' is displayed as
     an uppercase letter sigma.

     `sum' evaluates its summand <expr> and lower and upper limits <i_0>
     and <i_1>, `sum' quotes (does not evaluate) the index <i>.

     If the upper and lower limits differ by an integer, the summand
     <expr> is evaluated for each value of the summation index <i>, and
     the result is an explicit sum.

     Otherwise, the range of the index is indefinite.  Some rules are
     applied to simplify the summation.  When the global variable
     `simpsum' is `true', additional rules are applied.  In some cases,
     simplification yields a result which is not a summation;
     otherwise, the result is a noun form `'sum'.

     When the `evflag' (evaluation flag) `cauchysum' is `true', a
     product of summations is expressed as a Cauchy product, in which
     the index of the inner summation is a function of the index of the
     outer one, rather than varying independently.

     The global variable `genindex' is the alphabetic prefix used to
     generate the next index of summation, when an automatically
     generated index is needed.

     `gensumnum' is the numeric suffix used to generate the next index
     of summation, when an automatically generated index is needed.
     When `gensumnum' is `false', an automatically-generated index is
     only `genindex' with no numeric suffix.

     See also `sumcontract', `intosum', `bashindices', `niceindices',
     `nouns', `evflag', and `zeilberger'.

     Examples:

          (%i1) sum (i^2, i, 1, 7);
          (%o1)                          140
          (%i2) sum (a[i], i, 1, 7);
          (%o2)           a  + a  + a  + a  + a  + a  + a
                           7    6    5    4    3    2    1
          (%i3) sum (a(i), i, 1, 7);
          (%o3)    a(7) + a(6) + a(5) + a(4) + a(3) + a(2) + a(1)
          (%i4) sum (a(i), i, 1, n);
                                      n
                                     ====
                                     \
          (%o4)                       >    a(i)
                                     /
                                     ====
                                     i = 1
          (%i5) sum (2^i + i^2, i, 0, n);
                                    n
                                   ====
                                   \       i    2
          (%o5)                     >    (2  + i )
                                   /
                                   ====
                                   i = 0
          (%i6) sum (2^i + i^2, i, 0, n), simpsum;
                                        3      2
                             n + 1   2 n  + 3 n  + n
          (%o6)             2      + --------------- - 1
                                            6
          (%i7) sum (1/3^i, i, 1, inf);
                                      inf
                                      ====
                                      \     1
          (%o7)                        >    --
                                      /      i
                                      ====  3
                                      i = 1
          (%i8) sum (1/3^i, i, 1, inf), simpsum;
                                          1
          (%o8)                           -
                                          2
          (%i9) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf);
                                        inf
                                        ====
                                        \     1
          (%o9)                      30  >    --
                                        /      2
                                        ====  i
                                        i = 1
          (%i10) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf), simpsum;
                                            2
          (%o10)                       5 %pi
          (%i11) sum (integrate (x^k, x, 0, 1), k, 1, n);
                                      n
                                     ====
                                     \       1
          (%o11)                      >    -----
                                     /     k + 1
                                     ====
                                     k = 1
          (%i12) sum (if k <= 5 then a^k else b^k, k, 1, 10);
                    10    9    8    7    6    5    4    3    2
          (%o12)   b   + b  + b  + b  + b  + a  + a  + a  + a  + a

Function: sumcontract (<expr>)

     Combines all sums of an addition that have upper and lower bounds
     that differ by constants.  The result is an expression containing
     one summation for each set of such summations added to all
     appropriate extra terms that had to be extracted to form this sum.
     `sumcontract' combines all compatible sums and uses one of the
     indices from one of the sums if it can, and then try to form a
     reasonable index if it cannot use any supplied.

     It may be necessary to do an `intosum (<expr>)' before the
     `sumcontract'.

Option variable: sumexpand

     Default value: `false'

     When `sumexpand' is `true', products of sums and exponentiated
     sums simplify to nested sums.

     See also `cauchysum'.

     Examples:

          (%i1) sumexpand: true$
          (%i2) sum (f (i), i, 0, m) * sum (g (j), j, 0, n);
                               m      n
                              ====   ====
                              \      \
          (%o2)                >      >     f(i1) g(i2)
                              /      /
                              ====   ====
                              i1 = 0 i2 = 0
          (%i3) sum (f (i), i, 0, m)^2;
                               m      m
                              ====   ====
                              \      \
          (%o3)                >      >     f(i3) f(i4)
                              /      /
                              ====   ====
                              i3 = 0 i4 = 0

28.2 Introduction to Series

Maxima contains functions `taylor' and `powerseries' for finding the
series of differentiable functions.  It also has tools such as `nusum'
capable of finding the closed form of some series.  Operations such as
addition and multiplication work as usual on series.  This section
presents the global variables which control the expansion.

28.3 Functions and Variables for Series

Option variable: cauchysum

     Default value: `false'

     When multiplying together sums with `inf' as their upper limit, if
     `sumexpand' is `true' and `cauchysum' is `true' then the Cauchy
     product will be used rather than the usual product.  In the Cauchy
     product the index of the inner summation is a function of the
     index of the outer one rather than varying independently.

     Example:

          (%i1) sumexpand: false$
          (%i2) cauchysum: false$
          (%i3) s: sum (f(i), i, 0, inf) * sum (g(j), j, 0, inf);
                                inf         inf
                                ====        ====
                                \           \
          (%o3)                ( >    f(i))  >    g(j)
                                /           /
                                ====        ====
                                i = 0       j = 0
          (%i4) sumexpand: true$
          (%i5) cauchysum: true$
          (%i6) ''s;
                           inf     i1
                           ====   ====
                           \      \
          (%o6)             >      >     g(i1 - i2) f(i2)
                           /      /
                           ====   ====
                           i1 = 0 i2 = 0

Function: deftaylor (<f_1>(<x_1>), <expr_1>, ..., <f_n>(<x_n>), <expr_n>)

     For each function <f_i> of one variable <x_i>, `deftaylor' defines
     <expr_i> as the Taylor series about zero.  <expr_i> is typically a
     polynomial in <x_i> or a summation; more general expressions are
     accepted by `deftaylor' without complaint.

     `powerseries (<f_i>(<x_i>), <x_i>, 0)' returns the series defined
     by `deftaylor'.

     `deftaylor' returns a list of the functions <f_1>, ..., <f_n>.
     `deftaylor' evaluates its arguments.

     Example:

          (%i1) deftaylor (f(x), x^2 + sum(x^i/(2^i*i!^2), i, 4, inf));
          (%o1)                          [f]
          (%i2) powerseries (f(x), x, 0);
                                inf
                                ====      i1
                                \        x         2
          (%o2)                  >     -------- + x
                                /       i1    2
                                ====   2   i1!
                                i1 = 4
          (%i3) taylor (exp (sqrt (f(x))), x, 0, 4);
                                2         3          4
                               x    3073 x    12817 x
          (%o3)/T/     1 + x + -- + ------- + -------- + . . .
                               2     18432     307200

Option variable: maxtayorder

     Default value: `true'

     When `maxtayorder' is `true', then during algebraic manipulation
     of (truncated) Taylor series, `taylor' tries to retain as many
     terms as are known to be correct.

Function: niceindices (<expr>)

     Renames the indices of sums and products in <expr>.  `niceindices'
     attempts to rename each index to the value of
     `niceindicespref[1]', unless that name appears in the summand or
     multiplicand, in which case `niceindices' tries the succeeding
     elements of `niceindicespref' in turn, until an unused variable is
     found.  If the entire list is exhausted, additional indices are
     constructed by appending integers to the value of
     `niceindicespref[1]', e.g., `i0', `i1', `i2', ...

     `niceindices' returns an expression.  `niceindices' evaluates its
     argument.

     Example:

          (%i1) niceindicespref;
          (%o1)                  [i, j, k, l, m, n]
          (%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
                           inf    inf
                          /===\   ====
                           ! !    \
          (%o2)            ! !     >      f(bar i j + foo)
                           ! !    /
                          bar = 1 ====
                                  foo = 1
          (%i3) niceindices (%);
                               inf  inf
                              /===\ ====
                               ! !  \
          (%o3)                ! !   >    f(i j l + k)
                               ! !  /
                              l = 1 ====
                                    k = 1

Option variable: niceindicespref

     Default value: `[i, j, k, l, m, n]'

     `niceindicespref' is the list from which `niceindices' takes the
     names of indices for sums and products.

     The elements of `niceindicespref' are typically names of variables,
     although that is not enforced by `niceindices'.

     Example:

          (%i1) niceindicespref: [p, q, r, s, t, u]$
          (%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
                           inf    inf
                          /===\   ====
                           ! !    \
          (%o2)            ! !     >      f(bar i j + foo)
                           ! !    /
                          bar = 1 ====
                                  foo = 1
          (%i3) niceindices (%);
                               inf  inf
                              /===\ ====
                               ! !  \
          (%o3)                ! !   >    f(i j q + p)
                               ! !  /
                              q = 1 ====
                                    p = 1

Function: nusum (<expr>, <x>, <i_0>, <i_1>)

     Carries out indefinite hypergeometric summation of <expr> with
     respect to <x> using a decision procedure due to R.W. Gosper.
     <expr> and the result must be expressible as products of integer
     powers, factorials, binomials, and rational functions.

     The terms "definite" and "indefinite summation" are used
     analogously to "definite" and "indefinite integration".  To sum
     indefinitely means to give a symbolic result for the sum over
     intervals of variable length, not just e.g. 0 to inf.  Thus, since
     there is no formula for the general partial sum of the binomial
     series, `nusum' can't do it.

     `nusum' and `unsum' know a little about sums and differences of
     finite products.  See also `unsum'.

     Examples:

          (%i1) nusum (n*n!, n, 0, n);

          Dependent equations eliminated:  (1)
          (%o1)                     (n + 1)! - 1
          (%i2) nusum (n^4*4^n/binomial(2*n,n), n, 0, n);
                               4        3       2              n
                2 (n + 1) (63 n  + 112 n  + 18 n  - 22 n + 3) 4      2
          (%o2) ------------------------------------------------ - ------
                              693 binomial(2 n, n)                 3 11 7
          (%i3) unsum (%, n);
                                        4  n
                                       n  4
          (%o3)                   ----------------
                                  binomial(2 n, n)
          (%i4) unsum (prod (i^2, i, 1, n), n);
                              n - 1
                              /===\
                               ! !   2
          (%o4)              ( ! !  i ) (n - 1) (n + 1)
                               ! !
                              i = 1
          (%i5) nusum (%, n, 1, n);

          Dependent equations eliminated:  (2 3)
                                      n
                                    /===\
                                     ! !   2
          (%o5)                      ! !  i  - 1
                                     ! !
                                    i = 1

Function: pade (<taylor_series>, <numer_deg_bound>, <denom_deg_bound>)

     Returns a list of all rational functions which have the given
     Taylor series expansion where the sum of the degrees of the
     numerator and the denominator is less than or equal to the
     truncation level of the power series, i.e.  are "best"
     approximants, and which additionally satisfy the specified degree
     bounds.

     <taylor_series> is a univariate Taylor series.  <numer_deg_bound>
     and <denom_deg_bound> are positive integers specifying degree
     bounds on the numerator and denominator.

     <taylor_series> can also be a Laurent series, and the degree
     bounds can be `inf' which causes all rational functions whose total
     degree is less than or equal to the length of the power series to
     be returned.  Total degree is defined as `<numer_deg_bound> +
     <denom_deg_bound>'.  Length of a power series is defined as
     `"truncation level" + 1 - min(0, "order of series")'.

          (%i1) taylor (1 + x + x^2 + x^3, x, 0, 3);
                                        2    3
          (%o1)/T/             1 + x + x  + x  + . . .
          (%i2) pade (%, 1, 1);
                                           1
          (%o2)                       [- -----]
                                         x - 1
          (%i3) t: taylor(-(83787*x^10 - 45552*x^9 - 187296*x^8
                             + 387072*x^7 + 86016*x^6 - 1507328*x^5
                             + 1966080*x^4 + 4194304*x^3 - 25165824*x^2
                             + 67108864*x - 134217728)
                 /134217728, x, 0, 10);
                              2    3       4       5       6        7
                       x   3 x    x    15 x    23 x    21 x    189 x
          (%o3)/T/ 1 - - + ---- - -- - ----- + ----- - ----- - ------
                       2    16    32   1024    2048    32768   65536

                                            8         9          10
                                      5853 x    2847 x    83787 x
                                    + ------- + ------- - --------- + . . .
                                      4194304   8388608   134217728
          (%i4) pade (t, 4, 4);
          (%o4)                          []

     There is no rational function of degree 4 numerator/denominator,
     with this power series expansion.  You must in general have degree
     of the numerator and degree of the denominator adding up to at
     least the degree of the power series, in order to have enough
     unknown coefficients to solve.

          (%i5) pade (t, 5, 5);
                               5                4                 3
          (%o5) [- (520256329 x  - 96719020632 x  - 489651410240 x

                            2
           - 1619100813312 x  - 2176885157888 x - 2386516803584)

                         5                 4                  3
          /(47041365435 x  + 381702613848 x  + 1360678489152 x

                            2
           + 2856700692480 x  + 3370143559680 x + 2386516803584)]

Function: powerseries (<expr>, <x>, <a>)

     Returns the general form of the power series expansion for <expr>
     in the variable <x> about the point <a> (which may be `inf' for
     infinity):
                     inf
                     ====
                     \               n
                      >    b  (x - a)
                     /      n
                     ====
                     n = 0

     If `powerseries' is unable to expand <expr>, `taylor' may give the
     first several terms of the series.

     When `verbose' is `true', `powerseries' prints progress messages.

          (%i1) verbose: true$
          (%i2) powerseries (log(sin(x)/x), x, 0);
          can't expand
                                           log(sin(x))
          so we'll try again after applying the rule:
                                                  d
                                                / -- (sin(x))
                                                [ dx
                                  log(sin(x)) = i ----------- dx
                                                ]   sin(x)
                                                /
          in the first simplification we have returned:
                                       /
                                       [
                                       i cot(x) dx - log(x)
                                       ]
                                       /
                              inf
                              ====        i1  2 i1             2 i1
                              \      (- 1)   2     bern(2 i1) x
                               >     ------------------------------
                              /                i1 (2 i1)!
                              ====
                              i1 = 1
          (%o2)                -------------------------------------
                                                2

Option variable: psexpand

     Default value: `false'

     When `psexpand' is `true', an extended rational function
     expression is displayed fully expanded.  The switch `ratexpand'
     has the same effect.

     When `psexpand' is `false', a multivariate expression is displayed
     just as in the rational function package.

     When `psexpand' is  `multi', then terms with the same total degree
     in the variables are grouped together.

Function: revert (<expr>, <x>)
Function: revert2 (<expr>, <x>, <n>)

     These functions return the reversion of <expr>, a Taylor series
     about zero in the variable <x>.  `revert' returns a polynomial of
     degree equal to the highest power in <expr>.  `revert2' returns a
     polynomial of degree <n>, which may be greater than, equal to, or
     less than the degree of <expr>.

     `load ("revert")' loads these functions.

     Examples:

          (%i1) load ("revert")$
          (%i2) t: taylor (exp(x) - 1, x, 0, 6);
                             2    3    4    5     6
                            x    x    x    x     x
          (%o2)/T/      x + -- + -- + -- + --- + --- + . . .
                            2    6    24   120   720
          (%i3) revert (t, x);
                         6       5       4       3       2
                     10 x  - 12 x  + 15 x  - 20 x  + 30 x  - 60 x
          (%o3)/R/ - --------------------------------------------
                                          60
          (%i4) ratexpand (%);
                               6    5    4    3    2
                              x    x    x    x    x
          (%o4)             - -- + -- - -- + -- - -- + x
                              6    5    4    3    2
          (%i5) taylor (log(x+1), x, 0, 6);
                              2    3    4    5    6
                             x    x    x    x    x
          (%o5)/T/       x - -- + -- - -- + -- - -- + . . .
                             2    3    4    5    6
          (%i6) ratsimp (revert (t, x) - taylor (log(x+1), x, 0, 6));
          (%o6)                           0
          (%i7) revert2 (t, x, 4);
                                    4    3    2
                                   x    x    x
          (%o7)                  - -- + -- - -- + x
                                   4    3    2

Function: taylor (<expr>, <x>, <a>, <n>)
Function: taylor (<expr>, [<x_1>, <x_2>, ...], <a>, <n>)
Function: taylor (<expr>, [<x>, <a>, <n>, 'asymp])
Function: taylor (<expr>, [<x_1>, <x_2>, ...], [<a_1>, <a_2>, ...], [<n_1>, <n_2>, ...])
Function: taylor (<expr>, [<x_1>, <a_1>, <n_1>], [<x_2>, <a_2>, <n_2>], ...)

     `taylor (<expr>, <x>, <a>, <n>)' expands the expression <expr> in
     a truncated Taylor or Laurent series in the variable <x> around
     the point <a>, containing terms through `(<x> - <a>)^<n>'.

     If <expr> is of the form `<f>(<x>)/<g>(<x>)' and `<g>(<x>)' has no
     terms up to degree <n> then `taylor' attempts to expand `<g>(<x>)'
     up to degree `2 <n>'.  If there are still no nonzero terms,
     `taylor' doubles the degree of the expansion of `<g>(<x>)' so long
     as the degree of the expansion is less than or equal to `<n>
     2^taylordepth'.

     `taylor (<expr>, [<x_1>, <x_2>, ...], <a>, <n>)' returns a
     truncated power series of degree <n> in all variables <x_1>,
     <x_2>, ...  about the point `(<a>, <a>, ...)'.

     `taylor (<expr>, [<x_1>, <a_1>, <n_1>], [<x_2>, <a_2>, <n_2>],
     ...)' returns a truncated power series in the variables <x_1>,
     <x_2>, ... about the point `(<a_1>, <a_2>, ...)', truncated at
     <n_1>, <n_2>, ...

     `taylor (<expr>, [<x_1>, <x_2>, ...], [<a_1>, <a_2>, ...], [<n_1>,
     <n_2>, ...])' returns a truncated power series in the variables
     <x_1>, <x_2>, ... about the point `(<a_1>, <a_2>, ...)', truncated
     at <n_1>, <n_2>, ...

     `taylor (<expr>, [<x>, <a>, <n>, 'asymp])' returns an expansion of
     <expr> in negative powers of `<x> - <a>'.  The highest order term
     is `(<x> - <a>)^<-n>'.

     When `maxtayorder' is `true', then during algebraic manipulation
     of (truncated) Taylor series, `taylor' tries to retain as many
     terms as are known to be correct.

     When `psexpand' is `true', an extended rational function
     expression is displayed fully expanded.  The switch `ratexpand'
     has the same effect.  When `psexpand' is `false', a multivariate
     expression is displayed just as in the rational function package.
     When `psexpand' is  `multi', then terms with the same total degree
     in the variables are grouped together.

     See also the `taylor_logexpand' switch for controlling expansion.

     Examples:

          (%i1) taylor (sqrt (sin(x) + a*x + 1), x, 0, 3);
                                     2             2
                       (a + 1) x   (a  + 2 a + 1) x
          (%o1)/T/ 1 + --------- - -----------------
                           2               8

                                             3      2             3
                                         (3 a  + 9 a  + 9 a - 1) x
                                       + -------------------------- + . . .
                                                     48
          (%i2) %^2;
                                              3
                                             x
          (%o2)/T/           1 + (a + 1) x - -- + . . .
                                             6
          (%i3) taylor (sqrt (x + 1), x, 0, 5);
                                 2    3      4      5
                            x   x    x    5 x    7 x
          (%o3)/T/      1 + - - -- + -- - ---- + ---- + . . .
                            2   8    16   128    256
          (%i4) %^2;
          (%o4)/T/                  1 + x + . . .
          (%i5) product ((1 + x^i)^2.5, i, 1, inf)/(1 + x^2);
                                   inf
                                  /===\
                                   ! !    i     2.5
                                   ! !  (x  + 1)
                                   ! !
                                  i = 1
          (%o5)                   -----------------
                                        2
                                       x  + 1
          (%i6) ev (taylor(%, x,  0, 3), keepfloat);
                                         2           3
          (%o6)/T/    1 + 2.5 x + 3.375 x  + 6.5625 x  + . . .
          (%i7) taylor (1/log (x + 1), x, 0, 3);
                                         2       3
                           1   1   x    x    19 x
          (%o7)/T/         - + - - -- + -- - ----- + . . .
                           x   2   12   24    720
          (%i8) taylor (cos(x) - sec(x), x, 0, 5);
                                          4
                                     2   x
          (%o8)/T/                - x  - -- + . . .
                                         6
          (%i9) taylor ((cos(x) - sec(x))^3, x, 0, 5);
          (%o9)/T/                    0 + . . .
          (%i10) taylor (1/(cos(x) - sec(x))^3, x, 0, 5);
                                                         2          4
                      1     1       11      347    6767 x    15377 x
          (%o10)/T/ - -- + ---- + ------ - ----- - ------- - --------
                       6      4        2   15120   604800    7983360
                      x    2 x    120 x

                                                                    + . . .
          (%i11) taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6);
                         2  2       4      2   4
                        k  x    (3 k  - 4 k ) x
          (%o11)/T/ 1 - ----- - ----------------
                          2            24

                                              6       4       2   6
                                         (45 k  - 60 k  + 16 k ) x
                                       - -------------------------- + . . .
                                                    720
          (%i12) taylor ((x + 1)^n, x, 0, 4);
                                2       2     3      2         3
                              (n  - n) x    (n  - 3 n  + 2 n) x
          (%o12)/T/ 1 + n x + ----------- + --------------------
                                   2                 6

                                         4      3       2         4
                                       (n  - 6 n  + 11 n  - 6 n) x
                                     + ---------------------------- + . . .
                                                    24
          (%i13) taylor (sin (y + x), x, 0, 3, y, 0, 3);
                         3                 2
                        y                 y
          (%o13)/T/ y - -- + . . . + (1 - -- + . . .) x
                        6                 2

                              3                       2
                         y   y            2      1   y            3
                    + (- - + -- + . . .) x  + (- - + -- + . . .) x  + . . .
                         2   12                  6   12
          (%i14) taylor (sin (y + x), [x, y], 0, 3);
                               3        2      2      3
                              x  + 3 y x  + 3 y  x + y
          (%o14)/T/   y + x - ------------------------- + . . .
                                          6
          (%i15) taylor (1/sin (y + x), x, 0, 3, y, 0, 3);
                    1   y              1    1               1            2
          (%o15)/T/ - + - + . . . + (- -- + - + . . .) x + (-- + . . .) x
                    y   6               2   6                3
                                       y                    y

                                                     1            3
                                                + (- -- + . . .) x  + . . .
                                                      4
                                                     y
          (%i16) taylor (1/sin (y + x), [x, y], 0, 3);
                                       3         2       2        3
                      1     x + y   7 x  + 21 y x  + 21 y  x + 7 y
          (%o16)/T/ ----- + ----- + ------------------------------- + . . .
                    x + y     6                   360

Option variable: taylordepth

     Default value: 3

     If there are still no nonzero terms, `taylor' doubles the degree
     of the expansion of `<g>(<x>)' so long as the degree of the
     expansion is less than or equal to `<n> 2^taylordepth'.

Function: taylorinfo (<expr>)

     Returns information about the Taylor series <expr>.  The return
     value is a list of lists.  Each list comprises the name of a
     variable, the point of expansion, and the degree of the expansion.

     `taylorinfo' returns `false' if <expr> is not a Taylor series.

     Example:

          (%i1) taylor ((1 - y^2)/(1 - x), x, 0, 3, [y, a, inf]);
                            2                       2
          (%o1)/T/ - (y - a)  - 2 a (y - a) + (1 - a )

                   2                        2
           + (1 - a  - 2 a (y - a) - (y - a) ) x

                   2                        2   2
           + (1 - a  - 2 a (y - a) - (y - a) ) x

                   2                        2   3
           + (1 - a  - 2 a (y - a) - (y - a) ) x  + . . .
          (%i2) taylorinfo(%);
          (%o2)               [[y, a, inf], [x, 0, 3]]

Function: taylorp (<expr>)

     Returns `true' if <expr> is a Taylor series, and `false' otherwise.

Option variable: taylor_logexpand

     Default value: `true'

     `taylor_logexpand' controls expansions of logarithms in `taylor'
     series.

     When `taylor_logexpand' is `true', all logarithms are expanded
     fully so that zero-recognition problems involving logarithmic
     identities do not disturb the expansion process.  However, this
     scheme is not always mathematically correct since it ignores
     branch information.

     When `taylor_logexpand' is set to `false', then the only expansion
     of logarithms that occur is that necessary to obtain a formal
     power series.

Option variable: taylor_order_coefficients

     Default value: `true'

     `taylor_order_coefficients' controls the ordering of coefficients
     in a Taylor series.

     When `taylor_order_coefficients' is `true', coefficients of taylor
     series are ordered canonically.

Function: taylor_simplifier (<expr>)

     Simplifies coefficients of the power series <expr>.  `taylor'
     calls this function.

Option variable: taylor_truncate_polynomials

     Default value: `true'

     When `taylor_truncate_polynomials' is `true', polynomials are
     truncated based upon the input truncation levels.

     Otherwise, polynomials input to `taylor' are considered to have
     infinite precison.

Function: taytorat (<expr>)

     Converts <expr> from `taylor' form to canonical rational expression
     (CRE) form.  The effect is the same as `rat (ratdisrep (<expr>))',
     but faster.

Function: trunc (<expr>)

     Annotates the internal representation of the general expression
     <expr> so that it is displayed as if its sums were truncated
     Taylor series.  <expr> is not otherwise modified.

     Example:

          (%i1) expr: x^2 + x + 1;
                                      2
          (%o1)                      x  + x + 1
          (%i2) trunc (expr);
                                          2
          (%o2)                  1 + x + x  + . . .
          (%i3) is (expr = trunc (expr));
          (%o3)                         true

Function: unsum (<f>, <n>)

     Returns the first backward difference `<f>(<n>) - <f>(<n> - 1)'.
     Thus `unsum' in a sense is the inverse of `sum'.

     See also `nusum'.

     Examples:

          (%i1) g(p) := p*4^n/binomial(2*n,n);
                                               n
                                            p 4
          (%o1)               g(p) := ----------------
                                      binomial(2 n, n)
          (%i2) g(n^4);
                                        4  n
                                       n  4
          (%o2)                   ----------------
                                  binomial(2 n, n)
          (%i3) nusum (%, n, 0, n);
                               4        3       2              n
                2 (n + 1) (63 n  + 112 n  + 18 n  - 22 n + 3) 4      2
          (%o3) ------------------------------------------------ - ------
                              693 binomial(2 n, n)                 3 11 7
          (%i4) unsum (%, n);
                                        4  n
                                       n  4
          (%o4)                   ----------------
                                  binomial(2 n, n)

Option variable: verbose

     Default value: `false'

     When `verbose' is `true', `powerseries' prints progress messages.

28.4 Poisson series

Function: intopois (<a>)

     Converts <a> into a Poisson encoding.

Function: outofpois (<a>)

     Converts <a> from Poisson encoding to general representation.  If
     <a> is not in Poisson form, `outofpois' carries out the conversion,
     i.e., the return value is `outofpois (intopois (<a>))'.  This
     function is thus a canonical simplifier for sums of powers of sine
     and cosine terms of a particular type.

Function: poisdiff (<a>, <b>)

     Differentiates <a> with respect to <b>. <b> must occur only in the
     trig arguments or only in the coefficients.

Function: poisexpt (<a>, <b>)

     Functionally identical to `intopois (<a>^<b>)'.  <b> must be a
     positive integer.

Function: poisint (<a>, <b>)

     Integrates in a similarly restricted sense (to `poisdiff').
     Non-periodic terms in <b> are dropped if <b> is in the trig
     arguments.

Option variable: poislim

     Default value: 5

     `poislim' determines the domain of the coefficients in the
     arguments of the trig functions.  The initial value of 5
     corresponds to the interval [-2^(5-1)+1,2^(5-1)], or [-15,16], but
     it can be set to [-2^(n-1)+1, 2^(n-1)].

Function: poismap (<series>, <sinfn>, <cosfn>)

     will map the functions <sinfn> on the sine terms and <cosfn> on the
     cosine terms of the Poisson series given.  <sinfn> and <cosfn> are
     functions of two arguments which are a coefficient and a
     trigonometric part of a term in series respectively.

Function: poisplus (<a>, <b>)

     Is functionally identical to `intopois (a + b)'.

Function: poissimp (<a>)

     Converts <a> into a Poisson series for <a> in general
     representation.

Special symbol: poisson

     The symbol `/P/' follows the line label of Poisson series
     expressions.

Function: poissubst (<a>, <b>, <c>)

     Substitutes <a> for <b> in <c>.  <c> is a Poisson series.

     (1) Where <B> is a variable <u>, <v>, <w>, <x>, <y>, or <z>, then
     <a> must be an expression linear in those variables (e.g., `6*u +
     4*v').

     (2) Where <b> is other than those variables, then <a> must also be
     free of those variables, and furthermore, free of sines or cosines.

     `poissubst (<a>, <b>, <c>, <d>, <n>)' is a special type of
     substitution which operates on <a> and <b> as in type (1) above,
     but where <d> is a Poisson series, expands `cos(<d>)' and
     `sin(<d>)' to order <n> so as to provide the result of substituting
     `<a> + <d>' for <b> in <c>.  The idea is that <d> is an expansion
     in terms of a small parameter.  For example, `poissubst (u, v,
     cos(v), %e, 3)' yields `cos(u)*(1 - %e^2/2) - sin(u)*(%e -
     %e^3/6)'.

Function: poistimes (<a>, <b>)

     Is functionally identical to `intopois (<a>*<b>)'.

Function: poistrim ()

     is a reserved function name which (if the user has defined it)
     gets applied during Poisson multiplication.  It is a predicate
     function of 6 arguments which are the coefficients of the <u>,
     <v>, ..., <z> in a term.  Terms for which `poistrim' is `true'
     (for the coefficients of that term) are eliminated during
     multiplication.

Function: printpois (<a>)

     Prints a Poisson series in a readable format.  In common with
     `outofpois', it will convert <a> into a Poisson encoding first, if
     necessary.

29 Number Theory

Functions and Variables for Number Theory

29.1 Functions and Variables for Number Theory

Function: bern (<n>)

     Returns the <n>'th Bernoulli number for integer <n>.  Bernoulli
     numbers equal to zero are suppressed if `zerobern' is `false'.

     See also `burn'.

          (%i1) zerobern: true$
          (%i2) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
                            1  1       1      1        1
          (%o2)       [1, - -, -, 0, - --, 0, --, 0, - --]
                            2  6       30     42       30
          (%i3) zerobern: false$
          (%i4) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
                      1  1    1   5     691   7    3617  43867
          (%o4) [1, - -, -, - --, --, - ----, -, - ----, -----]
                      2  6    30  66    2730  6    510    798

Function: bernpoly (<x>, <n>)

     Returns the <n>'th Bernoulli polynomial in the variable <x>.

Function: bfzeta (<s>, <n>)

     Returns the Riemann zeta function for the argument <s>.  The
     return value is a big float (bfloat); <n> is the number of digits
     in the return value.

Function: bfhzeta (<s>, <h>, <n>)

     Returns the Hurwitz zeta function for the arguments <s> and <h>.
     The return value is a big float (bfloat); <n> is the number of
     digits in the return value.

     The Hurwitz zeta function is defined as

                                  inf
                                  ====
                                  \        1
                   zeta (s,h)  =   >    --------
                                  /            s
                                  ====  (k + h)
                                  k = 0

     `load ("bffac")' loads this function.

Function: burn (<n>)

     Returns a rational number, which is an approximation of the <n>'th
     Bernoulli number for integer <n>.  `burn' exploits the observation
     that (rational) Bernoulli numbers can be approximated by
     (transcendental) zetas with tolerable efficiency:

                             n - 1  1 - 2 n
                        (- 1)      2        zeta(2 n) (2 n)!
               B(2 n) = ------------------------------------
                                          2 n
                                       %pi

     `burn' may be more efficient than `bern' for large, isolated <n>
     as `bern' computes all the Bernoulli numbers up to index <n> before
     returning.  `burn' invokes the approximation for even integers <n>
     > 255.  For odd integers and <n> <= 255 the function `bern' is
     called.

     `load ("bffac")' loads this function.  See also `bern'.

Function: cf (<expr>)

     Converts <expr> into a continued fraction.  <expr> is an expression
     comprising continued fractions and square roots of integers.
     Operands in the expression may be combined with arithmetic
     operators.  Aside from continued fractions and square roots,
     factors in the expression must be integer or rational numbers.
     Maxima does not know about operations on continued fractions
     outside of `cf'.

     `cf' evaluates its arguments after binding `listarith' to `false'.
     `cf' returns a continued fraction, represented as a list.

     A continued fraction `a + 1/(b + 1/(c + ...))' is represented by
     the list `[a, b, c, ...]'.  The list elements `a', `b', `c', ...
     must evaluate to integers.  <expr> may also contain `sqrt (n)'
     where `n' is an integer.  In this case `cf' will give as many
     terms of the continued fraction as the value of the variable
     `cflength' times the period.

     A continued fraction can be evaluated to a number by evaluating
     the arithmetic representation returned by `cfdisrep'.  See also
     `cfexpand' for another way to evaluate a continued fraction.

     See also `cfdisrep', `cfexpand', and `cflength'.

     Examples:

        * <expr> is an expression comprising continued fractions and
          square roots of integers.

               (%i1) cf ([5, 3, 1]*[11, 9, 7] + [3, 7]/[4, 3, 2]);
               (%o1)               [59, 17, 2, 1, 1, 1, 27]
               (%i2) cf ((3/17)*[1, -2, 5]/sqrt(11) + (8/13));
               (%o2)        [0, 1, 1, 1, 3, 2, 1, 4, 1, 9, 1, 9, 2]

        * `cflength' controls how many periods of the continued fraction
          are computed for algebraic, irrational numbers.

               (%i1) cflength: 1$
               (%i2) cf ((1 + sqrt(5))/2);
               (%o2)                    [1, 1, 1, 1, 2]
               (%i3) cflength: 2$
               (%i4) cf ((1 + sqrt(5))/2);
               (%o4)               [1, 1, 1, 1, 1, 1, 1, 2]
               (%i5) cflength: 3$
               (%i6) cf ((1 + sqrt(5))/2);
               (%o6)           [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]

        * A continued fraction can be evaluated by evaluating the
          arithmetic representation returned by `cfdisrep'.

               (%i1) cflength: 3$
               (%i2) cfdisrep (cf (sqrt (3)))$
               (%i3) ev (%, numer);
               (%o3)                   1.731707317073171

        * Maxima does not know about operations on continued fractions
          outside of `cf'.

               (%i1) cf ([1,1,1,1,1,2] * 3);
               (%o1)                     [4, 1, 5, 2]
               (%i2) cf ([1,1,1,1,1,2]) * 3;
               (%o2)                  [3, 3, 3, 3, 3, 6]

Function: cfdisrep (<list>)

     Constructs and returns an ordinary arithmetic expression of the
     form `a + 1/(b + 1/(c + ...))' from the list representation of a
     continued fraction `[a, b, c, ...]'.

          (%i1) cf ([1, 2, -3] + [1, -2, 1]);
          (%o1)                     [1, 1, 1, 2]
          (%i2) cfdisrep (%);
                                            1
          (%o2)                     1 + ---------
                                              1
                                        1 + -----
                                                1
                                            1 + -
                                                2

Function: cfexpand (<x>)

     Returns a matrix of the numerators and denominators of the last
     (column 1) and next-to-last (column 2) convergents of the
     continued fraction <x>.

          (%i1) cf (rat (ev (%pi, numer)));

          `rat' replaced 3.141592653589793 by 103993/33102 =3.141592653011902
          (%o1)                  [3, 7, 15, 1, 292]
          (%i2) cfexpand (%);
                                   [ 103993  355 ]
          (%o2)                    [             ]
                                   [ 33102   113 ]
          (%i3) %[1,1]/%[2,1], numer;
          (%o3)                   3.141592653011902

Option variable: cflength

     Default value: 1

     `cflength' controls the number of terms of the continued fraction
     the function `cf' will give, as the value `cflength' times the
     period.  Thus the default is to give one period.

          (%i1) cflength: 1$
          (%i2) cf ((1 + sqrt(5))/2);
          (%o2)                    [1, 1, 1, 1, 2]
          (%i3) cflength: 2$
          (%i4) cf ((1 + sqrt(5))/2);
          (%o4)               [1, 1, 1, 1, 1, 1, 1, 2]
          (%i5) cflength: 3$
          (%i6) cf ((1 + sqrt(5))/2);
          (%o6)           [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]

Function: divsum (<n>, <k>)
Function: divsum (<n>)

     `divsum (<n>, <k>)' returns the sum of the divisors of <n> raised
     to the <k>'th power.

     `divsum (<n>)' returns the sum of the divisors of <n>.

          (%i1) divsum (12);
          (%o1)                          28
          (%i2) 1 + 2 + 3 + 4 + 6 + 12;
          (%o2)                          28
          (%i3) divsum (12, 2);
          (%o3)                          210
          (%i4) 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2;
          (%o4)                          210

Function: euler (<n>)

     Returns the <n>'th Euler number for nonnegative integer <n>.

     For the Euler-Mascheroni constant, see `%gamma'.

          (%i1) map (euler, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
          (%o1)    [1, 0, - 1, 0, 5, 0, - 61, 0, 1385, 0, - 50521]

Function: fib (<n>)

     Returns the <n>'th Fibonacci number.  `fib(0)' equal to 0 and
     `fib(1)' equal to 1, and `fib (-<n>)' equal to `(-1)^(<n> + 1) *
     fib(<n>)'.

     After calling `fib', `prevfib' is equal to `fib (<x> - 1)', the
     Fibonacci number preceding the last one computed.

          (%i1) map (fib, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
          (%o1)         [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

Function: fibtophi (<expr>)

     Expresses Fibonacci numbers in <expr> in terms of the constant
     `%phi', which is `(1 + sqrt(5))/2', approximately 1.61803399.

     Examples:

          (%i1) fibtophi (fib (n));
                                     n             n
                                 %phi  - (1 - %phi)
          (%o1)                  -------------------
                                     2 %phi - 1
          (%i2) fib (n-1) + fib (n) - fib (n+1);
          (%o2)          - fib(n + 1) + fib(n) + fib(n - 1)
          (%i3) fibtophi (%);
                      n + 1             n + 1       n             n
                  %phi      - (1 - %phi)        %phi  - (1 - %phi)
          (%o3) - --------------------------- + -------------------
                          2 %phi - 1                2 %phi - 1
                                                    n - 1             n - 1
                                                %phi      - (1 - %phi)
                                              + ---------------------------
                                                        2 %phi - 1
          (%i4) ratsimp (%);
          (%o4)                           0

Function: ifactors (<n>)

     For a positive integer <n> returns the factorization of <n>.  If
     `n=p1^e1..pk^nk' is the decomposition of <n> into prime factors,
     ifactors returns `[[p1, e1], ... , [pk, ek]]'.

     Factorization methods used are trial divisions by primes up to
     9973, Pollard's rho method and elliptic curve method.

          (%i1) ifactors(51575319651600);
          (%o1)     [[2, 4], [3, 2], [5, 2], [1583, 1], [9050207, 1]]
          (%i2) apply("*", map(lambda([u], u[1]^u[2]), %));
          (%o2)                        51575319651600

Function: igcdex (<n>, <k>)

     Returns a list `[<a>, <b>, <u>]' where <u> is the greatest common
     divisor of <n> and <k>, and <u> is equal to `<a> <n> + <b> <k>'.
     The arguments <n> and <k> must be integers.

     `igcdex' implements the Euclidean algorithm.  See also `gcdex'.

     The command `load(gcdex)' loads the function.

     Examples:

          (%i1) load(gcdex)$

          (%i2) igcdex(30,18);
          (%o2)                      [- 1, 2, 6]
          (%i3) igcdex(1526757668, 7835626735736);
          (%o3)            [845922341123, - 164826435, 4]
          (%i4) igcdex(fib(20), fib(21));
          (%o4)                   [4181, - 2584, 1]

Function: inrt (<x>, <n>)

     Returns the integer <n>'th root of the absolute value of <x>.

          (%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
          (%i2) map (lambda ([a], inrt (10^a, 3)), l);
          (%o2) [2, 4, 10, 21, 46, 100, 215, 464, 1000, 2154, 4641, 10000]

Function: inv_mod (<n>, <m>)

     Computes the inverse of <n> modulo <m>.  `inv_mod (n,m)' returns
     `false', if <n> is a zero divisor modulo <m>.

          (%i1) inv_mod(3, 41);
          (%o1)                           14
          (%i2) ratsimp(3^-1), modulus=41;
          (%o2)                           14
          (%i3) inv_mod(3, 42);
          (%o3)                          false

Function: isqrt (<x>)

     Returns the "integer square root" of the absolute value of <x>,
     which is an integer.

Function: jacobi (<p>, <q>)

     Returns the Jacobi symbol of <p> and <q>.

          (%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
          (%i2) map (lambda ([a], jacobi (a, 9)), l);
          (%o2)         [1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0]

Function: lcm (<expr_1>, ..., <expr_n>)

     Returns the least common multiple of its arguments.  The arguments
     may be general expressions as well as integers.

     `load ("functs")' loads this function.

Function: mod (<x>, <y>)

     If <x> and <y> are real numbers and <y> is nonzero, return `<x> -
     <y> * floor(<x> / <y>)'.  Further for all real <x>, we have `mod
     (<x>, 0) = <x>'.  For a discussion of the definition `mod (<x>, 0)
     = <x>', see Section 3.4, of "Concrete Mathematics," by Graham,
     Knuth, and Patashnik.  The function `mod (<x>, 1)' is a sawtooth
     function with period 1 with `mod (1, 1) = 0' and `mod (0, 1) = 0'.

     To find the principal argument (a number in the interval `(-%pi,
     %pi]') of a complex number, use the function `<x> |-> %pi - mod
     (%pi - <x>, 2*%pi)', where <x> is an argument.

     When <x> and <y> are constant expressions (`10 * %pi', for
     example), `mod' uses the same big float evaluation scheme that
     `floor' and `ceiling' uses.  Again, it's possible, although
     unlikely, that `mod' could return an erroneous value in such cases.

     For nonnumerical arguments <x> or <y>, `mod' knows several
     simplification rules:

          (%i1) mod (x, 0);
          (%o1)                           x
          (%i2) mod (a*x, a*y);
          (%o2)                      a mod(x, y)
          (%i3) mod (0, x);
          (%o3)                           0

Function: next_prime (<n>)

     Returns the smallest prime bigger than <n>.

          (%i1) next_prime(27);
          (%o1)                       29

Function: partfrac (<expr>, <var>)

     Expands the expression <expr> in partial fractions with respect to
     the main variable <var>.  `partfrac' does a complete partial
     fraction decomposition.  The algorithm employed is based on the
     fact that the denominators of the partial fraction expansion (the
     factors of the original denominator) are relatively prime.  The
     numerators can be written as linear combinations of denominators,
     and the expansion falls out.

          (%i1) 1/(1+x)^2 - 2/(1+x) + 2/(2+x);
                                2       2        1
          (%o1)               ----- - ----- + --------
                              x + 2   x + 1          2
                                              (x + 1)
          (%i2) ratsimp (%);
                                           x
          (%o2)                 - -------------------
                                   3      2
                                  x  + 4 x  + 5 x + 2
          (%i3) partfrac (%, x);
                                2       2        1
          (%o3)               ----- - ----- + --------
                              x + 2   x + 1          2
                                              (x + 1)

Function: power_mod (<a>, <n>, <m>)

     Uses a modular algorithm to compute `a^n mod m' where <a> and <n>
     are integers and <m> is a positive integer.  If <n> is negative,
     `inv_mod' is used to find the modular inverse.

          (%i1) power_mod(3, 15, 5);
          (%o1)                          2
          (%i2) mod(3^15,5);
          (%o2)                          2
          (%i3) power_mod(2, -1, 5);
          (%o3)                          3
          (%i4) inv_mod(2,5);
          (%o4)                          3

Function: primep (<n>)

     Primality test.  If `primep (<n>)' returns `false', <n> is a
     composite number and if it returns `true', <n> is a prime number
     with very high probability.

     For <n> less than 341550071728321 a deterministic version of
     Miller-Rabin's test is used.  If `primep (<n>)' returns `true',
     then <n> is a prime number.

     For <n> bigger than 341550071728321 `primep' uses
     `primep_number_of_tests' Miller-Rabin's pseudo-primality tests and
     one Lucas pseudo-primality test.  The probability that <n> will
     pass one Miller-Rabin test is less than 1/4.  Using the default
     value 25 for `primep_number_of_tests', the probability of <n>
     beeing composite is much smaller that 10^-15.

Option variable: primep_number_of_tests

     Default value: 25

     Number of Miller-Rabin's tests used in `primep'.

Function: prev_prime (<n>)

     Returns the greatest prime smaller than <n>.

          (%i1) prev_prime(27);
          (%o1)                       23

Function: qunit (<n>)

     Returns the principal unit of the real quadratic number field
     `sqrt (<n>)' where <n> is an integer, i.e., the element whose norm
     is unity.  This amounts to solving Pell's equation `a^2 - <n> b^2
     = 1'.

          (%i1) qunit (17);
          (%o1)                     sqrt(17) + 4
          (%i2) expand (% * (sqrt(17) - 4));
          (%o2)                           1

Function: totient (<n>)

     Returns the number of integers less than or equal to <n> which are
     relatively prime to <n>.

Option variable: zerobern

     Default value: `true'

     When `zerobern' is `false', `bern' excludes the Bernoulli numbers
     and `euler' excludes the Euler numbers which are equal to zero.
     See `bern' and `euler'.

Function: zeta (<n>)

     Returns the Riemann zeta function.  If <n> is a negative integer,
     0, or a positive even integer, the Riemann zeta function
     simplifies to an exact value.  For a positive even integer the
     option variable `zeta%pi' has to be `true' in addition (See
     `zeta%pi').  For a floating point or bigfloat number the Riemann
     zeta function is evaluated numerically.  Maxima returns a noun
     form `zeta (<n>)' for all other arguments, including rational
     noninteger, and complex arguments, or for even integers, if
     `zeta%pi' has the value `false'.

     `zeta(1)' is undefined, but Maxima knows the limit `limit(zeta(x),
     x, 1)' from above and below.

     The Riemann zeta function distributes over lists, matrices, and
     equations.

     See also `bfzeta' and `zeta%pi'.

     Examples:

          (%i1) zeta([-2, -1, 0, 0.5, 2, 3, 1+%i]);
                                                       2
                      1     1                       %pi
          (%o1) [0, - --, - -, - 1.460354508809586, ----, zeta(3),
                      12    2                        6
                                                              zeta(%i + 1)]
          (%i2) limit(zeta(x),x,1,plus);
          (%o2)                          inf
          (%i3) limit(zeta(x),x,1,minus);
          (%o3)                         minf

Option variable: zeta%pi

     Default value: `true'

     When `zeta%pi' is `true', `zeta' returns an expression
     proportional to `%pi^n' for even integer `n'.  Otherwise, `zeta'
     returns a noun form `zeta (n)' for even integer `n'.

     Examples:

          (%i1) zeta%pi: true$
          (%i2) zeta (4);
                                           4
                                        %pi
          (%o2)                         ----
                                         90
          (%i3) zeta%pi: false$
          (%i4) zeta (4);
          (%o4)                        zeta(4)

30 Symmetries


Introduction to Symmetries
Functions and Variables for Symmetries

30.1 Introduction to Symmetries

`sym' is a package for working with symmetric groups of polynomials.

   It was written for Macsyma-Symbolics by Annick Valibouze
(`http://www-calfor.lip6.fr/~avb/').  The algorithms are described in
the following papers:

  1. Fonctions syme'triques et changements de bases. Annick Valibouze.
     EUROCAL'87 (Leipzig, 1987), 323-332, Lecture Notes in Comput. Sci
     378.  Springer, Berlin, 1989.
     `http://www.stix.polytechnique.fr/publications/1984-1994.html'

  2. Re'solvantes et fonctions syme'triques. Annick Valibouze.
     Proceedings of the ACM-SIGSAM 1989 International Symposium on
     Symbolic and Algebraic Computation, ISSAC'89 (Portland, Oregon).
     ACM Press, 390-399, 1989.
     `http://www-calfor.lip6.fr/~avb/DonneesTelechargeables/MesArticles/issac89ACMValibouze.pdf'

  3. Symbolic computation with symmetric polynomials, an extension to
     Macsyma.  Annick Valibouze. Computers and Mathematics (MIT, USA,
     June 13-17, 1989), Springer-Verlag, New York Berlin, 308-320, 1989.
     `http://www.stix.polytechnique.fr/publications/1984-1994.html'

  4. The'orie de Galois Constructive. Annick Valibouze. Me'moire
     d'habilitation a` diriger les recherches (HDR), Universite' P. et
     M. Curie (Paris VI), 1994.

30.2 Functions and Variables for Symmetries

Function: comp2pui (<n>, <L>)

     implements passing from the complete symmetric functions given in
     the list <L> to the elementary symmetric functions from 0 to <n>.
     If the list <L> contains fewer than <n+1> elements, it will be
     completed with formal values of the type <h1>, <h2>, etc. If the
     first element of the list <L> exists, it specifies the size of the
     alphabet, otherwise the size is set to <n>.

          (%i1) comp2pui (3, [4, g]);
                                  2                    2
          (%o1)    [4, g, 2 h2 - g , 3 h3 - g h2 + g (g  - 2 h2)]

Function: ele2pui (<m>, <L>)

     goes from the elementary symmetric functions to the complete
     functions.  Similar to `comp2ele' and `comp2pui'.

     Other functions for changing bases: `comp2ele'.

Function: ele2comp (<m>, <L>)

     Goes from the elementary symmetric functions to the compete
     functions.  Similar to `comp2ele' and `comp2pui'.

     Other functions for changing bases: `comp2ele'.

Function: elem (<ele>, <sym>, <lvar>)

     decomposes the symmetric polynomial <sym>, in the variables
     contained in the list <lvar>, in terms of the elementary symmetric
     functions given in the list <ele>.  If the first element of <ele>
     is given, it will be the size of the alphabet, otherwise the size
     will be the degree of the polynomial <sym>.  If values are missing
     in the list <ele>, formal values of the type <e1>, <e2>, etc. will
     be added.  The polynomial <sym> may be given in three different
     forms: contracted (`elem' should then be 1, its default value),
     partitioned (`elem' should be 3), or extended (i.e. the entire
     polynomial, and `elem' should then be 2).  The function `pui' is
     used in the same way.

     On an alphabet of size 3 with <e1>, the first elementary symmetric
     function, with value 7, the symmetric polynomial in 3 variables
     whose contracted form (which here depends on only two of its
     variables) is <x^4-2*x*y> decomposes as follows in elementary
     symmetric functions:

          (%i1) elem ([3, 7], x^4 - 2*x*y, [x, y]);
          (%o1) 7 (e3 - 7 e2 + 7 (49 - e2)) + 21 e3

                                                   + (- 2 (49 - e2) - 2) e2
          (%i2) ratsimp (%);
                                        2
          (%o2)             28 e3 + 2 e2  - 198 e2 + 2401

     Other functions for changing bases: `comp2ele'.

Function: mon2schur (<L>)

     The list <L> represents the Schur function S_L: we have L = [i_1,
     i_2, ..., i_q], with i_1 <= i_2 <= ... <= i_q.  The Schur function
     S_[i_1, i_2, ..., i_q] is the minor of the infinite matrix
     h_[i-j], i <= 1, j <= 1, consisting of the q first rows and the
     columns 1 + i_1, 2 + i_2, ..., q + i_q.

     This Schur function can be written in terms of monomials by using
     `treinat' and `kostka'.  The form returned is a symmetric
     polynomial in a contracted representation in the variables
     x_1,x_2,...

          (%i1) mon2schur ([1, 1, 1]);
          (%o1)                       x1 x2 x3
          (%i2) mon2schur ([3]);
                                            2        3
          (%o2)                x1 x2 x3 + x1  x2 + x1
          (%i3) mon2schur ([1, 2]);
                                                2
          (%o3)                  2 x1 x2 x3 + x1  x2

     which means that for 3 variables this gives:

             2 x1 x2 x3 + x1^2 x2 + x2^2 x1 + x1^2 x3 + x3^2 x1
              + x2^2 x3 + x3^2 x2
     Other functions for changing bases: `comp2ele'.

Function: multi_elem (<l_elem>, <multi_pc>, <l_var>)

     decomposes a multi-symmetric polynomial in the multi-contracted
     form <multi_pc> in the groups of variables contained in the list
     of lists <l_var> in terms of the elementary symmetric functions
     contained in <l_elem>.

          (%i1) multi_elem ([[2, e1, e2], [2, f1, f2]], a*x + a^2 + x^3,
                [[x, y], [a, b]]);
                                                            3
          (%o1)         - 2 f2 + f1 (f1 + e1) - 3 e1 e2 + e1
          (%i2) ratsimp (%);
                                   2                       3
          (%o2)         - 2 f2 + f1  + e1 f1 - 3 e1 e2 + e1

     Other functions for changing bases: `comp2ele'.

Function: multi_pui

     is to the function `pui' what the function `multi_elem' is to the
     function `elem'.

          (%i1) multi_pui ([[2, p1, p2], [2, t1, t2]], a*x + a^2 + x^3,
                [[x, y], [a, b]]);
                                                      3
                                          3 p1 p2   p1
          (%o1)              t2 + p1 t1 + ------- - ---
                                             2       2

Function: pui (<L>, <sym>, <lvar>)

     decomposes the symmetric polynomial <sym>, in the variables in the
     list <lvar>, in terms of the power functions in the list <L>.  If
     the first element of <L> is given, it will be the size of the
     alphabet, otherwise the size will be the degree of the polynomial
     <sym>.  If values are missing in the list <L>, formal values of
     the type <p1>, <p2> , etc. will be added. The polynomial <sym> may
     be given in three different forms: contracted (`elem' should then
     be 1, its default value), partitioned (`elem' should be 3), or
     extended (i.e. the entire polynomial, and `elem' should then be
     2). The function `pui' is used in the same way.

          (%i1) pui;
          (%o1)                           1
          (%i2) pui ([3, a, b], u*x*y*z, [x, y, z]);
                                 2
                             a (a  - b) u   (a b - p3) u
          (%o2)              ------------ - ------------
                                  6              3
          (%i3) ratsimp (%);
                                                 3
                                (2 p3 - 3 a b + a ) u
          (%o3)                 ---------------------
                                          6
     Other functions for changing bases: `comp2ele'.

Function: pui2comp (<n>, <lpui>)

     renders the list of the first <n> complete functions (with the
     length first) in terms of the power functions given in the list
     <lpui>. If the list <lpui> is empty, the cardinal is <n>,
     otherwise it is its first element (as in `comp2ele' and
     `comp2pui').

          (%i1) pui2comp (2, []);
                                                 2
                                          p2 + p1
          (%o1)                   [2, p1, --------]
                                             2
          (%i2) pui2comp (3, [2, a1]);
                                                      2
                                           a1 (p2 + a1 )
                                   2  p3 + ------------- + a1 p2
                            p2 + a1              2
          (%o2)     [2, a1, --------, --------------------------]
                               2                  3
          (%i3) ratsimp (%);
                                      2                     3
                               p2 + a1   2 p3 + 3 a1 p2 + a1
          (%o3)        [2, a1, --------, --------------------]
                                  2               6
     Other functions for changing bases: `comp2ele'.

Function: pui2ele (<n>, <lpui>)

     effects the passage from power functions to the elementary
     symmetric functions.  If the flag `pui2ele' is `girard', it will
     return the list of elementary symmetric functions from 1 to <n>,
     and if the flag is `close', it will return the <n>-th elementary
     symmetric function.

     Other functions for changing bases: `comp2ele'.

Function: puireduc (<n>, <lpui>)

     <lpui> is a list whose first element is an integer <m>.
     `puireduc' gives the first <n> power functions in terms of the
     first <m>.

          (%i1) puireduc (3, [2]);
                                                   2
                                             p1 (p1  - p2)
          (%o1)          [2, p1, p2, p1 p2 - -------------]
                                                   2
          (%i2) ratsimp (%);
                                                     3
                                         3 p1 p2 - p1
          (%o2)              [2, p1, p2, -------------]
                                               2

Function: schur2comp (<P>, <l_var>)

     <P> is a polynomial in the variables of the list <l_var>.  Each of
     these variables represents a complete symmetric function.  In
     <l_var> the <i>-th complete symmetric function is represented by
     the concatenation of the letter `h' and the integer <i>: `h<i>'.
     This function expresses <P> in terms of Schur functions.

          (%i1) schur2comp (h1*h2 - h3, [h1, h2, h3]);
          (%o1)                         s
                                         1, 2
          (%i2) schur2comp (a*h3, [h3]);
          (%o2)                         s  a
                                         3

Function: cont2part (<pc>, <lvar>)

     returns the partitioned polynomial associated to the contracted
     form <pc> whose variables are in <lvar>.

          (%i1) pc: 2*a^3*b*x^4*y + x^5;
                                     3    4      5
          (%o1)                   2 a  b x  y + x
          (%i2) cont2part (pc, [x, y]);
                                             3
          (%o2)              [[1, 5, 0], [2 a  b, 4, 1]]

Function: contract (<psym>, <lvar>)

     returns a contracted form (i.e. a monomial orbit under the action
     of the symmetric group) of the polynomial <psym> in the variables
     contained in the list <lvar>.  The function `explose' performs the
     inverse operation.  The function `tcontract' tests the symmetry of
     the polynomial.

          (%i1) psym: explose (2*a^3*b*x^4*y, [x, y, z]);
                   3      4      3      4      3    4        3    4
          (%o1) 2 a  b y z  + 2 a  b x z  + 2 a  b y  z + 2 a  b x  z

                                                     3      4      3    4
                                                + 2 a  b x y  + 2 a  b x  y
          (%i2) contract (psym, [x, y, z]);
                                        3    4
          (%o2)                      2 a  b x  y

Function: explose (<pc>, <lvar>)

     returns the symmetric polynomial associated with the contracted
     form <pc>. The list <lvar> contains the variables.

          (%i1) explose (a*x + 1, [x, y, z]);
          (%o1)                  a z + a y + a x + 1

Function: part2cont (<ppart>, <lvar>)

     goes from the partitioned form to the contracted form of a
     symmetric polynomial.  The contracted form is rendered with the
     variables in <lvar>.

          (%i1) part2cont ([[2*a^3*b, 4, 1]], [x, y]);
                                        3    4
          (%o1)                      2 a  b x  y

Function: partpol (<psym>, <lvar>)

     <psym> is a symmetric polynomial in the variables of the list
     <lvar>. This function retturns its partitioned representation.

          (%i1) partpol (-a*(x + y) + 3*x*y, [x, y]);
          (%o1)               [[3, 1, 1], [- a, 1, 0]]

Function: tcontract (<pol>, <lvar>)

     tests if the polynomial <pol> is symmetric in the variables of the
     list <lvar>.  If so, it returns a contracted representation like
     the function `contract'.

Function: tpartpol (<pol>, <lvar>)

     tests if the polynomial <pol> is symmetric in the variables of the
     list <lvar>.  If so, it returns its partitioned representation like
     the function `partpol'.

Function: direct ([<p_1>, ..., <p_n>], <y>, <f>, [<lvar_1>, ..., <lvar_n>])

     calculates the direct image (see M. Giusti, D. Lazard et A.
     Valibouze, ISSAC 1988, Rome) associated to the function <f>, in
     the lists of variables <lvar_1>, ..., <lvar_n>, and in the
     polynomials <p_1>, ..., <p_n> in a variable <y>.  The arity of the
     function <f> is important for the calulation.  Thus, if the
     expression for <f> does not depend on some variable, it is useless
     to include this variable, and not including it will also
     considerably reduce the amount of computation.

          (%i1) direct ([z^2  - e1* z + e2, z^2  - f1* z + f2],
                        z, b*v + a*u, [[u, v], [a, b]]);
                 2
          (%o1) y  - e1 f1 y

                                           2            2             2   2
                            - 4 e2 f2 - (e1  - 2 e2) (f1  - 2 f2) + e1  f1
                          + -----------------------------------------------
                                                   2
          (%i2) ratsimp (%);
                        2                2                   2
          (%o2)        y  - e1 f1 y + (e1  - 4 e2) f2 + e2 f1
          (%i3) ratsimp (direct ([z^3-e1*z^2+e2*z-e3,z^2  - f1* z + f2],
                        z, b*v + a*u, [[u, v], [a, b]]));
                 6            5         2                        2    2   4
          (%o3) y  - 2 e1 f1 y  + ((2 e1  - 6 e2) f2 + (2 e2 + e1 ) f1 ) y

                                    3                               3   3
           + ((9 e3 + 5 e1 e2 - 2 e1 ) f1 f2 + (- 2 e3 - 2 e1 e2) f1 ) y

                   2       2        4    2
           + ((9 e2  - 6 e1  e2 + e1 ) f2

                              2       2       2                   2    4
           + (- 9 e1 e3 - 6 e2  + 3 e1  e2) f1  f2 + (2 e1 e3 + e2 ) f1 )

            2          2                      2     3          2
           y  + (((9 e1  - 27 e2) e3 + 3 e1 e2  - e1  e2) f1 f2

                           2            2    3                5
           + ((15 e2 - 2 e1 ) e3 - e1 e2 ) f1  f2 - 2 e2 e3 f1 ) y

                     2                   3           3     2   2    3
           + (- 27 e3  + (18 e1 e2 - 4 e1 ) e3 - 4 e2  + e1  e2 ) f2

                   2      3                   3    2   2
           + (27 e3  + (e1  - 9 e1 e2) e3 + e2 ) f1  f2

                             2    4        2   6
           + (e1 e2 e3 - 9 e3 ) f1  f2 + e3  f1

     Finding the polynomial whose roots are the sums a+u where a is a
     root of z^2 - e_1 z + e_2 and u is a root of z^2 - f_1 z + f_2.

          (%i1) ratsimp (direct ([z^2 - e1* z + e2, z^2 - f1* z + f2],
                                    z, a + u, [[u], [a]]));
                 4                    3             2
          (%o1) y  + (- 2 f1 - 2 e1) y  + (2 f2 + f1  + 3 e1 f1 + 2 e2

               2   2                              2               2
           + e1 ) y  + ((- 2 f1 - 2 e1) f2 - e1 f1  + (- 2 e2 - e1 ) f1

                            2                     2            2
           - 2 e1 e2) y + f2  + (e1 f1 - 2 e2 + e1 ) f2 + e2 f1  + e1 e2 f1

               2
           + e2

     `direct' accepts two flags: `elementaires' and `puissances'
     (default) which allow decomposing the symmetric polynomials
     appearing in the calculation into elementary symmetric functions,
     or power functions, respectively.

     Functions of `sym' used in this function:

     `multi_orbit' (so `orbit'), `pui_direct', `multi_elem' (so
     `elem'), `multi_pui' (so `pui'), `pui2ele', `ele2pui' (if the flag
     `direct' is in `puissances').

Function: multi_orbit (<P>, [<lvar_1>, <lvar_2>,..., <lvar_p>])

     <P> is a polynomial in the set of variables contained in the lists
     <lvar_1>, <lvar_2>, ..., <lvar_p>. This function returns the orbit
     of the polynomial <P> under the action of the product of the
     symmetric groups of the sets of variables represented in these <p>
     lists.

          (%i1) multi_orbit (a*x + b*y, [[x, y], [a, b]]);
          (%o1)                [b y + a x, a y + b x]
          (%i2) multi_orbit (x + y + 2*a, [[x, y], [a, b, c]]);
          (%o2)        [y + x + 2 c, y + x + 2 b, y + x + 2 a]
     Also see: `orbit' for the action of a single symmetric group.

Function: multsym (<ppart_1>, <ppart_2>, <n>)

     returns the product of the two symmetric polynomials in <n>
     variables by working only modulo the action of the symmetric group
     of order <n>. The polynomials are in their partitioned form.

     Given the 2 symmetric polynomials in <x>, <y>:  `3*(x + y) +
     2*x*y' and `5*(x^2 + y^2)' whose partitioned forms are `[[3, 1],
     [2, 1, 1]]' and `[[5, 2]]', their product will be

          (%i1) multsym ([[3, 1], [2, 1, 1]], [[5, 2]], 2);
          (%o1)         [[10, 3, 1], [15, 3, 0], [15, 2, 1]]
     that is `10*(x^3*y + y^3*x) + 15*(x^2*y + y^2*x) + 15*(x^3 + y^3)'.

     Functions for changing the representations of a symmetric
     polynomial:

     `contract', `cont2part', `explose', `part2cont', `partpol',
     `tcontract', `tpartpol'.

Function: orbit (<P>, <lvar>)

     computes the orbit of the polynomial <P> in the variables in the
     list <lvar> under the action of the symmetric group of the set of
     variables in the list <lvar>.

          (%i1) orbit (a*x + b*y, [x, y]);
          (%o1)                [a y + b x, b y + a x]
          (%i2) orbit (2*x + x^2, [x, y]);
                                  2         2
          (%o2)                 [y  + 2 y, x  + 2 x]
     See also `multi_orbit' for the action of a product of symmetric
     groups on a polynomial.

Function: pui_direct (<orbite>, [<lvar_1>, ..., <lvar_n>], [<d_1>, <d_2>, ..., <d_n>])

     Let <f> be a polynomial in <n> blocks of variables <lvar_1>, ...,
     <lvar_n>.  Let <c_i> be the number of variables in <lvar_i>, and
     <SC> be the product of <n> symmetric groups of degree <c_1>, ...,
     <c_n>. This group acts naturally on <f>.  The list <orbite> is the
     orbit, denoted `<SC>(<f>)', of the function <f> under the action
     of <SC>. (This list may be obtained by the function
     `multi_orbit'.)  The <di> are integers s.t.  c_1 <= d_1, c_2 <=
     d_2, ..., c_n <= d_n.

     Let <SD> be the product of the symmetric groups S_[d_1] x S_[d_2]
     x ... x S_[d_n].  The function `pui_direct' returns the first <n>
     power functions of `<SD>(<f>)' deduced from the power functions of
     `<SC>(<f>)', where <n> is the size of `<SD>(<f>)'.

     The result is in multi-contracted form w.r.t. <SD>, i.e. only one
     element is kept per orbit, under the action of <SD>.

          (%i1) l: [[x, y], [a, b]];
          (%o1)                   [[x, y], [a, b]]
          (%i2) pui_direct (multi_orbit (a*x + b*y, l), l, [2, 2]);
                                                 2  2
          (%o2)               [a x, 4 a b x y + a  x ]
          (%i3) pui_direct (multi_orbit (a*x + b*y, l), l, [3, 2]);
                                       2  2     2    2        3  3
          (%o3) [2 a x, 4 a b x y + 2 a  x , 3 a  b x  y + 2 a  x ,

              2  2  2  2      3    3        4  4
          12 a  b  x  y  + 4 a  b x  y + 2 a  x ,

              3  2  3  2      4    4        5  5
          10 a  b  x  y  + 5 a  b x  y + 2 a  x ,

              3  3  3  3       4  2  4  2      5    5        6  6
          40 a  b  x  y  + 15 a  b  x  y  + 6 a  b x  y + 2 a  x ]
          (%i4) pui_direct ([y + x + 2*c, y + x + 2*b, y + x + 2*a],
                [[x, y], [a, b, c]], [2, 3]);
                                       2              2
          (%o4) [3 x + 2 a, 6 x y + 3 x  + 4 a x + 4 a ,

                           2                   3        2       2        3
                        9 x  y + 12 a x y + 3 x  + 6 a x  + 12 a  x + 8 a ]

Function: kostka (<part_1>, <part_2>)

     written by P. Esperet, calculates the Kostka number of the
     partition <part_1> and <part_2>.

          (%i1) kostka ([3, 3, 3], [2, 2, 2, 1, 1, 1]);
          (%o1)                           6

Function: lgtreillis (<n>, <m>)

     returns the list of partitions of weight <n> and length <m>.

          (%i1) lgtreillis (4, 2);
          (%o1)                   [[3, 1], [2, 2]]
     Also see: `ltreillis', `treillis' and `treinat'.

Function: ltreillis (<n>, <m>)

     returns the list of partitions of weight <n> and length less than
     or equal to <m>.

          (%i1) ltreillis (4, 2);
          (%o1)               [[4, 0], [3, 1], [2, 2]]
     Also see: `lgtreillis', `treillis' and `treinat'.

Function: treillis (<n>)

     returns all partitions of weight <n>.

          (%i1) treillis (4);
          (%o1)    [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]

     See also: `lgtreillis', `ltreillis' and `treinat'.

Function: treinat (<part>)

     retruns the list of partitions inferior to the partition <part>
     w.r.t.  the natural order.

          (%i1) treinat ([5]);
          (%o1)                         [[5]]
          (%i2) treinat ([1, 1, 1, 1, 1]);
          (%o2) [[5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1],

                                                           [1, 1, 1, 1, 1]]
          (%i3) treinat ([3, 2]);
          (%o3)                 [[5], [4, 1], [3, 2]]

     See also: `lgtreillis', `ltreillis' and `treillis'.

Function: ele2polynome (<L>, <z>)

     returns the polynomial in <z> s.t. the elementary symmetric
     functions of its roots are in the list `<L> = [<n>, <e_1>, ...,
     <e_n>]', where <n> is the degree of the polynomial and <e_i> the
     <i>-th elementary symmetric function.

          (%i1) ele2polynome ([2, e1, e2], z);
                                    2
          (%o1)                    z  - e1 z + e2
          (%i2) polynome2ele (x^7 - 14*x^5 + 56*x^3  - 56*x + 22, x);
          (%o2)          [7, 0, - 14, 0, 56, 0, - 56, - 22]
          (%i3) ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x);
                            7       5       3
          (%o3)            x  - 14 x  + 56 x  - 56 x + 22
     The inverse: `polynome2ele (<P>, <z>)'.

     Also see: `polynome2ele', `pui2polynome'.

Function: polynome2ele (<P>, <x>)

     gives the list `<l> = [<n>, <e_1>, ..., <e_n>]' where <n> is the
     degree of the polynomial <P> in the variable <x> and <e_i> is the
     <i>-the elementary symmetric function of the roots of <P>.

          (%i1) polynome2ele (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x);
          (%o1)          [7, 0, - 14, 0, 56, 0, - 56, - 22]
          (%i2) ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x);
                            7       5       3
          (%o2)            x  - 14 x  + 56 x  - 56 x + 22
     The inverse: `ele2polynome (<l>, <x>)'

Function: prodrac (<L>, <k>)

     <L> is a list containing the elementary symmetric functions on a
     set <A>. `prodrac' returns the polynomial whose roots are the <k>
     by <k> products of the elements of <A>.

     Also see `somrac'.

Function: pui2polynome (<x>, <lpui>)

     calculates the polynomial in <x> whose power functions of the roots
     are given in the list <lpui>.

          (%i1) pui;
          (%o1)                           1
          (%i2) kill(labels);
          (%o0)                         done
          (%i1) polynome2ele (x^3 - 4*x^2 + 5*x - 1, x);
          (%o1)                     [3, 4, 5, 1]
          (%i2) ele2pui (3, %);
          (%o2)                     [3, 4, 6, 7]
          (%i3) pui2polynome (x, %);
                                  3      2
          (%o3)                  x  - 4 x  + 5 x - 1
     See also: `polynome2ele', `ele2polynome'.

Function: somrac (<L>, <k>)

     The list <L> contains elementary symmetric functions of a
     polynomial <P> . The function computes the polynomial whose roots
     are the <k> by <k> distinct sums of the roots of <P>.

     Also see `prodrac'.

Function: resolvante (<P>, <x>, <f>, [<x_1>,..., <x_d>])

     calculates the resolvent of the polynomial <P> in <x> of degree
     <n> >= <d> by the function <f> expressed in the variables <x_1>,
     ..., <x_d>.  For efficiency of computation it is important to not
     include in the list `[<x_1>, ..., <x_d>]' variables which do not
     appear in the transformation function <f>.

     To increase the efficiency of the computation one may set flags in
     `resolvante' so as to use appropriate algorithms:

     If the function <f> is unitary:
        * A polynomial in a single variable,

        *   linear,

        *   alternating,

        *   a sum,

        *   symmetric,

        *   a product,

        * the function of the Cayley resolvent (usable up to degree 5)

               (x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x1 -
                    (x1*x3 + x3*x5 + x5*x2 + x2*x4 + x4*x1))^2

          general,
     the flag of `resolvante' may be, respectively:
        *   unitaire,

        *   lineaire,

        *   alternee,

        *   somme,

        *   produit,

        *   cayley,

        *   generale.

          (%i1) resolvante: unitaire$
          (%i2) resolvante (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x, x^3 - 1,
                [x]);

          " resolvante unitaire " [7, 0, 28, 0, 168, 0, 1120, - 154, 7840,
                                   - 2772, 56448, - 33880,

          413952, - 352352, 3076668, - 3363360, 23114112, - 30494464,

          175230832, - 267412992, 1338886528, - 2292126760]
            3       6      3       9      6      3
          [x  - 1, x  - 2 x  + 1, x  - 3 x  + 3 x  - 1,

           12      9      6      3       15      12       9       6      3
          x   - 4 x  + 6 x  - 4 x  + 1, x   - 5 x   + 10 x  - 10 x  + 5 x

                 18      15       12       9       6      3
           - 1, x   - 6 x   + 15 x   - 20 x  + 15 x  - 6 x  + 1,

           21      18       15       12       9       6      3
          x   - 7 x   + 21 x   - 35 x   + 35 x  - 21 x  + 7 x  - 1]
          [- 7, 1127, - 6139, 431767, - 5472047, 201692519, - 3603982011]
                 7      6        5         4          3           2
          (%o2) y  + 7 y  - 539 y  - 1841 y  + 51443 y  + 315133 y

                                                        + 376999 y + 125253
          (%i3) resolvante: lineaire$
          (%i4) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]);

          " resolvante lineaire "
                 24       20         16            12             8
          (%o4) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                              4
                                                 + 344489984 y  + 655360000
          (%i5) resolvante: general$
          (%i6) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]);

          " resolvante generale "
                 24       20         16            12             8
          (%o6) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                              4
                                                 + 344489984 y  + 655360000
          (%i7) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3, x4]);

          " resolvante generale "
                 24       20         16            12             8
          (%o7) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                              4
                                                 + 344489984 y  + 655360000
          (%i8) direct ([x^4 - 1], x, x1 + 2*x2 + 3*x3, [[x1, x2, x3]]);
                 24       20         16            12             8
          (%o8) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                              4
                                                 + 344489984 y  + 655360000
          (%i9) resolvante :lineaire$
          (%i10) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]);

          " resolvante lineaire "
                                        4
          (%o10)                       y  - 1
          (%i11) resolvante: symetrique$
          (%i12) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]);

          " resolvante symetrique "
                                        4
          (%o12)                       y  - 1
          (%i13) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]);

          " resolvante symetrique "
                                     6      2
          (%o13)                    y  - 4 y  - 1
          (%i14) resolvante: alternee$
          (%i15) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]);

          " resolvante alternee "
                      12      8       6        4        2
          (%o15)     y   + 8 y  + 26 y  - 112 y  + 216 y  + 229
          (%i16) resolvante: produit$
          (%i17) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]);

          " resolvante produit "
                  35      33         29        28         27        26
          (%o17) y   - 7 y   - 1029 y   + 135 y   + 7203 y   - 756 y

                   24           23          22            21           20
           + 1323 y   + 352947 y   - 46305 y   - 2463339 y   + 324135 y

                    19           18             17              15
           - 30618 y   - 453789 y   - 40246444 y   + 282225202 y

                       14              12             11            10
           - 44274492 y   + 155098503 y   + 12252303 y   + 2893401 y

                        9            8            7             6
           - 171532242 y  + 6751269 y  + 2657205 y  - 94517766 y

                      5             3
           - 3720087 y  + 26040609 y  + 14348907
          (%i18) resolvante: symetrique$
          (%i19) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]);

          " resolvante symetrique "
                  35      33         29        28         27        26
          (%o19) y   - 7 y   - 1029 y   + 135 y   + 7203 y   - 756 y

                   24           23          22            21           20
           + 1323 y   + 352947 y   - 46305 y   - 2463339 y   + 324135 y

                    19           18             17              15
           - 30618 y   - 453789 y   - 40246444 y   + 282225202 y

                       14              12             11            10
           - 44274492 y   + 155098503 y   + 12252303 y   + 2893401 y

                        9            8            7             6
           - 171532242 y  + 6751269 y  + 2657205 y  - 94517766 y

                      5             3
           - 3720087 y  + 26040609 y  + 14348907
          (%i20) resolvante: cayley$
          (%i21) resolvante (x^5 - 4*x^2 + x + 1, x, a, []);

          " resolvante de Cayley "
                  6       5         4          3            2
          (%o21) x  - 40 x  + 4080 x  - 92928 x  + 3772160 x  + 37880832 x

                                                                 + 93392896

     For the Cayley resolvent, the 2 last arguments are neutral and the
     input polynomial must necessarily be of degree 5.

     See also:
     `resolvante_bipartite', `resolvante_produit_sym',
     `resolvante_unitaire', `resolvante_alternee1', `resolvante_klein',
     `resolvante_klein3', `resolvante_vierer', `resolvante_diedrale'.

Function: resolvante_alternee1 (<P>, <x>)

     calculates the transformation `<P>(<x>)' of degree <n> by the
     function product(x_i - x_j, 1 <= i < j <= n - 1).

     See also:
     `resolvante_produit_sym', `resolvante_unitaire',
     `resolvante' , `resolvante_klein', `resolvante_klein3',
     `resolvante_vierer', `resolvante_diedrale', `resolvante_bipartite'.

Function: resolvante_bipartite (<P>, <x>)

     calculates the transformation of `<P>(<x>)' of even degree <n> by
     the function x_1 x_2 ... x_[n/2] + x_[n/2 + 1] ... x_n.

          (%i1) resolvante_bipartite (x^6 + 108, x);
                        10        8           6             4
          (%o1)        y   - 972 y  + 314928 y  - 34012224 y

     See also:
     `resolvante_produit_sym', `resolvante_unitaire',
     `resolvante', `resolvante_klein', `resolvante_klein3',
     `resolvante_vierer', `resolvante_diedrale', `resolvante_alternee1'.

Function: resolvante_diedrale (<P>, <x>)

     calculates the transformation of `<P>(<x>)' by the function `<x_1>
     <x_2> + <x_3> <x_4>'.

          (%i1) resolvante_diedrale (x^5 - 3*x^4 + 1, x);
                 15       12       11       10        9         8         7
          (%o1) x   - 21 x   - 81 x   - 21 x   + 207 x  + 1134 x  + 2331 x

                  6         5          4          3          2
           - 945 x  - 4970 x  - 18333 x  - 29079 x  - 20745 x  - 25326 x

           - 697

     See also:
     `resolvante_produit_sym', `resolvante_unitaire',
     `resolvante_alternee1', `resolvante_klein', `resolvante_klein3',
     `resolvante_vierer', `resolvante'.

Function: resolvante_klein (<P>, <x>)

     calculates the transformation of `<P>(<x>)' by the function `<x_1>
     <x_2> <x_4> + <x_4>'.

     See also:
     `resolvante_produit_sym', `resolvante_unitaire',
     `resolvante_alternee1', `resolvante', `resolvante_klein3',
     `resolvante_vierer', `resolvante_diedrale'.

Function: resolvante_klein3 (<P>, <x>)

     calculates the transformation of `<P>(<x>)' by the function `<x_1>
     <x_2> <x_4> + <x_4>'.

     See also:
     `resolvante_produit_sym', `resolvante_unitaire',
     `resolvante_alternee1', `resolvante_klein', `resolvante',
     `resolvante_vierer', `resolvante_diedrale'.

Function: resolvante_produit_sym (<P>, <x>)

     calculates the list of all product resolvents of the polynomial
     `<P>(<x>)'.

          (%i1) resolvante_produit_sym (x^5 + 3*x^4 + 2*x - 1, x);
                  5      4             10      8       7       6       5
          (%o1) [y  + 3 y  + 2 y - 1, y   - 2 y  - 21 y  - 31 y  - 14 y

              4       3      2       10      8       7    6       5       4
           - y  + 14 y  + 3 y  + 1, y   + 3 y  + 14 y  - y  - 14 y  - 31 y

                 3      2       5      4
           - 21 y  - 2 y  + 1, y  - 2 y  - 3 y - 1, y - 1]
          (%i2) resolvante: produit$
          (%i3) resolvante (x^5 + 3*x^4 + 2*x - 1, x, a*b*c, [a, b, c]);

          " resolvante produit "
                 10      8       7    6        5       4       3     2
          (%o3) y   + 3 y  + 14 y  - y  - 14 y  - 31 y  - 21 y  - 2 y  + 1

     See also:
     `resolvante', `resolvante_unitaire',
     `resolvante_alternee1', `resolvante_klein',
     `resolvante_klein3', `resolvante_vierer',
     `resolvante_diedrale'.

Function: resolvante_unitaire (<P>, <Q>, <x>)

     computes the resolvent of the polynomial `<P>(<x>)' by the
     polynomial `<Q>(<x>)'.

     See also:
     `resolvante_produit_sym', `resolvante',
     `resolvante_alternee1', `resolvante_klein', `resolvante_klein3',
     `resolvante_vierer', `resolvante_diedrale'.

Function: resolvante_vierer (<P>, <x>)

     computes the transformation of `<P>(<x>)' by the function `<x_1>
     <x_2> - <x_3> <x_4>'.

     See also:
     `resolvante_produit_sym', `resolvante_unitaire',
     `resolvante_alternee1', `resolvante_klein', `resolvante_klein3',
     `resolvante', `resolvante_diedrale'.

Function: multinomial (<r>, <part>)

     where <r> is the weight of the partition <part>.  This function
     returns the associate multinomial coefficient: if the parts of
     <part> are <i_1>, <i_2>, ..., <i_k>, the result is `<r>!/(<i_1>!
     <i_2>! ... <i_k>!)'.

Function: permut (<L>)

     returns the list of permutations of the list <L>.

31 Groups


Functions and Variables for Groups

31.1 Functions and Variables for Groups

Function: todd_coxeter (<relations>, <subgroup>)
Function: todd_coxeter (<relations>)

     Find the order of G/H where G is the Free Group modulo
     <relations>, and H is the subgroup of G generated by <subgroup>.
     <subgroup> is an optional argument, defaulting to [].  In doing
     this it produces a multiplication table for the right action of G
     on G/H, where the cosets are enumerated [H,Hg2,Hg3,...].  This can
     be seen internally in the variable `todd_coxeter_state'.

     Example:

          (%i1) symet(n):=create_list(
                  if (j - i) = 1 then (p(i,j))^^3 else
                      if (not i = j) then (p(i,j))^^2 else
                          p(i,i) , j, 1, n-1, i, 1, j);
                                                                 <3>
          (%o1) symet(n) := create_list(if j - i = 1 then p(i, j)

                                          <2>
           else (if not i = j then p(i, j)    else p(i, i)), j, 1, n - 1,

          i, 1, j)
          (%i2) p(i,j) := concat(x,i).concat(x,j);
          (%o2)        p(i, j) := concat(x, i) . concat(x, j)
          (%i3) symet(5);
                   <2>           <3>    <2>           <2>           <3>
          (%o3) [x1   , (x1 . x2)   , x2   , (x1 . x3)   , (x2 . x3)   ,

                      <2>           <2>           <2>           <3>    <2>
                    x3   , (x1 . x4)   , (x2 . x4)   , (x3 . x4)   , x4   ]
          (%i4) todd_coxeter(%o3);

          Rows tried 426
          (%o4)                          120
          (%i5) todd_coxeter(%o3,[x1]);

          Rows tried 213
          (%o5)                          60
          (%i6) todd_coxeter(%o3,[x1,x2]);

          Rows tried 71
          (%o6)                          20

32 Runtime Environment


Introduction for Runtime Environment
Interrupts
Functions and Variables for Runtime Environment

32.1 Introduction for Runtime Environment

`maxima-init.mac' is a file which is loaded automatically when Maxima
starts.  You can use `maxima-init.mac' to customize your Maxima
environment.  `maxima-init.mac', if it exists, is typically placed in
the directory named by `maxima_userdir', although it can be in any
directory searched by the function `file_search'.

   Here is an example `maxima-init.mac' file:

     setup_autoload ("specfun.mac", ultraspherical, assoc_legendre_p);
     showtime:all;

   In this example, `setup_autoload' tells Maxima to load the specified
file (`specfun.mac') if any of the functions (`ultraspherical',
`assoc_legendre_p') are called but not yet defined.  Thus you needn't
remember to load the file before calling the functions.

   The statement `showtime: all' tells Maxima to set the `showtime'
variable.  The `maxima-init.mac' file can contain any other assignments
or other Maxima statements.

32.2 Interrupts

The user can stop a time-consuming computation with the ^C (control-C)
character.  The default action is to stop the computation and print
another user prompt.  In this case, it is not possible to restart a
stopped computation.

   If the Lisp variable `*debugger-hook*' is set to `nil', by executing

     :lisp (setq *debugger-hook* nil)

then upon receiving ^C, Maxima will enter the Lisp debugger, and the
user may use the debugger to inspect the Lisp environment.  The stopped
computation can be restarted by entering `continue' in the Lisp
debugger.  The means of returning to Maxima from the Lisp debugger
(other than running the computation to completion) is different for
each version of Lisp.

   On Unix systems, the character ^Z (control-Z) causes Maxima to stop
altogether, and control is returned to the shell prompt.  The `fg'
command causes Maxima to resume from the point at which it was stopped.

32.3 Functions and Variables for Runtime Environment

System variable: maxima_tempdir

     `maxima_tempdir' names the directory in which Maxima creates some
     temporary files.  In particular, temporary files for plotting are
     created in `maxima_tempdir'.

     The initial value of `maxima_tempdir' is the user's home
     directory, if Maxima can locate it; otherwise Maxima makes a guess
     about a suitable directory.

     `maxima_tempdir' may be assigned a string which names a directory.

System variable: maxima_userdir

     `maxima_userdir' names a directory which Maxima searches to find
     Maxima and Lisp files.  (Maxima searches some other directories as
     well; `file_search_maxima' and `file_search_lisp' are the complete
     lists.)

     The initial value of `maxima_userdir' is a subdirectory of the
     user's home directory, if Maxima can locate it; otherwise Maxima
     makes a guess about a suitable directory.

     `maxima_userdir' may be assigned a string which names a directory.
     However, assigning to `maxima_userdir' does not automatically
     change `file_search_maxima' and `file_search_lisp'; those
     variables must be changed separately.

Function: room ()
Function: room (true)
Function: room (false)

     Prints out a description of the state of storage and stack
     management in Maxima.  `room' calls the Lisp function of the same
     name.

        * `room ()' prints out a moderate description.

        * `room (true)' prints out a verbose description.

        * `room (false)' prints out a terse description.

Function: sstatus (<keyword>, <item>)

     When <keyword> is the symbol `feature', <item> is put on the list
     of system features.  After `sstatus (keyword, item)' is executed,
     `status (feature, item)' returns `true'.  If <keyword> is the
     symbol `nofeature', <item> is deleted from the list of system
     features.  This can be useful for package writers, to keep track
     of what features they have loaded in.

     See also `status'.

Function: status (`feature')
Function: status (`feature', <item>)

     Returns information about the presence or absence of certain
     system-dependent features.

        * `status (feature)' returns a list of system features. These
          include Lisp version, operating system type, etc. The list
          may vary from one Lisp type to another.

        * `status (feature, item)' returns `true' if <item> is on the
          list of items returned by `status (feature)' and `false'
          otherwise.  `status' quotes the argument <item>. The
          quote-quote operator `''' defeats quotation. A feature whose
          name contains a special character, such as a hyphen, must be
          given as a string argument. For example, `status (feature,
          "ansi-cl")'.

     See also `sstatus'.

     The variable `features' contains a list of features which apply to
     mathematical expressions. See `features' and `featurep' for more
     information.

Function: system (<command>)

     Executes <command> as a separate process.  The command is passed
     to the default shell for execution.  `system' is not supported by
     all operating systems, but generally exists in Unix and Unix-like
     environments.

     Supposing `_hist.out' is a list of frequencies which you wish to
     plot as a bar graph using `xgraph'.

          (%i1) (with_stdout("_hist.out",
                     for i:1 thru length(hist) do (
                       print(i,hist[i]))),
                 system("xgraph -bar -brw .7 -nl < _hist.out"));

     In order to make the plot be done in the background (returning
     control to Maxima) and remove the temporary file after it is done
     do:

          system("(xgraph -bar -brw .7 -nl < _hist.out;  rm -f _hist.out)&")

Function: time (%o1, %o2, %o3, ...)

     Returns a list of the times, in seconds, taken to compute the
     output lines `%o1', `%o2', `%o3', ... The time returned is Maxima's
     estimate of the internal computation time, not the elapsed time.
     `time' can only be applied to output line variables; for any other
     variables, `time' returns `unknown'.

     Set `showtime: true' to make Maxima print out the computation time
     and elapsed time with each output line.

Function: timedate ()
Function: timedate (<T>)

     `timedate()' with no argument returns a string representing the
     current time and date.  The string has the format `YYYY-MM-DD
     HH:MM:SS[+|-]ZZ:ZZ', where the fields are year, month, day, hours,
     minutes, seconds, and time zone offset in hours and minutes.

     `timedate(<T>)' returns the time <T> as a string with the format
     `YYYY-MM-DD HH:MM:SS[+|-]ZZ:ZZ'.  <T> is interpreted as the number
     of seconds since midnight, January 1, 1900, as returned by
     `absolute_real_time'.

     Example:

     `timedate' with no argument returns a string representing the
     current time and date.

          (%i1) d : timedate ();
          (%o1)                      2010-06-08 04:08:09+01:00
          (%i2) print ("timedate reports current time", d) $
          timedate reports current time 2010-06-08 04:08:09+01:00

     `timedate' with an argument returns a string representing the
     argument.

          (%i1) timedate (0);
          (%o1)                      1900-01-01 01:00:00+01:00
          (%i2) timedate (absolute_real_time () - 7*24*3600);
          (%o2)                      2010-06-01 04:19:51+01:00

Function: absolute_real_time ()

     Returns the number of seconds since midnight, January 1, 1900 UTC.
     The return value is an integer.

     See also `elapsed_real_time' and `elapsed_run_time'.

     Example:

          (%i1) absolute_real_time ();
          (%o1)                      3385045277
          (%i2) 1900 + absolute_real_time () / (365.25 * 24 * 3600);
          (%o2)                   2007.265612087104

Function: elapsed_real_time ()

     Returns the number of seconds (including fractions of a second)
     since Maxima was most recently started or restarted.  The return
     value is a floating-point number.

     See also `absolute_real_time' and `elapsed_run_time'.

     Example:

          (%i1) elapsed_real_time ();
          (%o1)                       2.559324
          (%i2) expand ((a + b)^500)$
          (%i3) elapsed_real_time ();
          (%o3)                       7.552087

Function: elapsed_run_time ()

     Returns an estimate of the number of seconds (including fractions
     of a second) which Maxima has spent in computations since Maxima
     was most recently started or restarted.  The return value is a
     floating-point number.

     See also `absolute_real_time' and `elapsed_real_time'.

     Example:

          (%i1) elapsed_run_time ();
          (%o1)                         0.04
          (%i2) expand ((a + b)^500)$
          (%i3) elapsed_run_time ();
          (%o3)                         1.26

33 Miscellaneous Options


Introduction to Miscellaneous Options
Share
Functions and Variables for Miscellaneous Options

33.1 Introduction to Miscellaneous Options

In this section various options are discussed which have a global effect
on the operation of Maxima.  Also various lists such as the list of all
user defined functions, are discussed.

33.2 Share

The Maxima "share" directory contains programs and other files of
interest to Maxima users, but not part of the core implementation of
Maxima.  These programs are typically loaded via `load' or
`setup_autoload'.

   `:lisp *maxima-sharedir*' displays the location of the share
directory within the user's file system.

   `printfile ("share.usg")' prints an out-of-date list of share
packages.  Users may find it more informative to browse the share
directory using a file system browser.

33.3 Functions and Variables for Miscellaneous Options

System variable: askexp

     When `asksign' is called, `askexp' is the expression `asksign' is
     testing.

     At one time, it was possible for a user to inspect `askexp' by
     entering a Maxima break with control-A.

Option variable: genindex

     Default value: `i'

     `genindex' is the alphabetic prefix used to generate the next
     variable of summation when necessary.

Option variable: gensumnum

     Default value: 0

     `gensumnum' is the numeric suffix used to generate the next
     variable of summation.  If it is set to `false' then the index
     will consist only of `genindex' with no numeric suffix.

Function: gensym ()
Function: gensym (<x>)

     `gensym()' creates and returns a fresh symbol.

     The name of the new-symbol is the concatenation of a prefix, which
     defaults to "g", and a suffix, which is the decimal representation
     of a number that defaults to the value of a Lisp internal counter.

     If <x> is supplied, and is a string, then that string is used as a
     prefix instead of "g" for this call to gensym only.

     If <x> is supplied, and is an integer, then that integer, instead
     of the value of the internal Lisp integer, is used as the suffix
     for this call to gensym only.

     If and only if no explicit suffix is supplied, the Lisp internal
     integer is incremented after it is used.

     Examples:

          (%i1) gensym();
          (%o1)                         g887
          (%i2) gensym("new");
          (%o2)                        new888
          (%i3) gensym(123);
          (%o3)                         g123

Option variable: packagefile

     Default value: `false'

     Package designers who use `save'  or `translate'  to create
     packages (files) for others to use may want to set `packagefile:
     true' to prevent information from being added to Maxima's
     information-lists (e.g.  `values',  `functions' ) except where
     necessary when the file is loaded in.  In this way, the contents
     of the package will not get in the user's way when he adds his own
     data.  Note that this will not solve the problem of possible name
     conflicts.  Also note that the flag simply affects what is output
     to the package file.  Setting the flag to `true' is also useful for
     creating Maxima init files.

Function: remvalue (<name_1>, ..., <name_n>)
Function: remvalue (all)

     Removes the values of user variables <name_1>, ..., <name_n>
     (which can be subscripted) from the system.

     `remvalue (all)' removes the values of all variables in `values',
     the list of all variables given names by the user (as opposed to
     those which are automatically assigned by Maxima).

     See also `values'.

Function: rncombine (<expr>)

     Transforms <expr> by combining all terms of <expr> that have
     identical denominators or denominators that differ from each other
     by numerical factors only.  This is slightly different from the
     behavior of `combine', which collects terms that have identical
     denominators.

     Setting `pfeformat: true' and using `combine' yields results
     similar to those that can be obtained with `rncombine', but
     `rncombine' takes the additional step of cross-multiplying
     numerical denominator factors.  This results in neater forms, and
     the possibility of recognizing some cancellations.

     `load(rncomb)' loads this function.

Function: setup_autoload (<filename>, <function_1>, ..., <function_n>)

     Specifies that if any of <function_1>, ..., <function_n> are
     referenced and not yet defined, <filename> is loaded via `load'.
     <filename> usually contains definitions for the functions
     specified, although that is not enforced.

     `setup_autoload' does not work for array functions.

     `setup_autoload' quotes its arguments.

     Example:

          (%i1) legendre_p (1, %pi);
          (%o1)                  legendre_p(1, %pi)
          (%i2) setup_autoload ("specfun.mac", legendre_p, ultraspherical);
          (%o2)                         done
          (%i3) ultraspherical (2, 1/2, %pi);
          Warning - you are redefining the Macsyma function ultraspherical
          Warning - you are redefining the Macsyma function legendre_p
                                      2
                           3 (%pi - 1)
          (%o3)            ------------ + 3 (%pi - 1) + 1
                                2
          (%i4) legendre_p (1, %pi);
          (%o4)                          %pi
          (%i5) legendre_q (1, %pi);
                                        %pi + 1
                                %pi log(-------)
                                        1 - %pi
          (%o5)                 ---------------- - 1
                                       2

Function: tcl_output (<list>, <i0>, <skip>)
Function: tcl_output (<list>, <i0>)
Function: tcl_output ([<list_1>, ..., <list_n>], <i>)

     Prints elements of a list enclosed by curly braces `{ }', suitable
     as part of a program in the Tcl/Tk language.

     `tcl_output (<list>, <i0>, <skip>)' prints <list>, beginning with
     element <i0> and printing elements `<i0> + <skip>', `<i0> + 2
     <skip>', etc.

     `tcl_output (<list>, <i0>)' is equivalent to `tcl_output (<list>,
     <i0>, 2)'.

     `tcl_output ([<list_1>, ..., <list_n>], <i>)' prints the <i>'th
     elements of <list_1>, ..., <list_n>.

     Examples:

          (%i1) tcl_output ([1, 2, 3, 4, 5, 6], 1, 3)$

           {1.000000000     4.000000000
           }
          (%i2) tcl_output ([1, 2, 3, 4, 5, 6], 2, 3)$

           {2.000000000     5.000000000
           }
          (%i3) tcl_output ([3/7, 5/9, 11/13, 13/17], 1)$

           {((RAT SIMP) 3 7) ((RAT SIMP) 11 13)
           }
          (%i4) tcl_output ([x1, y1, x2, y2, x3, y3], 2)$

           {$Y1 $Y2 $Y3
           }
          (%i5) tcl_output ([[1, 2, 3], [11, 22, 33]], 1)$

           {SIMP 1.000000000     11.00000000
           }

34 Rules and Patterns


Introduction to Rules and Patterns
Functions and Variables for Rules and Patterns

34.1 Introduction to Rules and Patterns

This section describes user-defined pattern matching and simplification
rules.  There are two groups of functions which implement somewhat
different pattern matching schemes.  In one group are `tellsimp',
`tellsimpafter', `defmatch', `defrule', `apply1', `applyb1', and
`apply2'.  In the other group are `let' and `letsimp'.  Both schemes
define patterns in terms of pattern variables declared by
`matchdeclare'.

   Pattern-matching rules defined by `tellsimp' and `tellsimpafter' are
applied automatically by the Maxima simplifier.  Rules defined by
`defmatch', `defrule', and `let' are applied by an explicit function
call.

   There are additional mechanisms for rules applied to polynomials by
`tellrat', and for commutative and noncommutative algebra in `affine'
package.

34.2 Functions and Variables for Rules and Patterns

Function: apply1 (<expr>, <rule_1>, ..., <rule_n>)

     Repeatedly applies <rule_1> to <expr> until it fails, then
     repeatedly applies the same rule to all subexpressions of <expr>,
     left to right, until <rule_1> has failed on all subexpressions.
     Call the result of transforming <expr> in this manner <expr_2>.
     Then <rule_2> is applied in the same fashion starting at the top
     of <expr_2>.  When <rule_n> fails on the final subexpression, the
     result is returned.

     `maxapplydepth' is the depth of the deepest subexpressions
     processed by `apply1' and `apply2'.

     See also `applyb1', `apply2', and `let'.

Function: apply2 (<expr>, <rule_1>, ..., <rule_n>)

     If <rule_1> fails on a given subexpression, then <rule_2> is
     repeatedly applied, etc.  Only if all rules fail on a given
     subexpression is the whole set of rules repeatedly applied to the
     next subexpression.  If one of the rules succeeds, then the same
     subexpression is reprocessed, starting with the first rule.

     `maxapplydepth' is the depth of the deepest subexpressions
     processed by `apply1' and `apply2'.

     See also `apply1' and `let'.

Function: applyb1 (<expr>, <rule_1>, ..., <rule_n>)

     Repeatedly applies <rule_1> to the deepest subexpression of <expr>
     until it fails, then repeatedly applies the same rule one level
     higher (i.e., larger subexpressions), until <rule_1> has failed on
     the top-level expression.  Then <rule_2> is applied in the same
     fashion to the result of <rule_1>.  After <rule_n> has been
     applied to the top-level expression, the result is returned.

     `applyb1' is similar to `apply1' but works from the bottom up
     instead of from the top down.

     `maxapplyheight' is the maximum height which `applyb1' reaches
     before giving up.

     See also `apply1', `apply2', and `let'.

Option variable: current_let_rule_package

     Default value: `default_let_rule_package'

     `current_let_rule_package' is the name of the rule package that is
     used by functions in the `let' package (`letsimp', etc.) if no
     other rule package is specified.  This variable may be assigned
     the name of any rule package defined via the `let' command.

     If a call such as `letsimp (expr, rule_pkg_name)' is made, the
     rule package `rule_pkg_name' is used for that function call only,
     and the value of `current_let_rule_package' is not changed.

Option variable: default_let_rule_package

     Default value: `default_let_rule_package'

     `default_let_rule_package' is the name of the rule package used
     when one is not explicitly set by the user with `let' or by
     changing the value of `current_let_rule_package'.

Function: defmatch (<progname>, <pattern>, <x_1>, ..., <x_n>)
Function: defmatch (<progname>, <pattern>)

     Defines a function `<progname>(<expr>, <x_1>, ..., <x_n>)' which
     tests <expr> to see if it matches <pattern>.

     <pattern> is an expression containing the pattern arguments <x_1>,
     ..., <x_n> (if any) and some pattern variables (if any).  The
     pattern arguments are given explicitly as arguments to `defmatch'
     while the pattern variables are declared by the `matchdeclare'
     function.  Any variable not declared as a pattern variable in
     `matchdeclare' or as a pattern argument in `defmatch' matches only
     itself.

     The first argument to the created function <progname> is an
     expression to be matched against the pattern and the other
     arguments are the actual arguments which correspond to the dummy
     variables <x_1>, ..., <x_n> in the pattern.

     If the match is successful, <progname> returns a list of equations
     whose left sides are the pattern arguments and pattern variables,
     and whose right sides are the subexpressions which the pattern
     arguments and variables matched.  The pattern variables, but not
     the pattern arguments, are assigned the subexpressions they match.
     If the match fails, <progname> returns `false'.

     A literal pattern (that is, a pattern which contains neither
     pattern arguments nor pattern variables) returns `true' if the
     match succeeds.

     See also `matchdeclare', `defrule', `tellsimp', and
     `tellsimpafter'.

     Examples:

     Define a function `linearp(expr, x)' which tests `expr' to see if
     it is of the form `a*x + b' such that `a' and `b' do not contain
     `x' and `a' is nonzero.  This match function matches expressions
     which are linear in any variable, because the pattern argument `x'
     is given to `defmatch'.

          (%i1) matchdeclare (a, lambda ([e], e#0 and freeof(x, e)), b,
                              freeof(x));
          (%o1)                         done
          (%i2) defmatch (linearp, a*x + b, x);
          (%o2)                        linearp
          (%i3) linearp (3*z + (y + 1)*z + y^2, z);
                                   2
          (%o3)              [b = y , a = y + 4, x = z]
          (%i4) a;
          (%o4)                         y + 4
          (%i5) b;
                                          2
          (%o5)                          y
          (%i6) x;
          (%o6)                           x

     Define a function `linearp(expr)' which tests `expr' to see if it
     is of the form `a*x + b' such that `a' and `b' do not contain `x'
     and `a' is nonzero.  This match function only matches expressions
     linear in `x', not any other variable, because no pattern argument
     is given to `defmatch'.

          (%i1) matchdeclare (a, lambda ([e], e#0 and freeof(x, e)), b,
                              freeof(x));
          (%o1)                         done
          (%i2) defmatch (linearp, a*x + b);
          (%o2)                        linearp
          (%i3) linearp (3*z + (y + 1)*z + y^2);
          (%o3)                         false
          (%i4) linearp (3*x + (y + 1)*x + y^2);
                                       2
          (%o4)                  [b = y , a = y + 4]

     Define a function `checklimits(expr)' which tests `expr' to see if
     it is a definite integral.

          (%i1) matchdeclare ([a, f], true);
          (%o1)                         done
          (%i2) constinterval (l, h) := constantp (h - l);
          (%o2)        constinterval(l, h) := constantp(h - l)
          (%i3) matchdeclare (b, constinterval (a));
          (%o3)                         done
          (%i4) matchdeclare (x, atom);
          (%o4)                         done
          (%i5) simp : false;
          (%o5)                         false
          (%i6) defmatch (checklimits, 'integrate (f, x, a, b));
          (%o6)                      checklimits
          (%i7) simp : true;
          (%o7)                         true
          (%i8) 'integrate (sin(t), t, %pi + x, 2*%pi + x);
                                 x + 2 %pi
                                /
                                [
          (%o8)                 I          sin(t) dt
                                ]
                                /
                                 x + %pi
          (%i9) checklimits (%);
          (%o9)    [b = x + 2 %pi, a = x + %pi, x = t, f = sin(t)]

Function: defrule (<rulename>, <pattern>, <replacement>)

     Defines and names a replacement rule for the given pattern.  If
     the rule named <rulename> is applied to an expression (by
     `apply1', `applyb1', or `apply2'), every subexpression matching
     the pattern will be replaced by the replacement.  All variables in
     the replacement which have been assigned values by the pattern
     match are assigned those values in the replacement which is then
     simplified.

     The rules themselves can be treated as functions which transform
     an expression by one operation of the pattern match and
     replacement.  If the match fails, the rule function returns
     `false'.

Function: disprule (<rulename_1>, ..., <rulename_2>)
Function: disprule (all)

     Display rules with the names <rulename_1>, ..., <rulename_n>, as
     returned by `defrule', `tellsimp', or `tellsimpafter', or a
     pattern defined by `defmatch'.  Each rule is displayed with an
     intermediate expression label (`%t').

     `disprule (all)' displays all rules.

     `disprule' quotes its arguments.  `disprule' returns the list of
     intermediate expression labels corresponding to the displayed
     rules.

     See also `letrules', which displays rules defined by `let'.

     Examples:

          (%i1) tellsimpafter (foo (x, y), bar (x) + baz (y));
          (%o1)                   [foorule1, false]
          (%i2) tellsimpafter (x + y, special_add (x, y));
          (%o2)                   [+rule1, simplus]
          (%i3) defmatch (quux, mumble (x));
          (%o3)                         quux
          (%i4) disprule (foorule1, "+rule1", quux);
          (%t4)        foorule1 : foo(x, y) -> baz(y) + bar(x)

          (%t5)          +rule1 : y + x -> special_add(x, y)

          (%t6)                quux : mumble(x) -> []

          (%o6)                    [%t4, %t5, %t6]
          (%i6) ''%;
          (%o6) [foorule1 : foo(x, y) -> baz(y) + bar(x),
               +rule1 : y + x -> special_add(x, y), quux : mumble(x) -> []]

Function: let (<prod>, <repl>, <predname>, <arg_1>, ..., <arg_n>)
Function: let ([<prod>, <repl>, <predname>, <arg_1>, ..., <arg_n>], <package_name>)

     Defines a substitution rule for `letsimp' such that <prod> is
     replaced by <repl>.  <prod> is a product of positive or negative
     powers of the following terms:

        * Atoms which `letsimp' will search for literally unless
          previous to calling `letsimp' the `matchdeclare' function is
          used to associate a predicate with the atom.  In this case
          `letsimp' will match the atom to any term of a product
          satisfying the predicate.

        * Kernels such as `sin(x)', `n!', `f(x,y)', etc.  As with atoms
          above `letsimp' will look for a literal match unless
          `matchdeclare' is used to associate a predicate with the
          argument of the kernel.

     A term to a positive power will only match a term having at least
     that power.  A term to a negative power on the other hand will
     only match a term with a power at least as negative.  In the case
     of negative powers in <prod> the switch `letrat' must be set to
     `true'.  See also `letrat'.

     If a predicate is included in the `let' function followed by a
     list of arguments, a tentative match (i.e. one that would be
     accepted if the predicate were omitted) is accepted only if
     `predname (arg_1', ..., arg_n')' evaluates to `true' where
     <arg_i'> is the value matched to <arg_i>.  The <arg_i> may be the
     name of any atom or the argument of any kernel appearing in <prod>.
     <repl> may be any rational expression.  If any of the atoms or
     arguments from <prod> appear in <repl> the appropriate
     substitutions are made.   The global flag `letrat' controls the
     simplification of quotients by `letsimp'.  When `letrat' is
     `false', `letsimp' simplifies the numerator and denominator of
     <expr> separately, and does not simplify the quotient.
     Substitutions such as `n!/n' goes to `(n-1)!' then fail.  When
     `letrat' is `true', then the numerator, denominator, and the
     quotient are simplified in that order.

     These substitution functions allow you to work with several rule
     packages at once.  Each rule package can contain any number of
     `let' rules and is referenced by a user-defined name.  The command
     `let ([<prod>, <repl>, <predname>, <arg_1>, ..., <arg_n>],
     <package_name>)' adds the rule <predname> to the rule package
     <package_name>.  The command `letsimp (<expr>, <package_name>)'
     applies the rules in <package_name>.  `letsimp (<expr>,
     <package_name1>, <package_name2>, ...)' is equivalent to `letsimp
     (<expr>, <package_name1>)' followed by `letsimp (%,
     <package_name2>)', ...

     `current_let_rule_package' is the name of the rule package that is
     presently being used.  This variable may be assigned the name of
     any rule package defined via the `let' command.  Whenever any of
     the functions comprising the `let' package are called with no
     package name, the package named by `current_let_rule_package' is
     used.  If a call such as `letsimp (<expr>, <rule_pkg_name>)' is
     made, the rule package <rule_pkg_name> is used for that `letsimp'
     command only, and `current_let_rule_package' is not changed.  If
     not otherwise specified, `current_let_rule_package' defaults to
     `default_let_rule_package'.

          (%i1) matchdeclare ([a, a1, a2], true)$
          (%i2) oneless (x, y) := is (x = y-1)$
          (%i3) let (a1*a2!, a1!, oneless, a2, a1);
          (%o3)         a1 a2! --> a1! where oneless(a2, a1)
          (%i4) letrat: true$
          (%i5) let (a1!/a1, (a1-1)!);
                                  a1!
          (%o5)                   --- --> (a1 - 1)!
                                  a1
          (%i6) letsimp (n*m!*(n-1)!/m);
          (%o6)                      (m - 1)! n!
          (%i7) let (sin(a)^2, 1 - cos(a)^2);
                                  2               2
          (%o7)                sin (a) --> 1 - cos (a)
          (%i8) letsimp (sin(x)^4);
                                  4           2
          (%o8)                cos (x) - 2 cos (x) + 1

Option variable: letrat

     Default value: `false'

     When `letrat' is `false', `letsimp' simplifies the numerator and
     denominator of a ratio separately, and does not simplify the
     quotient.

     When `letrat' is `true', the numerator, denominator, and their
     quotient are simplified in that order.

          (%i1) matchdeclare (n, true)$
          (%i2) let (n!/n, (n-1)!);
                                   n!
          (%o2)                    -- --> (n - 1)!
                                   n
          (%i3) letrat: false$
          (%i4) letsimp (a!/a);
                                         a!
          (%o4)                          --
                                         a
          (%i5) letrat: true$
          (%i6) letsimp (a!/a);
          (%o6)                       (a - 1)!

Function: letrules ()
Function: letrules (<package_name>)

     Displays the rules in a rule package.  `letrules ()' displays the
     rules in the current rule package.  `letrules (<package_name>)'
     displays the rules in <package_name>.

     The current rule package is named by `current_let_rule_package'.
     If not otherwise specified, `current_let_rule_package' defaults to
     `default_let_rule_package'.

     See also `disprule', which displays rules defined by `tellsimp' and
     `tellsimpafter'.

Function: letsimp (<expr>)
Function: letsimp (<expr>, <package_name>)
Function: letsimp (<expr>, <package_name_1>, ..., <package_name_n>)

     Repeatedly applies the substitution rules defined by `let' until
     no further change is made to <expr>.

     `letsimp (<expr>)' uses the rules from `current_let_rule_package'.

     `letsimp (<expr>, <package_name>)' uses the rules from
     <package_name> without changing `current_let_rule_package'.

     `letsimp (<expr>, <package_name_1>, ..., <package_name_n>)' is
     equivalent to `letsimp (<expr>, <package_name_1>', followed by
     `letsimp (%, <package_name_2>)', and so on.

Option variable: let_rule_packages

     Default value: `[default_let_rule_package]'

     `let_rule_packages' is a list of all user-defined let rule packages
     plus the default package `default_let_rule_package'.

Function: matchdeclare (<a_1>, <pred_1>, ..., <a_n>, <pred_n>)

     Associates a predicate <pred_k> with a variable or list of
     variables <a_k> so that <a_k> matches expressions for which the
     predicate returns anything other than `false'.

     A predicate is the name of a function, or a lambda expression, or
     a function call or lambda call missing the last argument, or
     `true' or `all'.  Any expression matches `true' or `all'.  If the
     predicate is specified as a function call or lambda call, the
     expression to be tested is appended to the list of arguments; the
     arguments are evaluated at the time the match is evaluated.
     Otherwise, the predicate is specified as a function name or lambda
     expression, and the expression to be tested is the sole argument.
     A predicate function need not be defined when `matchdeclare' is
     called; the predicate is not evaluated until a match is attempted.

     A predicate may return a Boolean expression as well as `true' or
     `false'.  Boolean expressions are evaluated by `is' within the
     constructed rule function, so it is not necessary to call `is'
     within the predicate.

     If an expression satisfies a match predicate, the match variable
     is assigned the expression, except for match variables which are
     operands of addition `+' or multiplication `*'.  Only addition and
     multiplication are handled specially; other n-ary operators (both
     built-in and user-defined) are treated like ordinary functions.

     In the case of addition and multiplication, the match variable may
     be assigned a single expression which satisfies the match
     predicate, or a sum or product (respectively) of such expressions.
     Such multiple-term matching is greedy: predicates are evaluated in
     the order in which their associated variables appear in the match
     pattern, and a term which satisfies more than one predicate is
     taken by the first predicate which it satisfies.  Each predicate
     is tested against all operands of the sum or product before the
     next predicate is evaluated.  In addition, if 0 or 1
     (respectively) satisfies a match predicate, and there are no other
     terms which satisfy the predicate, 0 or 1 is assigned to the match
     variable associated with the predicate.

     The algorithm for processing addition and multiplication patterns
     makes some match results (for example, a pattern in which a "match
     anything" variable appears) dependent on the ordering of terms in
     the match pattern and in the expression to be matched.  However,
     if all match predicates are mutually exclusive, the match result
     is insensitive to ordering, as one match predicate cannot accept
     terms matched by another.

     Calling `matchdeclare' with a variable <a> as an argument changes
     the `matchdeclare' property for <a>, if one was already declared;
     only the most recent `matchdeclare' is in effect when a rule is
     defined.  Later changes to the `matchdeclare' property (via
     `matchdeclare' or `remove') do not affect existing rules.

     `propvars (matchdeclare)' returns the list of all variables for
     which there is a `matchdeclare' property.  `printprops (<a>,
     matchdeclare)' returns the predicate for variable `a'.
     `printprops (all, matchdeclare)' returns the list of predicates
     for all `matchdeclare' variables.  `remove (<a>, matchdeclare)'
     removes the `matchdeclare' property from <a>.

     The functions `defmatch', `defrule', `tellsimp', `tellsimpafter',
     and `let' construct rules which test expressions against patterns.

     `matchdeclare' quotes its arguments.  `matchdeclare' always
     returns `done'.

     Examples:

     A predicate is the name of a function, or a lambda expression, or
     a function call or lambda call missing the last argument, or
     `true' or `all'.

          (%i1) matchdeclare (aa, integerp);
          (%o1)                         done
          (%i2) matchdeclare (bb, lambda ([x], x > 0));
          (%o2)                         done
          (%i3) matchdeclare (cc, freeof (%e, %pi, %i));
          (%o3)                         done
          (%i4) matchdeclare (dd, lambda ([x, y], gcd (x, y) = 1) (1728));
          (%o4)                         done
          (%i5) matchdeclare (ee, true);
          (%o5)                         done
          (%i6) matchdeclare (ff, all);
          (%o6)                         done

     If an expression satisfies a match predicate, the match variable
     is assigned the expression.

          (%i1) matchdeclare (aa, integerp, bb, atom);
          (%o1)                         done
          (%i2) defrule (r1, bb^aa, ["integer" = aa, "atom" = bb]);
                              aa
          (%o2)        r1 : bb   -> [integer = aa, atom = bb]
          (%i3) r1 (%pi^8);
          (%o3)               [integer = 8, atom = %pi]

     In the case of addition and multiplication, the match variable may
     be assigned a single expression which satisfies the match
     predicate, or a sum or product (respectively) of such expressions.

          (%i1) matchdeclare (aa, atom, bb, lambda ([x], not atom(x)));
          (%o1)                         done
          (%i2) defrule (r1, aa + bb, ["all atoms" = aa, "all nonatoms" =
                         bb]);
          bb + aa partitions `sum'
          (%o2)  r1 : bb + aa -> [all atoms = aa, all nonatoms = bb]
          (%i3) r1 (8 + a*b + sin(x));
          (%o3)     [all atoms = 8, all nonatoms = sin(x) + a b]
          (%i4) defrule (r2, aa * bb, ["all atoms" = aa, "all nonatoms" =
                         bb]);
          bb aa partitions `product'
          (%o4)   r2 : aa bb -> [all atoms = aa, all nonatoms = bb]
          (%i5) r2 (8 * (a + b) * sin(x));
          (%o5)    [all atoms = 8, all nonatoms = (b + a) sin(x)]

     When matching arguments of `+' and `*', if all match predicates
     are mutually exclusive, the match result is insensitive to
     ordering, as one match predicate cannot accept terms matched by
     another.

          (%i1) matchdeclare (aa, atom, bb, lambda ([x], not atom(x)));
          (%o1)                         done
          (%i2) defrule (r1, aa + bb, ["all atoms" = aa, "all nonatoms" =
                         bb]);
          bb + aa partitions `sum'
          (%o2)  r1 : bb + aa -> [all atoms = aa, all nonatoms = bb]
          (%i3) r1 (8 + a*b + %pi + sin(x) - c + 2^n);
                                                               n
          (%o3) [all atoms = %pi + 8, all nonatoms = sin(x) + 2  - c + a b]
          (%i4) defrule (r2, aa * bb, ["all atoms" = aa, "all nonatoms" =
                         bb]);
          bb aa partitions `product'
          (%o4)   r2 : aa bb -> [all atoms = aa, all nonatoms = bb]
          (%i5) r2 (8 * (a + b) * %pi * sin(x) / c * 2^n);
                                                            n
                                                   (b + a) 2  sin(x)
          (%o5) [all atoms = 8 %pi, all nonatoms = -----------------]
                                                           c

     The functions `propvars' and `printprops' return information about
     match variables.

          (%i1) matchdeclare ([aa, bb, cc], atom, [dd, ee], integerp);
          (%o1)                         done
          (%i2) matchdeclare (ff, floatnump, gg, lambda ([x], x > 100));
          (%o2)                         done
          (%i3) propvars (matchdeclare);
          (%o3)             [aa, bb, cc, dd, ee, ff, gg]
          (%i4) printprops (ee, matchdeclare);
          (%o4)                    [integerp(ee)]
          (%i5) printprops (gg, matchdeclare);
          (%o5)              [lambda([x], x > 100, gg)]
          (%i6) printprops (all, matchdeclare);
          (%o6) [lambda([x], x > 100, gg), floatnump(ff), integerp(ee),
                                integerp(dd), atom(cc), atom(bb), atom(aa)]

Option variable: maxapplydepth

     Default value: 10000

     `maxapplydepth' is the maximum depth to which `apply1' and
     `apply2' will delve.

Option variable: maxapplyheight

     Default value: 10000

     `maxapplyheight' is the maximum height to which `applyb1' will
     reach before giving up.

Function: remlet (<prod>, <name>)
Function: remlet ()
Function: remlet (all)
Function: remlet (all, <name>)

     Deletes the substitution rule, <prod> -> repl, most recently
     defined by the `let' function.  If name is supplied the rule is
     deleted from the rule package name.

     `remlet()' and `remlet(all)' delete all substitution rules from the
     current rule package.  If the name of a rule package is supplied,
     e.g.  `remlet (all, <name>)', the rule package <name> is also
     deleted.

     If a substitution is to be changed using the same product,
     `remlet' need not be called, just redefine the substitution using
     the same product (literally) with the `let' function and the new
     replacement and/or predicate name.  Should `remlet (<prod>)' now be
     called the original substitution rule is revived.

     See also `remrule', which removes a rule defined by `tellsimp' or
     `tellsimpafter'.

Function: remrule (<op>, <rulename>)
Function: remrule (<op>, all)

     Removes rules defined by `tellsimp' or `tellsimpafter'.

     `remrule (<op>, <rulename>)' removes the rule with the name
     <rulename> from the operator <op>.  When <op> is a built-in or
     user-defined operator (as defined by `infix', `prefix', etc.),
     <op> and <rulename> must be enclosed in double quote marks.

     `remrule (<op>, all)' removes all rules for the operator <op>.

     See also `remlet', which removes a rule defined by `let'.

     Examples:

          (%i1) tellsimp (foo (aa, bb), bb - aa);
          (%o1)                   [foorule1, false]
          (%i2) tellsimpafter (aa + bb, special_add (aa, bb));
          (%o2)                   [+rule1, simplus]
          (%i3) infix ("@@");
          (%o3)                          @@
          (%i4) tellsimp (aa @@ bb, bb/aa);
          (%o4)                   [@@rule1, false]
          (%i5) tellsimpafter (quux (%pi, %e), %pi - %e);
          (%o5)                  [quuxrule1, false]
          (%i6) tellsimpafter (quux (%e, %pi), %pi + %e);
          (%o6)             [quuxrule2, quuxrule1, false]
          (%i7) [foo (aa, bb), aa + bb, aa @@ bb, quux (%pi, %e),
                 quux (%e, %pi)];
                                               bb
          (%o7) [bb - aa, special_add(aa, bb), --, %pi - %e, %pi + %e]
                                               aa
          (%i8) remrule (foo, foorule1);
          (%o8)                          foo
          (%i9) remrule ("+", ?\+rule1);
          (%o9)                           +
          (%i10) remrule ("@@", ?\@\@rule1);
          (%o10)                         @@
          (%i11) remrule (quux, all);
          (%o11)                        quux
          (%i12) [foo (aa, bb), aa + bb, aa @@ bb, quux (%pi, %e),
                  quux (%e, %pi)];
          (%o12) [foo(aa, bb), bb + aa, aa @@ bb, quux(%pi, %e),
                                                   quux(%e, %pi)]

Function: tellsimp (<pattern>, <replacement>)

     is similar to `tellsimpafter' but places new information before
     old so that it is applied before the built-in simplification rules.

     `tellsimp' is used when it is important to modify the expression
     before the simplifier works on it, for instance if the simplifier
     "knows" something about the expression, but what it returns is not
     to your liking.  If the simplifier "knows" something about the
     main operator of the expression, but is simply not doing enough for
     you, you probably want to use `tellsimpafter'.

     The pattern may not be a sum, product, single variable, or number.

     The system variable `rules' is the list of rules defined by
     `defrule', `defmatch', `tellsimp', and `tellsimpafter'.

     Examples:

          (%i1) matchdeclare (x, freeof (%i));
          (%o1)                         done
          (%i2) %iargs: false$
          (%i3) tellsimp (sin(%i*x), %i*sinh(x));
          (%o3)                 [sinrule1, simp-%sin]
          (%i4) trigexpand (sin (%i*y + x));
          (%o4)         sin(x) cos(%i y) + %i cos(x) sinh(y)
          (%i5) %iargs:true$
          (%i6) errcatch(0^0);
           0
          0  has been generated
          (%o6)                          []
          (%i7) ev (tellsimp (0^0, 1), simp: false);
          (%o7)                  [^rule1, simpexpt]
          (%i8) 0^0;
          (%o8)                           1
          (%i9) remrule ("^", %th(2)[1]);
          (%o9)                           ^
          (%i10) tellsimp (sin(x)^2, 1 - cos(x)^2);
          (%o10)                 [^rule2, simpexpt]
          (%i11) (1 + sin(x))^2;
                                                2
          (%o11)                    (sin(x) + 1)
          (%i12) expand (%);
                                             2
          (%o12)               2 sin(x) - cos (x) + 2
          (%i13) sin(x)^2;
                                            2
          (%o13)                     1 - cos (x)
          (%i14) kill (rules);
          (%o14)                        done
          (%i15) matchdeclare (a, true);
          (%o15)                        done
          (%i16) tellsimp (sin(a)^2, 1 - cos(a)^2);
          (%o16)                 [^rule3, simpexpt]
          (%i17) sin(y)^2;
                                            2
          (%o17)                     1 - cos (y)

Function: tellsimpafter (<pattern>, <replacement>)

     Defines a simplification rule which the Maxima simplifier applies
     after built-in simplification rules.  <pattern> is an expression,
     comprising pattern variables (declared by `matchdeclare') and
     other atoms and operators, considered literals for the purpose of
     pattern matching.  <replacement> is substituted for an actual
     expression which matches <pattern>; pattern variables in
     <replacement> are assigned the values matched in the actual
     expression.

     <pattern> may be any nonatomic expression in which the main
     operator is not a pattern variable; the simplification rule is
     associated with the main operator.  The names of functions (with
     one exception, described below), lists, and arrays may appear in
     <pattern> as the main operator only as literals (not pattern
     variables); this rules out expressions such as `aa(x)' and `bb[y]'
     as patterns, if `aa' and `bb' are pattern variables.  Names of
     functions, lists, and arrays which are pattern variables may
     appear as operators other than the main operator in <pattern>.

     There is one exception to the above rule concerning names of
     functions.  The name of a subscripted function in an expression
     such as `aa[x](y)' may be a pattern variable, because the main
     operator is not `aa' but rather the Lisp atom `mqapply'.  This is
     a consequence of the representation of expressions involving
     subscripted functions.

     Simplification rules are applied after evaluation (if not
     suppressed through quotation or the flag `noeval').  Rules
     established by `tellsimpafter' are applied in the order they were
     defined, and after any built-in rules.  Rules are applied
     bottom-up, that is, applied first to subexpressions before
     application to the whole expression.  It may be necessary to
     repeatedly simplify a result (for example, via the quote-quote
     operator `''' or the flag `infeval') to ensure that all rules are
     applied.

     Pattern variables are treated as local variables in simplification
     rules.  Once a rule is defined, the value of a pattern variable
     does not affect the rule, and is not affected by the rule.  An
     assignment to a pattern variable which results from a successful
     rule match does not affect the current assignment (or lack of it)
     of the pattern variable.  However, as with all atoms in Maxima,
     the properties of pattern variables (as declared by `put' and
     related functions) are global.

     The rule constructed by `tellsimpafter' is named after the main
     operator of <pattern>.  Rules for built-in operators, and
     user-defined operators defined by `infix', `prefix', `postfix',
     `matchfix', and `nofix', have names which are Lisp identifiers.
     Rules for other functions have names which are Maxima identifiers.

     The treatment of noun and verb forms is slightly confused.  If a
     rule is defined for a noun (or verb) form and a rule for the
     corresponding verb (or noun) form already exists, the
     newly-defined rule applies to both forms (noun and verb).  If a
     rule for the corresponding verb (or noun) form does not exist, the
     newly-defined rule applies only to the noun (or verb) form.

     The rule constructed by `tellsimpafter' is an ordinary Lisp
     function.  If the name of the rule is `$foorule1', the construct
     `:lisp (trace $foorule1)' traces the function, and `:lisp
     (symbol-function '$foorule1)' displays its definition.

     `tellsimpafter' quotes its arguments.  `tellsimpafter' returns the
     list of rules for the main operator of <pattern>, including the
     newly established rule.

     See also `matchdeclare', `defmatch', `defrule', `tellsimp', `let',
     `kill', `remrule', and `clear_rules'.

     Examples:

     <pattern> may be any nonatomic expression in which the main
     operator is not a pattern variable.

          (%i1) matchdeclare (aa, atom, [ll, mm], listp, xx, true)$
          (%i2) tellsimpafter (sin (ll), map (sin, ll));
          (%o2)                 [sinrule1, simp-%sin]
          (%i3) sin ([1/6, 1/4, 1/3, 1/2, 1]*%pi);
                              1  sqrt(2)  sqrt(3)
          (%o3)              [-, -------, -------, 1, 0]
                              2     2        2
          (%i4) tellsimpafter (ll^mm, map ("^", ll, mm));
          (%o4)                  [^rule1, simpexpt]
          (%i5) [a, b, c]^[1, 2, 3];
                                          2   3
          (%o5)                      [a, b , c ]
          (%i6) tellsimpafter (foo (aa (xx)), aa (foo (xx)));
          (%o6)                   [foorule1, false]
          (%i7) foo (bar (u - v));
          (%o7)                    bar(foo(u - v))

     Rules are applied in the order they were defined.  If two rules
     can match an expression, the rule which was defined first is
     applied.

          (%i1) matchdeclare (aa, integerp);
          (%o1)                         done
          (%i2) tellsimpafter (foo (aa), bar_1 (aa));
          (%o2)                   [foorule1, false]
          (%i3) tellsimpafter (foo (aa), bar_2 (aa));
          (%o3)              [foorule2, foorule1, false]
          (%i4) foo (42);
          (%o4)                       bar_1(42)

     Pattern variables are treated as local variables in simplification
     rules.  (Compare to `defmatch', which treats pattern variables as
     global variables.)

          (%i1) matchdeclare (aa, integerp, bb, atom);
          (%o1)                         done
          (%i2) tellsimpafter (foo(aa, bb), bar('aa=aa, 'bb=bb));
          (%o2)                   [foorule1, false]
          (%i3) bb: 12345;
          (%o3)                         12345
          (%i4) foo (42, %e);
          (%o4)                 bar(aa = 42, bb = %e)
          (%i5) bb;
          (%o5)                         12345

     As with all atoms, properties of pattern variables are global even
     though values are local.  In this example, an assignment property
     is declared via `define_variable'.  This is a property of the atom
     `bb' throughout Maxima.

          (%i1) matchdeclare (aa, integerp, bb, atom);
          (%o1)                         done
          (%i2) tellsimpafter (foo(aa, bb), bar('aa=aa, 'bb=bb));
          (%o2)                   [foorule1, false]
          (%i3) foo (42, %e);
          (%o3)                 bar(aa = 42, bb = %e)
          (%i4) define_variable (bb, true, boolean);
          (%o4)                         true
          (%i5) foo (42, %e);
          Error: bb was declared mode boolean, has value: %e
           -- an error.  Quitting.  To debug this try debugmode(true);

     Rules are named after main operators.  Names of rules for built-in
     and user-defined operators are Lisp identifiers, while names for
     other functions are Maxima identifiers.

          (%i1) tellsimpafter (foo (%pi + %e), 3*%pi);
          (%o1)                   [foorule1, false]
          (%i2) tellsimpafter (foo (%pi * %e), 17*%e);
          (%o2)              [foorule2, foorule1, false]
          (%i3) tellsimpafter (foo (%i ^ %e), -42*%i);
          (%o3)         [foorule3, foorule2, foorule1, false]
          (%i4) tellsimpafter (foo (9) + foo (13), quux (22));
          (%o4)                   [+rule1, simplus]
          (%i5) tellsimpafter (foo (9) * foo (13), blurf (22));
          (%o5)                  [*rule1, simptimes]
          (%i6) tellsimpafter (foo (9) ^ foo (13), mumble (22));
          (%o6)                  [^rule1, simpexpt]
          (%i7) rules;
          (%o7) [foorule1, foorule2, foorule3, +rule1, *rule1, ^rule1]
          (%i8) foorule_name: first (%o1);
          (%o8)                       foorule1
          (%i9) plusrule_name: first (%o4);
          (%o9)                        +rule1
          (%i10) remrule (foo, foorule1);
          (%o10)                         foo
          (%i11) remrule ("^", ?\^rule1);
          (%o11)                          ^
          (%i12) rules;
          (%o12)        [foorule2, foorule3, +rule1, *rule1]

     A worked example: anticommutative multiplication.

          (%i1) gt (i, j) := integerp(j) and i < j;
          (%o1)           gt(i, j) := integerp(j) and i < j
          (%i2) matchdeclare (i, integerp, j, gt(i));
          (%o2)                         done
          (%i3) tellsimpafter (s[i]^^2, 1);
          (%o3)                 [^^rule1, simpncexpt]
          (%i4) tellsimpafter (s[i] . s[j], -s[j] . s[i]);
          (%o4)                   [.rule1, simpnct]
          (%i5) s[1] . (s[1] + s[2]);
          (%o5)                    s  . (s  + s )
                                    1     2    1
          (%i6) expand (%);
          (%o6)                      1 - s  . s
                                          2    1
          (%i7) factor (expand (sum (s[i], i, 0, 9)^^5));
          (%o7) 100 (s  + s  + s  + s  + s  + s  + s  + s  + s  + s )
                      9    8    7    6    5    4    3    2    1    0

Function: clear_rules ()

     Executes `kill (rules)' and then resets the next rule number to 1
     for addition `+', multiplication `*', and exponentiation `^'.

35 Sets


Introduction to Sets
Functions and Variables for Sets

35.1 Introduction to Sets

Maxima provides set functions, such as intersection and union, for
finite sets that are defined by explicit enumeration.  Maxima treats
lists and sets as distinct objects. This feature makes it possible to
work with sets that have members that are either lists or sets.

   In addition to functions for finite sets, Maxima provides some
functions related to combinatorics; these include the Stirling numbers
of the first and second kind, the Bell numbers, multinomial
coefficients, partitions of nonnegative integers, and a few others.
Maxima also defines a Kronecker delta function.

35.2 Functions and Variables for Sets

Function: adjoin (<x>, <a>)

     Returns the union of the set <a> with `{<x>}'.

     `adjoin' complains if <a> is not a literal set.

     `adjoin(<x>, <a>)' and `union(set(<x>), <a>)' are equivalent;
     however, `adjoin' may be somewhat faster than `union'.

     See also `disjoin'.

     Examples:

          (%i1) adjoin (c, {a, b});
          (%o1)                       {a, b, c}
          (%i2) adjoin (a, {a, b});
          (%o2)                        {a, b}

Function: belln (<n>)

     Represents the n-th Bell number.  `belln(n)' is the number of
     partitions of a set with <n> members.

     For nonnegative integers <n>, `belln(<n>)' simplifies to the n-th
     Bell number.  `belln' does not simplify for any other arguments.

     `belln' distributes over equations, lists, matrices, and sets.

     Examples:

     `belln' applied to nonnegative integers.

          (%i1) makelist (belln (i), i, 0, 6);
          (%o1)               [1, 1, 2, 5, 15, 52, 203]
          (%i2) is (cardinality (set_partitions ({})) = belln (0));
          (%o2)                         true
          (%i3) is (cardinality (set_partitions ({1, 2, 3, 4, 5, 6})) =
                                 belln (6));
          (%o3)                         true

     `belln' applied to arguments which are not nonnegative integers.

          (%i1) [belln (x), belln (sqrt(3)), belln (-9)];
          (%o1)        [belln(x), belln(sqrt(3)), belln(- 9)]

Function: cardinality (<a>)

     Returns the number of distinct elements of the set <a>.

     `cardinality' ignores redundant elements even when simplification
     is disabled.

     Examples:

          (%i1) cardinality ({});
          (%o1)                           0
          (%i2) cardinality ({a, a, b, c});
          (%o2)                           3
          (%i3) simp : false;
          (%o3)                         false
          (%i4) cardinality ({a, a, b, c});
          (%o4)                           3

Function: cartesian_product (<b_1>, ... , <b_n>)

     Returns a set of lists of the form `[<x_1>, ..., <x_n>]', where
     <x_1>, ..., <x_n> are elements of the sets <b_1>, ... , <b_n>,
     respectively.

     `cartesian_product' complains if any argument is not a literal set.

     Examples:

          (%i1) cartesian_product ({0, 1});
          (%o1)                      {[0], [1]}
          (%i2) cartesian_product ({0, 1}, {0, 1});
          (%o2)           {[0, 0], [0, 1], [1, 0], [1, 1]}
          (%i3) cartesian_product ({x}, {y}, {z});
          (%o3)                      {[x, y, z]}
          (%i4) cartesian_product ({x}, {-1, 0, 1});
          (%o4)              {[x, - 1], [x, 0], [x, 1]}

Function: disjoin (<x>, <a>)

     Returns the set <a> without the member <x>.  If <x> is not a
     member of <a>, return <a> unchanged.

     `disjoin' complains if <a> is not a literal set.

     `disjoin(<x>, <a>)', `delete(<x>, <a>)', and `setdifference(<a>,
     set(<x>))' are all equivalent.  Of these, `disjoin' is generally
     faster than the others.

     Examples:

          (%i1) disjoin (a, {a, b, c, d});
          (%o1)                       {b, c, d}
          (%i2) disjoin (a + b, {5, z, a + b, %pi});
          (%o2)                      {5, %pi, z}
          (%i3) disjoin (a - b, {5, z, a + b, %pi});
          (%o3)                  {5, %pi, b + a, z}

Function: disjointp (<a>, <b>)

     Returns `true' if and only if the sets <a> and <b> are disjoint.

     `disjointp' complains if either <a> or <b> is not a literal set.

     Examples:

          (%i1) disjointp ({a, b, c}, {1, 2, 3});
          (%o1)                         true
          (%i2) disjointp ({a, b, 3}, {1, 2, 3});
          (%o2)                         false

Function: divisors (<n>)

     Represents the set of divisors of <n>.

     `divisors(<n>)' simplifies to a set of integers when <n> is a
     nonzero integer.  The set of divisors includes the members 1 and
     <n>.  The divisors of a negative integer are the divisors of its
     absolute value.

     `divisors' distributes over equations, lists, matrices, and sets.

     Examples:

     We can verify that 28 is a perfect number: the sum of its divisors
     (except for itself) is 28.

          (%i1) s: divisors(28);
          (%o1)                 {1, 2, 4, 7, 14, 28}
          (%i2) lreduce ("+", args(s)) - 28;
          (%o2)                          28

     `divisors' is a simplifying function.  Substituting 8 for `a' in
     `divisors(a)' yields the divisors without reevaluating
     `divisors(8)'.

          (%i1) divisors (a);
          (%o1)                      divisors(a)
          (%i2) subst (8, a, %);
          (%o2)                     {1, 2, 4, 8}

     `divisors' distributes over equations, lists, matrices, and sets.

          (%i1) divisors (a = b);
          (%o1)               divisors(a) = divisors(b)
          (%i2) divisors ([a, b, c]);
          (%o2)        [divisors(a), divisors(b), divisors(c)]
          (%i3) divisors (matrix ([a, b], [c, d]));
                            [ divisors(a)  divisors(b) ]
          (%o3)             [                          ]
                            [ divisors(c)  divisors(d) ]
          (%i4) divisors ({a, b, c});
          (%o4)        {divisors(a), divisors(b), divisors(c)}

Function: elementp (<x>, <a>)

     Returns `true' if and only if <x> is a member of the set <a>.

     `elementp' complains if <a> is not a literal set.

     Examples:

          (%i1) elementp (sin(1), {sin(1), sin(2), sin(3)});
          (%o1)                         true
          (%i2) elementp (sin(1), {cos(1), cos(2), cos(3)});
          (%o2)                         false

Function: emptyp (<a>)

     Return `true' if and only if <a> is the empty set or the empty
     list.

     Examples:

          (%i1) map (emptyp, [{}, []]);
          (%o1)                     [true, true]
          (%i2) map (emptyp, [a + b, {{}}, %pi]);
          (%o2)                 [false, false, false]

Function: equiv_classes (<s>, <F>)

     Returns a set of the equivalence classes of the set <s> with
     respect to the equivalence relation <F>.

     <F> is a function of two variables defined on the Cartesian
     product of <s> with <s>.  The return value of <F> is either `true'
     or `false', or an expression <expr> such that `is(<expr>)' is
     either `true' or `false'.

     When <F> is not an equivalence relation, `equiv_classes' accepts
     it without complaint, but the result is generally incorrect in
     that case.

     Examples:

     The equivalence relation is a lambda expression which returns
     `true' or `false'.

          (%i1) equiv_classes ({1, 1.0, 2, 2.0, 3, 3.0},
                                  lambda ([x, y], is (equal (x, y))));
          (%o1)            {{1, 1.0}, {2, 2.0}, {3, 3.0}}

     The equivalence relation is the name of a relational function
     which `is' evaluates to `true' or `false'.

          (%i1) equiv_classes ({1, 1.0, 2, 2.0, 3, 3.0}, equal);
          (%o1)            {{1, 1.0}, {2, 2.0}, {3, 3.0}}

     The equivalence classes are numbers which differ by a multiple of
     3.

          (%i1) equiv_classes ({1, 2, 3, 4, 5, 6, 7},
                               lambda ([x, y], remainder (x - y, 3) = 0));
          (%o1)              {{1, 4, 7}, {2, 5}, {3, 6}}

Function: every (<f>, <s>)
Function: every (<f>, <L_1>, ..., <L_n>)

     Returns `true' if the predicate <f> is `true' for all given
     arguments.

     Given one set as the second argument, `every(<f>, <s>)' returns
     `true' if `is(<f>(<a_i>))' returns `true' for all <a_i> in <s>.
     `every' may or may not evaluate <f> for all <a_i> in <s>.  Since
     sets are unordered, `every' may evaluate `<f>(<a_i>)' in any order.

     Given one or more lists as arguments, `every(<f>, <L_1>, ...,
     <L_n>)' returns `true' if `is(<f>(<x_1>, ..., <x_n>))' returns
     `true' for all <x_1>, ..., <x_n> in <L_1>, ..., <L_n>,
     respectively.  `every' may or may not evaluate <f> for every
     combination <x_1>, ..., <x_n>.  `every' evaluates lists in the
     order of increasing index.

     Given an empty set `{}' or empty lists `[]' as arguments, `every'
     returns `false'.

     When the global flag `maperror' is `true', all lists <L_1>, ...,
     <L_n> must have equal lengths.  When `maperror' is `false', list
     arguments are effectively truncated to the length of the shortest
     list.

     Return values of the predicate <f> which evaluate (via `is') to
     something other than `true' or `false' are governed by the global
     flag `prederror'.  When `prederror' is `true', such values are
     treated as `false', and the return value from `every' is `false'.
     When `prederror' is `false', such values are treated as `unknown',
     and the return value from `every' is `unknown'.

     Examples:

     `every' applied to a single set.  The predicate is a function of
     one argument.

          (%i1) every (integerp, {1, 2, 3, 4, 5, 6});
          (%o1)                         true
          (%i2) every (atom, {1, 2, sin(3), 4, 5 + y, 6});
          (%o2)                         false

     `every' applied to two lists.  The predicate is a function of two
     arguments.

          (%i1) every ("=", [a, b, c], [a, b, c]);
          (%o1)                         true
          (%i2) every ("#", [a, b, c], [a, b, c]);
          (%o2)                         false

     Return values of the predicate <f> which evaluate to something
     other than `true' or `false' are governed by the global flag
     `prederror'.

          (%i1) prederror : false;
          (%o1)                         false
          (%i2) map (lambda ([a, b], is (a < b)), [x, y, z],
                             [x^2, y^2, z^2]);
          (%o2)              [unknown, unknown, unknown]
          (%i3) every ("<", [x, y, z], [x^2, y^2, z^2]);
          (%o3)                        unknown
          (%i4) prederror : true;
          (%o4)                         true
          (%i5) every ("<", [x, y, z], [x^2, y^2, z^2]);
          (%o5)                         false

Function: extremal_subset (<s>, <f>, max)
Function: extremal_subset (<s>, <f>, min)

     Returns the subset of <s> for which the function <f> takes on
     maximum or minimum values.

     `extremal_subset(<s>, <f>, max)' returns the subset of the set or
     list <s> for which the real-valued function <f> takes on its
     maximum value.

     `extremal_subset(<s>, <f>, min)' returns the subset of the set or
     list <s> for which the real-valued function <f> takes on its
     minimum value.

     Examples:

          (%i1) extremal_subset ({-2, -1, 0, 1, 2}, abs, max);
          (%o1)                       {- 2, 2}
          (%i2) extremal_subset ({sqrt(2), 1.57, %pi/2}, sin, min);
          (%o2)                       {sqrt(2)}

Function: flatten (<expr>)

     Collects arguments of subexpressions which have the same operator
     as <expr> and constructs an expression from these collected
     arguments.

     Subexpressions in which the operator is different from the main
     operator of `expr' are copied without modification, even if they,
     in turn, contain some subexpressions in which the operator is the
     same as for `expr'.

     It may be possible for `flatten' to construct expressions in which
     the number of arguments differs from the declared arguments for an
     operator; this may provoke an error message from the simplifier or
     evaluator.  `flatten' does not try to detect such situations.

     Expressions with special representations, for example, canonical
     rational expressions (CRE), cannot be flattened; in such cases,
     `flatten' returns its argument unchanged.

     Examples:

     Applied to a list, `flatten' gathers all list elements that are
     lists.

          (%i1) flatten ([a, b, [c, [d, e], f], [[g, h]], i, j]);
          (%o1)            [a, b, c, d, e, f, g, h, i, j]

     Applied to a set, `flatten' gathers all members of set elements
     that are sets.

          (%i1) flatten ({a, {b}, {{c}}});
          (%o1)                       {a, b, c}
          (%i2) flatten ({a, {[a], {a}}});
          (%o2)                       {a, [a]}

     `flatten' is similar to the effect of declaring the main operator
     n-ary.  However, `flatten' has no effect on subexpressions which
     have an operator different from the main operator, while an n-ary
     declaration affects those.

          (%i1) expr: flatten (f (g (f (f (x)))));
          (%o1)                     f(g(f(f(x))))
          (%i2) declare (f, nary);
          (%o2)                         done
          (%i3) ev (expr);
          (%o3)                      f(g(f(x)))

     `flatten' treats subscripted functions the same as any other
     operator.

          (%i1) flatten (f[5] (f[5] (x, y), z));
          (%o1)                      f (x, y, z)
                                      5

     It may be possible for `flatten' to construct expressions in which
     the number of arguments differs from the declared arguments for an
     operator;

          (%i1) 'mod (5, 'mod (7, 4));
          (%o1)                   mod(5, mod(7, 4))
          (%i2) flatten (%);
          (%o2)                     mod(5, 7, 4)
          (%i3) ''%, nouns;
          Wrong number of arguments to mod
           -- an error.  Quitting.  To debug this try debugmode(true);

Function: full_listify (<a>)

     Replaces every set operator in <a> by a list operator, and returns
     the result.  `full_listify' replaces set operators in nested
     subexpressions, even if the main operator is not `set'.

     `listify' replaces only the main operator.

     Examples:

          (%i1) full_listify ({a, b, {c, {d, e, f}, g}});
          (%o1)               [a, b, [c, [d, e, f], g]]
          (%i2) full_listify (F (G ({a, b, H({c, d, e})})));
          (%o2)              F(G([a, b, H([c, d, e])]))

Function: fullsetify (<a>)

     When <a> is a list, replaces the list operator with a set operator,
     and applies `fullsetify' to each member which is a set.  When <a>
     is not a list, it is returned unchanged.

     `setify' replaces only the main operator.

     Examples:

     In line `(%o2)', the argument of `f' isn't converted to a set
     because the main operator of `f([b])' isn't a list.

          (%i1) fullsetify ([a, [a]]);
          (%o1)                       {a, {a}}
          (%i2) fullsetify ([a, f([b])]);
          (%o2)                      {a, f([b])}

Function: identity (<x>)

     Returns <x> for any argument <x>.

     Examples:

     `identity' may be used as a predicate when the arguments are
     already Boolean values.

          (%i1) every (identity, [true, true]);
          (%o1)                         true

Function: integer_partitions (<n>)
Function: integer_partitions (<n>, <len>)

     Returns integer partitions of <n>, that is, lists of integers
     which sum to <n>.

     `integer_partitions(<n>)' returns the set of all partitions of the
     integer <n>.  Each partition is a list sorted from greatest to
     least.

     `integer_partitions(<n>, <len>)' returns all partitions that have
     length <len> or less; in this case, zeros are appended to each
     partition with fewer than <len> terms to make each partition have
     exactly <len> terms.  Each partition is a list sorted from
     greatest to least.

     A list [a_1, ..., a_m] is a partition of a nonnegative integer n
     when (1) each a_i is a nonzero integer, and (2) a_1 + ... + a_m =
     n. Thus 0 has no partitions.

     Examples:

          (%i1) integer_partitions (3);
          (%o1)               {[1, 1, 1], [2, 1], [3]}
          (%i2) s: integer_partitions (25)$
          (%i3) cardinality (s);
          (%o3)                         1958
          (%i4) map (lambda ([x], apply ("+", x)), s);
          (%o4)                         {25}
          (%i5) integer_partitions (5, 3);
          (%o5) {[2, 2, 1], [3, 1, 1], [3, 2, 0], [4, 1, 0], [5, 0, 0]}
          (%i6) integer_partitions (5, 2);
          (%o6)               {[3, 2], [4, 1], [5, 0]}

     To find all partitions that satisfy a condition, use the function
     `subset'; here is an example that finds all partitions of 10 that
     consist of prime numbers.

          (%i1) s: integer_partitions (10)$
          (%i2) cardinality (s);
          (%o2)                          42
          (%i3) xprimep(x) := integerp(x) and (x > 1) and primep(x)$
          (%i4) subset (s, lambda ([x], every (xprimep, x)));
          (%o4) {[2, 2, 2, 2, 2], [3, 3, 2, 2], [5, 3, 2], [5, 5], [7, 3]}

Function: intersect (<a_1>, ..., <a_n>)

     `intersect' is the same as `intersection', which see.

Function: intersection (<a_1>, ..., <a_n>)

     Returns a set containing the elements that are common to the sets
     <a_1> through <a_n>.

     `intersection' complains if any argument is not a literal set.

     Examples:

          (%i1) S_1 : {a, b, c, d};
          (%o1)                     {a, b, c, d}
          (%i2) S_2 : {d, e, f, g};
          (%o2)                     {d, e, f, g}
          (%i3) S_3 : {c, d, e, f};
          (%o3)                     {c, d, e, f}
          (%i4) S_4 : {u, v, w};
          (%o4)                       {u, v, w}
          (%i5) intersection (S_1, S_2);
          (%o5)                          {d}
          (%i6) intersection (S_2, S_3);
          (%o6)                       {d, e, f}
          (%i7) intersection (S_1, S_2, S_3);
          (%o7)                          {d}
          (%i8) intersection (S_1, S_2, S_3, S_4);
          (%o8)                          {}

Function: kron_delta (<x1>, <x2>, ..., <xp>)

     Represents the Kronecker delta function.

     `kron_delta' simplifies to 1 when <xi> and <yj> are equal for all
     pairs of arguments, and it simplifies to 0 when <xi> and <yj> are
     not equal for some pair of arguments. Equality is determined using
     `is(equal(xi,xj))' and inequality by `is(notequal(xi,xj))'. For
     exactly one argument, `kron_delta' signals an error.

     Examples:

          (%i1) kron_delta(a,a);
          (%o1)                                  1
          (%i2) kron_delta(a,b,a,b);
          (%o2)                          kron_delta(a, b)
          (%i3) kron_delta(a,a,b,a+1);
          (%o3)                                  0
          (%i4) assume(equal(x,y));
          (%o4)                            [equal(x, y)]
          (%i5) kron_delta(x,y);
          (%o5)                                  1

Function: listify (<a>)

     Returns a list containing the members of <a> when <a> is a set.
     Otherwise, `listify' returns <a>.

     `full_listify' replaces all set operators in <a> by list operators.

     Examples:

          (%i1) listify ({a, b, c, d});
          (%o1)                     [a, b, c, d]
          (%i2) listify (F ({a, b, c, d}));
          (%o2)                    F({a, b, c, d})

Function: lreduce (<F>, <s>)
Function: lreduce (<F>, <s>, <s_0>)

     Extends the binary function <F> to an n-ary function by
     composition, where <s> is a list.

     `lreduce(<F>, <s>)' returns `F(... F(F(s_1, s_2), s_3), ... s_n)'.
     When the optional argument <s_0> is present, the result is
     equivalent to `lreduce(<F>, cons(<s_0>, <s>))'.

     The function <F> is first applied to the leftmost list elements,
     thus the name "lreduce".

     See also `rreduce', `xreduce', and `tree_reduce'.

     Examples:

     `lreduce' without the optional argument.

          (%i1) lreduce (f, [1, 2, 3]);
          (%o1)                     f(f(1, 2), 3)
          (%i2) lreduce (f, [1, 2, 3, 4]);
          (%o2)                  f(f(f(1, 2), 3), 4)

     `lreduce' with the optional argument.

          (%i1) lreduce (f, [1, 2, 3], 4);
          (%o1)                  f(f(f(4, 1), 2), 3)

     `lreduce' applied to built-in binary operators.  `/' is the
     division operator.

          (%i1) lreduce ("^", args ({a, b, c, d}));
                                         b c d
          (%o1)                       ((a ) )
          (%i2) lreduce ("/", args ({a, b, c, d}));
                                          a
          (%o2)                         -----
                                        b c d

Function: makeset (<expr>, <x>, <s>)

     Returns a set with members generated from the expression <expr>,
     where <x> is a list of variables in <expr>, and <s> is a set or
     list of lists.  To generate each set member, <expr> is evaluated
     with the variables <x> bound in parallel to a member of <s>.

     Each member of <s> must have the same length as <x>.  The list of
     variables <x> must be a list of symbols, without subscripts.  Even
     if there is only one symbol, <x> must be a list of one element,
     and each member of <s> must be a list of one element.

     See also `makelist'.

     Examples:

          (%i1) makeset (i/j, [i, j], [[1, a], [2, b], [3, c], [4, d]]);
                                     1  2  3  4
          (%o1)                     {-, -, -, -}
                                     a  b  c  d
          (%i2) S : {x, y, z}$
          (%i3) S3 : cartesian_product (S, S, S);
          (%o3) {[x, x, x], [x, x, y], [x, x, z], [x, y, x], [x, y, y],
          [x, y, z], [x, z, x], [x, z, y], [x, z, z], [y, x, x],
          [y, x, y], [y, x, z], [y, y, x], [y, y, y], [y, y, z],
          [y, z, x], [y, z, y], [y, z, z], [z, x, x], [z, x, y],
          [z, x, z], [z, y, x], [z, y, y], [z, y, z], [z, z, x],
          [z, z, y], [z, z, z]}
          (%i4) makeset (i + j + k, [i, j, k], S3);
          (%o4) {3 x, 3 y, y + 2 x, 2 y + x, 3 z, z + 2 x, z + y + x,
                                                 z + 2 y, 2 z + x, 2 z + y}
          (%i5) makeset (sin(x), [x], {[1], [2], [3]});
          (%o5)               {sin(1), sin(2), sin(3)}

Function: moebius (<n>)

     Represents the Moebius function.

     When <n> is product of k distinct primes, `moebius(<n>)'
     simplifies to (-1)^k; when <n> = 1, it simplifies to 1; and it
     simplifies to 0 for all other positive integers.

     `moebius' distributes over equations, lists, matrices, and sets.

     Examples:

          (%i1) moebius (1);
          (%o1)                           1
          (%i2) moebius (2 * 3 * 5);
          (%o2)                          - 1
          (%i3) moebius (11 * 17 * 29 * 31);
          (%o3)                           1
          (%i4) moebius (2^32);
          (%o4)                           0
          (%i5) moebius (n);
          (%o5)                      moebius(n)
          (%i6) moebius (n = 12);
          (%o6)                    moebius(n) = 0
          (%i7) moebius ([11, 11 * 13, 11 * 13 * 15]);
          (%o7)                      [- 1, 1, 1]
          (%i8) moebius (matrix ([11, 12], [13, 14]));
                                     [ - 1  0 ]
          (%o8)                      [        ]
                                     [ - 1  1 ]
          (%i9) moebius ({21, 22, 23, 24});
          (%o9)                      {- 1, 0, 1}

Function: multinomial_coeff (<a_1>, ..., <a_n>)
Function: multinomial_coeff ()

     Returns the multinomial coefficient.

     When each <a_k> is a nonnegative integer, the multinomial
     coefficient gives the number of ways of placing `<a_1> + ... +
     <a_n>' distinct objects into n boxes with <a_k> elements in the
     k'th box. In general, `multinomial_coeff (<a_1>, ..., <a_n>)'
     evaluates to `(<a_1> + ... + <a_n>)!/(<a_1>! ... <a_n>!)'.

     `multinomial_coeff()' (with no arguments) evaluates to 1.

     `minfactorial' may be able to simplify the value returned by
     `multinomial_coeff'.

     Examples:

          (%i1) multinomial_coeff (1, 2, x);
                                      (x + 3)!
          (%o1)                       --------
                                        2 x!
          (%i2) minfactorial (%);
                               (x + 1) (x + 2) (x + 3)
          (%o2)                -----------------------
                                          2
          (%i3) multinomial_coeff (-6, 2);
                                       (- 4)!
          (%o3)                       --------
                                      2 (- 6)!
          (%i4) minfactorial (%);
          (%o4)                          10

Function: num_distinct_partitions (<n>)
Function: num_distinct_partitions (<n>, list)

     Returns the number of distinct integer partitions of <n> when <n>
     is a nonnegative integer.  Otherwise, `num_distinct_partitions'
     returns a noun expression.

     `num_distinct_partitions(<n>, list)' returns a list of the number
     of distinct partitions of 1, 2, 3, ..., <n>.

     A distinct partition of <n> is a list of distinct positive
     integers k_1, ..., k_m such that <n> = k_1 + ... + k_m.

     Examples:

          (%i1) num_distinct_partitions (12);
          (%o1)                          15
          (%i2) num_distinct_partitions (12, list);
          (%o2)      [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15]
          (%i3) num_distinct_partitions (n);
          (%o3)              num_distinct_partitions(n)

Function: num_partitions (<n>)
Function: num_partitions (<n>, list)

     Returns the number of integer partitions of <n> when <n> is a
     nonnegative integer.  Otherwise, `num_partitions' returns a noun
     expression.

     `num_partitions(<n>, list)' returns a list of the number of
     integer partitions of 1, 2, 3, ..., <n>.

     For a nonnegative integer <n>, `num_partitions(<n>)' is equal to
     `cardinality(integer_partitions(<n>))'; however, `num_partitions'
     does not actually construct the set of partitions, so it is much
     faster.

     Examples:

          (%i1) num_partitions (5) = cardinality (integer_partitions (5));
          (%o1)                         7 = 7
          (%i2) num_partitions (8, list);
          (%o2)            [1, 1, 2, 3, 5, 7, 11, 15, 22]
          (%i3) num_partitions (n);
          (%o3)                   num_partitions(n)

Function: partition_set (<a>, <f>)

     Partitions the set <a> according to the predicate <f>.

     `partition_set' returns a list of two sets.  The first set
     comprises the elements of <a> for which <f> evaluates to `false',
     and the second comprises any other elements of <a>.
     `partition_set' does not apply `is' to the return value of <f>.

     `partition_set' complains if <a> is not a literal set.

     See also `subset'.

     Examples:

          (%i1) partition_set ({2, 7, 1, 8, 2, 8}, evenp);
          (%o1)                   [{1, 7}, {2, 8}]
          (%i2) partition_set ({x, rat(y), rat(y) + z, 1},
                               lambda ([x], ratp(x)));
          (%o2)/R/              [{1, x}, {y, y + z}]

Function: permutations (<a>)

     Returns a set of all distinct permutations of the members of the
     list or set <a>. Each permutation is a list, not a set.

     When <a> is a list, duplicate members of <a> are included in the
     permutations.

     `permutations' complains if <a> is not a literal list or set.

     See also `random_permutation'.

     Examples:

          (%i1) permutations ([a, a]);
          (%o1)                       {[a, a]}
          (%i2) permutations ([a, a, b]);
          (%o2)           {[a, a, b], [a, b, a], [b, a, a]}

Function: powerset (<a>)
Function: powerset (<a>, <n>)

     Returns the set of all subsets of <a>, or a subset of that set.

     `powerset(<a>)' returns the set of all subsets of the set <a>.
     `powerset(<a>)' has `2^cardinality(<a>)' members.

     `powerset(<a>, <n>)' returns the set of all subsets of <a> that
     have cardinality <n>.

     `powerset' complains if <a> is not a literal set, or if <n> is not
     a nonnegative integer.

     Examples:

          (%i1) powerset ({a, b, c});
          (%o1) {{}, {a}, {a, b}, {a, b, c}, {a, c}, {b}, {b, c}, {c}}
          (%i2) powerset ({w, x, y, z}, 4);
          (%o2)                    {{w, x, y, z}}
          (%i3) powerset ({w, x, y, z}, 3);
          (%o3)     {{w, x, y}, {w, x, z}, {w, y, z}, {x, y, z}}
          (%i4) powerset ({w, x, y, z}, 2);
          (%o4)   {{w, x}, {w, y}, {w, z}, {x, y}, {x, z}, {y, z}}
          (%i5) powerset ({w, x, y, z}, 1);
          (%o5)                 {{w}, {x}, {y}, {z}}
          (%i6) powerset ({w, x, y, z}, 0);
          (%o6)                         {{}}

Function: random_permutation (<a>)

     Returns a random permutation of the set or list <a>, as
     constructed by the Knuth shuffle algorithm.

     The return value is a new list, which is distinct from the
     argument even if all elements happen to be the same.  However, the
     elements of the argument are not copied.

     Examples:

          (%i1) random_permutation ([a, b, c, 1, 2, 3]);
          (%o1)                  [c, 1, 2, 3, a, b]
          (%i2) random_permutation ([a, b, c, 1, 2, 3]);
          (%o2)                  [b, 3, 1, c, a, 2]
          (%i3) random_permutation ({x + 1, y + 2, z + 3});
          (%o3)                 [y + 2, z + 3, x + 1]
          (%i4) random_permutation ({x + 1, y + 2, z + 3});
          (%o4)                 [x + 1, y + 2, z + 3]

Function: rreduce (<F>, <s>)
Function: rreduce (<F>, <s>, <s_{n + 1}>)

     Extends the binary function <F> to an n-ary function by
     composition, where <s> is a list.

     `rreduce(<F>, <s>)' returns `F(s_1, ... F(s_{n - 2}, F(s_{n - 1},
     s_n)))'.  When the optional argument <s_{n + 1}> is present, the
     result is equivalent to `rreduce(<F>, endcons(<s_{n + 1}>, <s>))'.

     The function <F> is first applied to the rightmost list elements,
     thus the name "rreduce".

     See also `lreduce', `tree_reduce', and `xreduce'.

     Examples:

     `rreduce' without the optional argument.

          (%i1) rreduce (f, [1, 2, 3]);
          (%o1)                     f(1, f(2, 3))
          (%i2) rreduce (f, [1, 2, 3, 4]);
          (%o2)                  f(1, f(2, f(3, 4)))

     `rreduce' with the optional argument.

          (%i1) rreduce (f, [1, 2, 3], 4);
          (%o1)                  f(1, f(2, f(3, 4)))

     `rreduce' applied to built-in binary operators.  `/' is the
     division operator.

          (%i1) rreduce ("^", args ({a, b, c, d}));
                                           d
                                          c
                                         b
          (%o1)                         a
          (%i2) rreduce ("/", args ({a, b, c, d}));
                                         a c
          (%o2)                          ---
                                         b d

Function: setdifference (<a>, <b>)

     Returns a set containing the elements in the set <a> that are not
     in the set <b>.

     `setdifference' complains if either <a> or <b> is not a literal
     set.

     Examples:

          (%i1) S_1 : {a, b, c, x, y, z};
          (%o1)                  {a, b, c, x, y, z}
          (%i2) S_2 : {aa, bb, c, x, y, zz};
          (%o2)                 {aa, bb, c, x, y, zz}
          (%i3) setdifference (S_1, S_2);
          (%o3)                       {a, b, z}
          (%i4) setdifference (S_2, S_1);
          (%o4)                     {aa, bb, zz}
          (%i5) setdifference (S_1, S_1);
          (%o5)                          {}
          (%i6) setdifference (S_1, {});
          (%o6)                  {a, b, c, x, y, z}
          (%i7) setdifference ({}, S_1);
          (%o7)                          {}

Function: setequalp (<a>, <b>)

     Returns `true' if sets <a> and <b> have the same number of elements
     and `is(<x> = <y>)' is `true' for `x' in the elements of <a> and
     `y' in the elements of <b>, considered in the order determined by
     `listify'.  Otherwise, `setequalp' returns `false'.

     Examples:

          (%i1) setequalp ({1, 2, 3}, {1, 2, 3});
          (%o1)                         true
          (%i2) setequalp ({a, b, c}, {1, 2, 3});
          (%o2)                         false
          (%i3) setequalp ({x^2 - y^2}, {(x + y) * (x - y)});
          (%o3)                         false

Function: setify (<a>)

     Constructs a set from the elements of the list <a>. Duplicate
     elements of the list <a> are deleted and the elements are sorted
     according to the predicate `orderlessp'.

     `setify' complains if <a> is not a literal list.

     Examples:

          (%i1) setify ([1, 2, 3, a, b, c]);
          (%o1)                  {1, 2, 3, a, b, c}
          (%i2) setify ([a, b, c, a, b, c]);
          (%o2)                       {a, b, c}
          (%i3) setify ([7, 13, 11, 1, 3, 9, 5]);
          (%o3)                {1, 3, 5, 7, 9, 11, 13}

Function: setp (<a>)

     Returns `true' if and only if <a> is a Maxima set.

     `setp' returns `true' for unsimplified sets (that is, sets with
     redundant members) as well as simplified sets.

     `setp' is equivalent to the Maxima function `setp(a) := not
     atom(a) and op(a) = 'set'.

     Examples:

          (%i1) simp : false;
          (%o1)                         false
          (%i2) {a, a, a};
          (%o2)                       {a, a, a}
          (%i3) setp (%);
          (%o3)                         true

Function: set_partitions (<a>)
Function: set_partitions (<a>, <n>)

     Returns the set of all partitions of <a>, or a subset of that set.

     `set_partitions(<a>, <n>)' returns a set of all decompositions of
     <a> into <n> nonempty disjoint subsets.

     `set_partitions(<a>)' returns the set of all partitions.

     `stirling2' returns the cardinality of the set of partitions of a
     set.

     A set of sets P is a partition of a set S when

       1. each member of P is a nonempty set,

       2. distinct members of P are disjoint,

       3. the union of the members of P equals S.

     Examples:

     The empty set is a partition of itself, the conditions 1 and 2
     being vacuously true.

          (%i1) set_partitions ({});
          (%o1)                         {{}}

     The cardinality of the set of partitions of a set can be found
     using `stirling2'.

          (%i1) s: {0, 1, 2, 3, 4, 5}$
          (%i2) p: set_partitions (s, 3)$
          (%i3) cardinality(p) = stirling2 (6, 3);
          (%o3)                        90 = 90

     Each member of `p' should have <n> = 3 members; let's check.

          (%i1) s: {0, 1, 2, 3, 4, 5}$
          (%i2) p: set_partitions (s, 3)$
          (%i3) map (cardinality, p);
          (%o3)                          {3}

     Finally, for each member of `p', the union of its members should
     equal `s'; again let's check.

          (%i1) s: {0, 1, 2, 3, 4, 5}$
          (%i2) p: set_partitions (s, 3)$
          (%i3) map (lambda ([x], apply (union, listify (x))), p);
          (%o3)                 {{0, 1, 2, 3, 4, 5}}

Function: some (<f>, <a>)
Function: some (<f>, <L_1>, ..., <L_n>)

     Returns `true' if the predicate <f> is `true' for one or more
     given arguments.

     Given one set as the second argument, `some(<f>, <s>)' returns
     `true' if `is(<f>(<a_i>))' returns `true' for one or more <a_i> in
     <s>.  `some' may or may not evaluate <f> for all <a_i> in <s>.
     Since sets are unordered, `some' may evaluate `<f>(<a_i>)' in any
     order.

     Given one or more lists as arguments, `some(<f>, <L_1>, ...,
     <L_n>)' returns `true' if `is(<f>(<x_1>, ..., <x_n>))' returns
     `true' for one or more <x_1>, ..., <x_n> in <L_1>, ..., <L_n>,
     respectively.  `some' may or may not evaluate <f> for some
     combinations <x_1>, ..., <x_n>.  `some' evaluates lists in the
     order of increasing index.

     Given an empty set `{}' or empty lists `[]' as arguments, `some'
     returns `false'.

     When the global flag `maperror' is `true', all lists <L_1>, ...,
     <L_n> must have equal lengths.  When `maperror' is `false', list
     arguments are effectively truncated to the length of the shortest
     list.

     Return values of the predicate <f> which evaluate (via `is') to
     something other than `true' or `false' are governed by the global
     flag `prederror'.  When `prederror' is `true', such values are
     treated as `false'.  When `prederror' is `false', such values are
     treated as `unknown'.

     Examples:

     `some' applied to a single set.  The predicate is a function of
     one argument.

          (%i1) some (integerp, {1, 2, 3, 4, 5, 6});
          (%o1)                         true
          (%i2) some (atom, {1, 2, sin(3), 4, 5 + y, 6});
          (%o2)                         true

     `some' applied to two lists.  The predicate is a function of two
     arguments.

          (%i1) some ("=", [a, b, c], [a, b, c]);
          (%o1)                         true
          (%i2) some ("#", [a, b, c], [a, b, c]);
          (%o2)                         false

     Return values of the predicate <f> which evaluate to something
     other than `true' or `false' are governed by the global flag
     `prederror'.

          (%i1) prederror : false;
          (%o1)                         false
          (%i2) map (lambda ([a, b], is (a < b)), [x, y, z],
                     [x^2, y^2, z^2]);
          (%o2)              [unknown, unknown, unknown]
          (%i3) some ("<", [x, y, z], [x^2, y^2, z^2]);
          (%o3)                        unknown
          (%i4) some ("<", [x, y, z], [x^2, y^2, z + 1]);
          (%o4)                         true
          (%i5) prederror : true;
          (%o5)                         true
          (%i6) some ("<", [x, y, z], [x^2, y^2, z^2]);
          (%o6)                         false
          (%i7) some ("<", [x, y, z], [x^2, y^2, z + 1]);
          (%o7)                         true

Function: stirling1 (<n>, <m>)

     Represents the Stirling number of the first kind.

     When <n> and <m> are nonnegative integers, the magnitude of
     `stirling1 (<n>, <m>)' is the number of permutations of a set with
     <n> members that have <m> cycles.  For details, see Graham, Knuth
     and Patashnik Concrete Mathematics.  Maxima uses a recursion
     relation to define `stirling1 (<n>, <m>)' for <m> less than 0; it
     is undefined for <n> less than 0 and for non-integer arguments.

     `stirling1' is a simplifying function.  Maxima knows the following
     identities.

       1. stirling1(0, n) = kron_delta(0, n) (Ref. [1])

       2. stirling1(n, n) = 1 (Ref. [1])

       3. stirling1(n, n - 1) = binomial(n, 2) (Ref. [1])

       4. stirling1(n + 1, 0) = 0 (Ref. [1])

       5. stirling1(n + 1, 1) = n! (Ref. [1])

       6. stirling1(n + 1, 2) = 2^n  - 1 (Ref. [1])

     These identities are applied when the arguments are literal
     integers or symbols declared as integers, and the first argument
     is nonnegative.  `stirling1' does not simplify for non-integer
     arguments.

     References:

     [1] Donald Knuth, The Art of Computer Programming, third edition,
     Volume 1, Section 1.2.6, Equations 48, 49, and 50.

     Examples:

          (%i1) declare (n, integer)$
          (%i2) assume (n >= 0)$
          (%i3) stirling1 (n, n);
          (%o3)                           1

     `stirling1' does not simplify for non-integer arguments.

          (%i1) stirling1 (sqrt(2), sqrt(2));
          (%o1)              stirling1(sqrt(2), sqrt(2))

     Maxima applies identities to `stirling1'.

          (%i1) declare (n, integer)$
          (%i2) assume (n >= 0)$
          (%i3) stirling1 (n + 1, n);
                                      n (n + 1)
          (%o3)                       ---------
                                          2
          (%i4) stirling1 (n + 1, 1);
          (%o4)                          n!

Function: stirling2 (<n>, <m>)

     Represents the Stirling number of the second kind.

     When <n> and <m> are nonnegative integers, `stirling2 (<n>, <m>)'
     is the number of ways a set with cardinality <n> can be
     partitioned into <m> disjoint subsets.  Maxima uses a recursion
     relation to define `stirling2 (<n>, <m>)' for <m> less than 0; it
     is undefined for <n> less than 0 and for non-integer arguments.

     `stirling2' is a simplifying function.  Maxima knows the following
     identities.

       1. stirling2(0, n) = kron_delta(0, n) (Ref. [1])

       2. stirling2(n, n) = 1 (Ref. [1])

       3. stirling2(n, n - 1) = binomial(n, 2) (Ref. [1])

       4. stirling2(n + 1, 1) = 1 (Ref. [1])

       5. stirling2(n + 1, 2) = 2^n  - 1 (Ref. [1])

       6. stirling2(n, 0) = kron_delta(n, 0) (Ref. [2])

       7. stirling2(n, m) = 0 when m > n (Ref. [2])

       8. stirling2(n, m) = sum((-1)^(m - k) binomial(m k) k^n,i,1,m) /
          m!  when m and n are integers, and n is nonnegative. (Ref.
          [3])

     These identities are applied when the arguments are literal
     integers or symbols declared as integers, and the first argument
     is nonnegative.  `stirling2' does not simplify for non-integer
     arguments.

     References:

     [1] Donald Knuth. The Art of Computer Programming, third edition,
     Volume 1, Section 1.2.6, Equations 48, 49, and 50.

     [2] Graham, Knuth, and Patashnik. Concrete Mathematics, Table 264.

     [3] Abramowitz and Stegun. Handbook of Mathematical Functions,
     Section 24.1.4.

     Examples:

          (%i1) declare (n, integer)$
          (%i2) assume (n >= 0)$
          (%i3) stirling2 (n, n);
          (%o3)                           1

     `stirling2' does not simplify for non-integer arguments.

          (%i1) stirling2 (%pi, %pi);
          (%o1)                  stirling2(%pi, %pi)

     Maxima applies identities to `stirling2'.

          (%i1) declare (n, integer)$
          (%i2) assume (n >= 0)$
          (%i3) stirling2 (n + 9, n + 8);
                                   (n + 8) (n + 9)
          (%o3)                    ---------------
                                          2
          (%i4) stirling2 (n + 1, 2);
                                        n
          (%o4)                        2  - 1

Function: subset (<a>, <f>)

     Returns the subset of the set <a> that satisfies the predicate <f>.

     `subset' returns a set which comprises the elements of <a> for
     which <f> returns anything other than `false'.  `subset' does not
     apply `is' to the return value of <f>.

     `subset' complains if <a> is not a literal set.

     See also `partition_set'.

     Examples:

          (%i1) subset ({1, 2, x, x + y, z, x + y + z}, atom);
          (%o1)                     {1, 2, x, z}
          (%i2) subset ({1, 2, 7, 8, 9, 14}, evenp);
          (%o2)                      {2, 8, 14}

Function: subsetp (<a>, <b>)

     Returns `true' if and only if the set <a> is a subset of <b>.

     `subsetp' complains if either <a> or <b> is not a literal set.

     Examples:

          (%i1) subsetp ({1, 2, 3}, {a, 1, b, 2, c, 3});
          (%o1)                         true
          (%i2) subsetp ({a, 1, b, 2, c, 3}, {1, 2, 3});
          (%o2)                         false

Function: symmdifference (<a_1>, ..., <a_n>)

     Returns the symmetric difference of sets <a_1>, ..., <a_n>.

     Given two arguments, `symmdifference (<a>, <b>)' is the same as
     `union (setdifference (<a>, <b>), setdifference (<b>, <a>))'.

     `symmdifference' complains if any argument is not a literal set.

     Examples:

          (%i1) S_1 : {a, b, c};
          (%o1)                       {a, b, c}
          (%i2) S_2 : {1, b, c};
          (%o2)                       {1, b, c}
          (%i3) S_3 : {a, b, z};
          (%o3)                       {a, b, z}
          (%i4) symmdifference ();
          (%o4)                          {}
          (%i5) symmdifference (S_1);
          (%o5)                       {a, b, c}
          (%i6) symmdifference (S_1, S_2);
          (%o6)                        {1, a}
          (%i7) symmdifference (S_1, S_2, S_3);
          (%o7)                        {1, b, z}
          (%i8) symmdifference ({}, S_1, S_2, S_3);
          (%o8)                        {1,b, z}

Function: tree_reduce (<F>, <s>)
Function: tree_reduce (<F>, <s>, <s_0>)

     Extends the binary function <F> to an n-ary function by
     composition, where <s> is a set or list.

     `tree_reduce' is equivalent to the following: Apply <F> to
     successive pairs of elements to form a new list `[<F>(<s_1>,
     <s_2>), <F>(<s_3>, <s_4>), ...]', carrying the final element
     unchanged if there are an odd number of elements.  Then repeat
     until the list is reduced to a single element, which is the return
     value.

     When the optional argument <s_0> is present, the result is
     equivalent `tree_reduce(<F>, cons(<s_0>, <s>)'.

     For addition of floating point numbers, `tree_reduce' may return a
     sum that has a smaller rounding error than either `rreduce' or
     `lreduce'.

     The elements of <s> and the partial results may be arranged in a
     minimum-depth binary tree, thus the name "tree_reduce".

     Examples:

     `tree_reduce' applied to a list with an even number of elements.

          (%i1) tree_reduce (f, [a, b, c, d]);
          (%o1)                  f(f(a, b), f(c, d))

     `tree_reduce' applied to a list with an odd number of elements.

          (%i1) tree_reduce (f, [a, b, c, d, e]);
          (%o1)               f(f(f(a, b), f(c, d)), e)

Function: union (<a_1>, ..., <a_n>)

     Returns the union of the sets <a_1> through <a_n>.

     `union()' (with no arguments) returns the empty set.

     `union' complains if any argument is not a literal set.

     Examples:

          (%i1) S_1 : {a, b, c + d, %e};
          (%o1)                   {%e, a, b, d + c}
          (%i2) S_2 : {%pi, %i, %e, c + d};
          (%o2)                 {%e, %i, %pi, d + c}
          (%i3) S_3 : {17, 29, 1729, %pi, %i};
          (%o3)                {17, 29, 1729, %i, %pi}
          (%i4) union ();
          (%o4)                          {}
          (%i5) union (S_1);
          (%o5)                   {%e, a, b, d + c}
          (%i6) union (S_1, S_2);
          (%o6)              {%e, %i, %pi, a, b, d + c}
          (%i7) union (S_1, S_2, S_3);
          (%o7)       {17, 29, 1729, %e, %i, %pi, a, b, d + c}
          (%i8) union ({}, S_1, S_2, S_3);
          (%o8)       {17, 29, 1729, %e, %i, %pi, a, b, d + c}

Function: xreduce (<F>, <s>)
Function: xreduce (<F>, <s>, <s_0>)

     Extends the function <F> to an n-ary function by composition, or,
     if <F> is already n-ary, applies <F> to <s>.  When <F> is not
     n-ary, `xreduce' is the same as `lreduce'.  The argument <s> is a
     list.

     Functions known to be n-ary include addition `+', multiplication
     `*', `and', `or', `max', `min', and `append'.  Functions may also
     be declared n-ary by `declare(<F>, nary)'.  For these functions,
     `xreduce' is expected to be faster than either `rreduce' or
     `lreduce'.

     When the optional argument <s_0> is present, the result is
     equivalent to `xreduce(<s>, cons(<s_0>, <s>))'.

     Floating point addition is not exactly associative; be that as it
     may, `xreduce' applies Maxima's n-ary addition when <s> contains
     floating point numbers.

     Examples:

     `xreduce' applied to a function known to be n-ary.  `F' is called
     once, with all arguments.

          (%i1) declare (F, nary);
          (%o1)                         done
          (%i2) F ([L]) := L;
          (%o2)                      F([L]) := L
          (%i3) xreduce (F, [a, b, c, d, e]);
          (%o3)         [[[[[("[", simp), a], b], c], d], e]

     `xreduce' applied to a function not known to be n-ary.  `G' is
     called several times, with two arguments each time.

          (%i1) G ([L]) := L;
          (%o1)                      G([L]) := L
          (%i2) xreduce (G, [a, b, c, d, e]);
          (%o2)         [[[[[("[", simp), a], b], c], d], e]
          (%i3) lreduce (G, [a, b, c, d, e]);
          (%o3)                 [[[[a, b], c], d], e]

36 Function Definition


Introduction to Function Definition
Function
Macros
Functions and Variables for Function Definition

36.1 Introduction to Function Definition

36.2 Function

36.3 Macros

Function: buildq (<L>, <expr>)

     Substitutes variables named by the list <L> into the expression
     <expr>, in parallel, without evaluating <expr>.  The resulting
     expression is simplified, but not evaluated, after `buildq'
     carries out the substitution.

     The elements of <L> are symbols or assignment expressions
     `<symbol>: <value>', evaluated in parallel.  That is, the binding
     of a variable on the right-hand side of an assignment is the
     binding of that variable in the context from which `buildq' was
     called, not the binding of that variable in the variable list <L>.
     If some variable in <L> is not given an explicit assignment, its
     binding in `buildq' is the same as in the context from which
     `buildq' was called.

     Then the variables named by <L> are substituted into <expr> in
     parallel.  That is, the substitution for every variable is
     determined before any substitution is made, so the substitution
     for one variable has no effect on any other.

     If any variable <x> appears as `splice (<x>)' in <expr>, then <x>
     must be bound to a list, and the list is spliced (interpolated)
     into <expr> instead of substituted.

     Any variables in <expr> not appearing in <L> are carried into the
     result verbatim, even if they have bindings in the context from
     which `buildq' was called.

     Examples

     `a' is explicitly bound to `x', while `b' has the same binding
     (namely 29) as in the calling context, and `c' is carried through
     verbatim.  The resulting expression is not evaluated until the
     explicit evaluation `''%'.

          (%i1) (a: 17, b: 29, c: 1729)$
          (%i2) buildq ([a: x, b], a + b + c);
          (%o2)                      x + c + 29
          (%i3) ''%;
          (%o3)                       x + 1758

     `e' is bound to a list, which appears as such in the arguments of
     `foo', and interpolated into the arguments of `bar'.

          (%i1) buildq ([e: [a, b, c]], foo (x, e, y));
          (%o1)                 foo(x, [a, b, c], y)
          (%i2) buildq ([e: [a, b, c]], bar (x, splice (e), y));
          (%o2)                  bar(x, a, b, c, y)

     The result is simplified after substitution.  If simplification
     were applied before substitution, these two results would be the
     same.

          (%i1) buildq ([e: [a, b, c]], splice (e) + splice (e));
          (%o1)                    2 c + 2 b + 2 a
          (%i2) buildq ([e: [a, b, c]], 2 * splice (e));
          (%o2)                        2 a b c

     The variables in <L> are bound in parallel; if bound sequentially,
     the first result would be `foo (b, b)'.  Substitutions are carried
     out in parallel; compare the second result with the result of
     `subst', which carries out substitutions sequentially.

          (%i1) buildq ([a: b, b: a], foo (a, b));
          (%o1)                       foo(b, a)
          (%i2) buildq ([u: v, v: w, w: x, x: y, y: z, z: u],
                        bar (u, v, w, x, y, z));
          (%o2)                 bar(v, w, x, y, z, u)
          (%i3) subst ([u=v, v=w, w=x, x=y, y=z, z=u],
                       bar (u, v, w, x, y, z));
          (%o3)                 bar(u, u, u, u, u, u)

     Construct a list of equations with some variables or expressions
     on the left-hand side and their values on the right-hand side.
     `macroexpand' shows the expression returned by `show_values'.

          (%i1) show_values ([L]) ::= buildq ([L], map ("=", 'L, L));
          (%o1)   show_values([L]) ::= buildq([L], map("=", 'L, L))
          (%i2) (a: 17, b: 29, c: 1729)$
          (%i3) show_values (a, b, c - a - b);
          (%o3)          [a = 17, b = 29, c - b - a = 1683]
          (%i4) macroexpand (show_values (a, b, c - a - b));
          (%o4)    map(=, '([a, b, c - b - a]), [a, b, c - b - a])

     Given a function of several arguments, create another function for
     which some of the arguments are fixed.

          (%i1) curry (f, [a]) :=
                  buildq ([f, a], lambda ([[x]], apply (f, append (a, x))))$
          (%i2) by3 : curry ("*", 3);
          (%o2)        lambda([[x]], apply(*, append([3], x)))
          (%i3) by3 (a + b);
          (%o3)                       3 (b + a)

Function: macroexpand (<expr>)

     Returns the macro expansion of <expr> without evaluating it, when
     `expr' is a macro function call.  Otherwise, `macroexpand' returns
     <expr>.

     If the expansion of <expr> yields another macro function call,
     that macro function call is also expanded.

     `macroexpand' quotes its argument.  However, if the expansion of a
     macro function call has side effects, those side effects are
     executed.

     See also `::=', `macros', and `macroexpand1'.

     Examples

          (%i1) g (x) ::= x / 99;
                                              x
          (%o1)                      g(x) ::= --
                                              99
          (%i2) h (x) ::= buildq ([x], g (x - a));
          (%o2)            h(x) ::= buildq([x], g(x - a))
          (%i3) a: 1234;
          (%o3)                         1234
          (%i4) macroexpand (h (y));
                                        y - a
          (%o4)                         -----
                                         99
          (%i5) h (y);
                                      y - 1234
          (%o5)                       --------
                                         99

Function: macroexpand1 (<expr>)

     Returns the macro expansion of <expr> without evaluating it, when
     `expr' is a macro function call.  Otherwise, `macroexpand1'
     returns <expr>.

     `macroexpand1' quotes its argument.  However, if the expansion of
     a macro function call has side effects, those side effects are
     executed.

     If the expansion of <expr> yields another macro function call,
     that macro function call is not expanded.

     See also `::=', `macros', and `macroexpand'.

     Examples

          (%i1) g (x) ::= x / 99;
                                              x
          (%o1)                      g(x) ::= --
                                              99
          (%i2) h (x) ::= buildq ([x], g (x - a));
          (%o2)            h(x) ::= buildq([x], g(x - a))
          (%i3) a: 1234;
          (%o3)                         1234
          (%i4) macroexpand1 (h (y));
          (%o4)                       g(y - a)
          (%i5) h (y);
                                      y - 1234
          (%o5)                       --------
                                         99

Global variable: macros

     Default value: `[]'

     `macros' is the list of user-defined macro functions.  The macro
     function definition operator `::=' puts a new macro function onto
     this list, and `kill', `remove', and `remfunction' remove macro
     functions from the list.

     See also `infolists'.

Function: splice (<a>)

     Splices (interpolates) the list named by the atom <a> into an
     expression, but only if `splice' appears within `buildq';
     otherwise, `splice' is treated as an undefined function.  If
     appearing within `buildq' as <a> alone (without `splice'), <a> is
     substituted (not interpolated) as a list into the result.  The
     argument of `splice' can only be an atom; it cannot be a literal
     list or an expression which yields a list.

     Typically `splice' supplies the arguments for a function or
     operator.  For a function `f', the expression `f (splice (<a>))'
     within `buildq' expands to `f (<a>[1], <a>[2], <a>[3], ...)'.  For
     an operator `o', the expression `"o" (splice (<a>)' within
     `buildq' expands to `"o" (<a>[1], <a>[2], <a>[3], ...)', where `o'
     may be any type of operator (typically one which takes multiple
     arguments).  Note that the operator must be enclosed in double
     quotes `"'.

     Examples

          (%i1) buildq ([x: [1, %pi, z - y]], foo (splice (x)) / length (x));
                                 foo(1, %pi, z - y)
          (%o1)                -----------------------
                               length([1, %pi, z - y])
          (%i2) buildq ([x: [1, %pi]], "/" (splice (x)));
                                          1
          (%o2)                          ---
                                         %pi
          (%i3) matchfix ("<>", "<>");
          (%o3)                          <>
          (%i4) buildq ([x: [1, %pi, z - y]], "<>" (splice (x)));
          (%o4)                   <>1, %pi, z - y<>

36.4 Functions and Variables for Function Definition

Function: apply (<F>, [<x_1>, ..., <x_n>])

     Constructs and evaluates an expression `<F>(<arg_1>, ...,
     <arg_n>)'.

     `apply' does not attempt to distinguish array functions from
     ordinary functions; when <F> is the name of an array function,
     `apply' evaluates `<F>(...)' (that is, a function call with
     parentheses instead of square brackets).  `arrayapply' evaluates a
     function call with square brackets in this case.

     Examples:

     `apply' evaluates its arguments.  In this example, `min' is
     applied to the value of `L'.

          (%i1) L : [1, 5, -10.2, 4, 3];
          (%o1)                 [1, 5, - 10.2, 4, 3]
          (%i2) apply (min, L);
          (%o2)                        - 10.2

     `apply' evaluates arguments, even if the function <F> quotes them.

          (%i1) F (x) := x / 1729;
                                             x
          (%o1)                     F(x) := ----
                                            1729
          (%i2) fname : F;
          (%o2)                           F
          (%i3) dispfun (F);
                                             x
          (%t3)                     F(x) := ----
                                            1729

          (%o3)                         [%t3]
          (%i4) dispfun (fname);
          fname is not the name of a user function.
           -- an error.  Quitting.  To debug this try debugmode(true);
          (%i5) apply (dispfun, [fname]);
                                             x
          (%t5)                     F(x) := ----
                                            1729
          (%o5)                         [%t5]

     `apply' evaluates the function name <F>.  Single quote `'' defeats
     evaluation.  `demoivre' is the name of a global variable and also
     a function.

          (%i1) demoivre;
          (%o1)                         false
          (%i2) demoivre (exp (%i * x));
          (%o2)                  %i sin(x) + cos(x)
          (%i3) apply (demoivre, [exp (%i * x)]);
          demoivre evaluates to false
          Improper name or value in functional position.
           -- an error.  Quitting.  To debug this try debugmode(true);
          (%i4) apply ('demoivre, [exp (%i * x)]);
          (%o4)                  %i sin(x) + cos(x)

Function: block ([<v_1>, ..., <v_m>], <expr_1>, ..., <expr_n>)
Function: block (<expr_1>, ..., <expr_n>)

     `block' evaluates <expr_1>, ..., <expr_n> in sequence and returns
     the value of the last expression evaluated.  The sequence can be
     modified by the `go', `throw', and `return' functions.  The last
     expression is <expr_n> unless `return' or an expression containing
     `throw' is evaluated.  Some variables <v_1>, ..., <v_m> can be
     declared local to the block; these are distinguished from global
     variables of the same names.  If no variables are declared local
     then the list may be omitted.  Within the block, any variable
     other than <v_1>, ..., <v_m> is a global variable.

     `block' saves the current values of the variables <v_1>, ...,
     <v_m> (if any) upon entry to the block, then unbinds the variables
     so that they evaluate to themselves.  The local variables may be
     bound to arbitrary values within the block but when the block is
     exited the saved values are restored, and the values assigned
     within the block are lost.

     The declaration `local(<v_1>, ..., <v_m>)' within `block' saves
     the properties associated with the symbols <v_1>, ..., <v_m>,
     removes any properties before evaluating other expressions, and
     restores any saved properties on exit from the block.  Some
     declarations are implemented as properties of a symbol, including
     `:=', `array', `dependencies', `atvalue', `matchdeclare',
     `atomgrad', `constant', `nonscalar', `assume', and some others.
     The effect of `local' is to make such declarations effective only
     within the block; otherwise declarations within a block are
     actually global declarations.

     `block' may appear within another `block'.  Local variables are
     established each time a new `block' is evaluated.  Local variables
     appear to be global to any enclosed blocks.  If a variable is
     non-local in a block, its value is the value most recently
     assigned by an enclosing block, if any, otherwise, it is the value
     of the variable in the global environment.  This policy may
     coincide with the usual understanding of "dynamic scope".

     The value of the block is the value of the last statement or the
     value of the argument to the function `return' which may be used
     to exit explicitly from the block. The function `go' may be used
     to transfer control to the statement of the block that is tagged
     with the argument to `go'.  To tag a statement, precede it by an
     atomic argument as another statement in the block.  For example:
     `block ([x], x:1, loop, x: x+1, ..., go(loop), ...)'.  The
     argument to `go' must be the name of a tag appearing within the
     block.  One cannot use `go' to transfer to a tag in a block other
     than the one containing the `go'.

     Blocks typically appear on the right side of a function definition
     but can be used in other places as well.

Function: break (<expr_1>, ..., <expr_n>)

     Evaluates and prints <expr_1>, ..., <expr_n> and then causes a
     Maxima break at which point the user can examine and change his
     environment.  Upon typing `exit;' the computation resumes.

Function: catch (<expr_1>, ..., <expr_n>)

     Evaluates <expr_1>, ..., <expr_n> one by one; if any leads to the
     evaluation of an expression of the form `throw (arg)', then the
     value of the `catch' is the value of `throw (arg)', and no further
     expressions are evaluated.  This "non-local return" thus goes
     through any depth of nesting to the nearest enclosing `catch'.  If
     there is no `catch' enclosing a `throw', an error message is
     printed.

     If the evaluation of the arguments does not lead to the evaluation
     of any `throw' then the value of `catch' is the value of <expr_n>.

          (%i1) lambda ([x], if x < 0 then throw(x) else f(x))$
          (%i2) g(l) := catch (map (''%, l))$
          (%i3) g ([1, 2, 3, 7]);
          (%o3)               [f(1), f(2), f(3), f(7)]
          (%i4) g ([1, 2, -3, 7]);
          (%o4)                          - 3

     The function `g' returns a list of `f' of each element of `l' if
     `l' consists only of non-negative numbers; otherwise, `g' "catches"
     the first negative element of `l' and "throws" it up.

Function: compfile (<filename>, <f_1>, ..., <f_n>)
Function: compfile (<filename>, functions)
Function: compfile (<filename>, all)

     Translates Maxima functions into Lisp and writes the translated
     code into the file <filename>.

     `compfile(<filename>, <f_1>, ..., <f_n>)' translates the specified
     functions.  `compfile (<filename>, functions)' and `compfile
     (<filename>, all)' translate all user-defined functions.

     The Lisp translations are not evaluated, nor is the output file
     processed by the Lisp compiler.  `translate' creates and evaluates
     Lisp translations.  `compile_file' translates Maxima into Lisp,
     and then executes the Lisp compiler.

     See also `translate', `translate_file', and `compile_file'.

Function: compile (<f_1>, ..., <f_n>)
Function: compile (functions)
Function: compile (all)

     Translates Maxima functions <f_1>, ..., <f_n> into Lisp, evaluates
     the Lisp translations, and calls the Lisp function `COMPILE' on
     each translated function.  `compile' returns a list of the names
     of the compiled functions.

     `compile (all)' or `compile (functions)' compiles all user-defined
     functions.

     `compile' quotes its arguments; the quote-quote operator `'''
     defeats quotation.

Function: define (<f>(<x_1>, ..., <x_n>), <expr>)
Function: define (<f>[<x_1>, ..., <x_n>], <expr>)
Function: define (funmake (<f>, [<x_1>, ..., <x_n>]), <expr>)
Function: define (arraymake (<f>, [<x_1>, ..., <x_n>]), <expr>)
Function: define (ev (<expr_1>), <expr_2>)

     Defines a function named <f> with arguments <x_1>, ..., <x_n> and
     function body <expr>.  `define' always evaluates its second
     argument (unless explicitly quoted).  The function so defined may
     be an ordinary Maxima function (with arguments enclosed in
     parentheses) or an array function (with arguments enclosed in
     square brackets).

     When the last or only function argument <x_n> is a list of one
     element, the function defined by `define' accepts a variable
     number of arguments.  Actual arguments are assigned one-to-one to
     formal arguments <x_1>, ..., <x_(n - 1)>, and any further actual
     arguments, if present, are assigned to <x_n> as a list.

     When the first argument of `define' is an expression of the form
     `<f>(<x_1>, ..., <x_n>)' or `<f>[<x_1>, ..., <x_n>]', the function
     arguments are evaluated but <f> is not evaluated, even if there is
     already a function or variable by that name.

     When the first argument is an expression with operator `funmake',
     `arraymake', or `ev', the first argument is evaluated; this allows
     for the function name to be computed, as well as the body.

     All function definitions appear in the same namespace; defining a
     function `f' within another function `g' does not automatically
     limit the scope of `f' to `g'.  However, `local(f)' makes the
     definition of function `f' effective only within the block or
     other compound expression in which `local' appears.

     If some formal argument <x_k> is a quoted symbol (after
     evaluation), the function defined by `define' does not evaluate
     the corresponding actual argument.  Otherwise all actual arguments
     are evaluated.

     See also `:=' and `::='.

     Examples:

     `define' always evaluates its second argument (unless explicitly
     quoted).

          (%i1) expr : cos(y) - sin(x);
          (%o1)                    cos(y) - sin(x)
          (%i2) define (F1 (x, y), expr);
          (%o2)              F1(x, y) := cos(y) - sin(x)
          (%i3) F1 (a, b);
          (%o3)                    cos(b) - sin(a)
          (%i4) F2 (x, y) := expr;
          (%o4)                   F2(x, y) := expr
          (%i5) F2 (a, b);
          (%o5)                    cos(y) - sin(x)

     The function defined by `define' may be an ordinary Maxima
     function or an array function.

          (%i1) define (G1 (x, y), x.y - y.x);
          (%o1)               G1(x, y) := x . y - y . x
          (%i2) define (G2 [x, y], x.y - y.x);
          (%o2)                G2     := x . y - y . x
                                 x, y

     When the last or only function argument <x_n> is a list of one
     element, the function defined by `define' accepts a variable
     number of arguments.

          (%i1) define (H ([L]), '(apply ("+", L)));
          (%o1)                H([L]) := apply("+", L)
          (%i2) H (a, b, c);
          (%o2)                       c + b + a

     When the first argument is an expression with operator `funmake',
     `arraymake', or `ev', the first argument is evaluated.

          (%i1) [F : I, u : x];
          (%o1)                        [I, x]
          (%i2) funmake (F, [u]);
          (%o2)                         I(x)
          (%i3) define (funmake (F, [u]), cos(u) + 1);
          (%o3)                  I(x) := cos(x) + 1
          (%i4) define (arraymake (F, [u]), cos(u) + 1);
          (%o4)                   I  := cos(x) + 1
                                   x
          (%i5) define (foo (x, y), bar (y, x));
          (%o5)                foo(x, y) := bar(y, x)
          (%i6) define (ev (foo (x, y)), sin(x) - cos(y));
          (%o6)             bar(y, x) := sin(x) - cos(y)

Function: define_variable (<name>, <default_value>, <mode>)

     Introduces a global variable into the Maxima environment.
     `define_variable' is useful in user-written packages, which are
     often translated or compiled.

     `define_variable' carries out the following steps:

       1. `mode_declare (<name>, <mode>)' declares the mode of <name> to
          the translator.  See `mode_declare' for a list of the
          possible modes.

       2. If the variable is unbound, <default_value> is assigned to
          <name>.

       3. `declare (<name>, special)' declares it special.

       4. Associates <name> with a test function to ensure that <name>
          is only assigned values of the declared mode.

     The `value_check' property can be assigned to any variable which
     has been defined via `define_variable' with a mode other than
     `any'.  The `value_check' property is a lambda expression or the
     name of a function of one variable, which is called when an
     attempt is made to assign a value to the variable.  The argument
     of the `value_check' function is the would-be assigned value.

     `define_variable' evaluates `default_value', and quotes `name' and
     `mode'.  `define_variable' returns the current value of `name',
     which is `default_value' if `name' was unbound before, and
     otherwise it is the previous value of `name'.

     Examples:

     `foo' is a Boolean variable, with the initial value `true'.

          (%i1) define_variable (foo, true, boolean);
          (%o1)                         true
          (%i2) foo;
          (%o2)                         true
          (%i3) foo: false;
          (%o3)                         false
          (%i4) foo: %pi;
          Error: foo was declared mode boolean, has value: %pi
           -- an error.  Quitting.  To debug this try debugmode(true);
          (%i5) foo;
          (%o5)                         false

     `bar' is an integer variable, which must be prime.

          (%i1) define_variable (bar, 2, integer);
          (%o1)                           2
          (%i2) qput (bar, prime_test, value_check);
          (%o2)                      prime_test
          (%i3) prime_test (y) := if not primep(y) then
                                     error (y, "is not prime.");
          (%o3) prime_test(y) := if not primep(y)

                                             then error(y, "is not prime.")
          (%i4) bar: 1439;
          (%o4)                         1439
          (%i5) bar: 1440;
          1440 is not prime.
          #0: prime_test(y=1440)
           -- an error.  Quitting.  To debug this try debugmode(true);
          (%i6) bar;
          (%o6)                         1439

     `baz_quux' is a variable which cannot be assigned a value.  The
     mode `any_check' is like `any', but `any_check' enables the
     `value_check' mechanism, and `any' does not.

          (%i1) define_variable (baz_quux, 'baz_quux, any_check);
          (%o1)                       baz_quux
          (%i2) F: lambda ([y], if y # 'baz_quux then
                           error ("Cannot assign to `baz_quux'."));
          (%o2) lambda([y], if y # 'baz_quux

                                  then error(Cannot assign to `baz_quux'.))
          (%i3) qput (baz_quux, ''F, value_check);
          (%o3) lambda([y], if y # 'baz_quux

                                  then error(Cannot assign to `baz_quux'.))
          (%i4) baz_quux: 'baz_quux;
          (%o4)                       baz_quux
          (%i5) baz_quux: sqrt(2);
          Cannot assign to `baz_quux'.
          #0: lambda([y],if y # 'baz_quux then
                           error("Cannot assign to `baz_quux'."))(y=sqrt(2))
           -- an error.  Quitting.  To debug this try debugmode(true);
          (%i6) baz_quux;
          (%o6)                       baz_quux

Function: dispfun (<f_1>, ..., <f_n>)
Function: dispfun (all)

     Displays the definition of the user-defined functions <f_1>, ...,
     <f_n>.  Each argument may be the name of a macro (defined with
     `::='), an ordinary function (defined with `:=' or `define'), an
     array function (defined with `:=' or `define', but enclosing
     arguments in square brackets `[ ]'), a subscripted function,
     (defined with `:=' or `define', but enclosing some arguments in
     square brackets and others in parentheses `( )') one of a family
     of subscripted functions selected by a particular subscript value,
     or a subscripted function defined with a constant subscript.

     `dispfun (all)' displays all user-defined functions as given by
     the `functions', `arrays', and `macros' lists, omitting
     subscripted functions defined with constant subscripts.

     `dispfun' creates an intermediate expression label (`%t1', `%t2',
     etc.)  for each displayed function, and assigns the function
     definition to the label.  In contrast, `fundef' returns the
     function definition.

     `dispfun' quotes its arguments; the quote-quote operator `'''
     defeats quotation.  `dispfun' returns the list of intermediate
     expression labels corresponding to the displayed functions.

     Examples:

          (%i1) m(x, y) ::= x^(-y);
                                               - y
          (%o1)                   m(x, y) ::= x
          (%i2) f(x, y) :=  x^(-y);
                                               - y
          (%o2)                    f(x, y) := x
          (%i3) g[x, y] :=  x^(-y);
                                              - y
          (%o3)                     g     := x
                                     x, y
          (%i4) h[x](y) :=  x^(-y);
                                              - y
          (%o4)                     h (y) := x
                                     x
          (%i5) i[8](y) :=  8^(-y);
                                              - y
          (%o5)                     i (y) := 8
                                     8
          (%i6) dispfun (m, f, g, h, h[5], h[10], i[8]);
                                               - y
          (%t6)                   m(x, y) ::= x
                                               - y
          (%t7)                    f(x, y) := x

                                              - y
          (%t8)                     g     := x
                                     x, y

                                              - y
          (%t9)                     h (y) := x
                                     x

                                              1
          (%t10)                     h (y) := --
                                      5        y
                                              5

                                               1
          (%t11)                    h  (y) := ---
                                     10         y
                                              10

                                              - y
          (%t12)                    i (y) := 8
                                     8

          (%o12)       [%t6, %t7, %t8, %t9, %t10, %t11, %t12]
          (%i12) ''%;
                               - y              - y            - y
          (%o12) [m(x, y) ::= x   , f(x, y) := x   , g     := x   ,
                                                      x, y
                            - y           1              1             - y
                  h (y) := x   , h (y) := --, h  (y) := ---, i (y) := 8   ]
                   x              5        y   10         y   8
                                          5             10

Function: fullmap (<f>, <expr_1>, ...)

     Similar to `map', but `fullmap' keeps mapping down all
     subexpressions until the main operators are no longer the same.

     `fullmap' is used by the Maxima simplifier for certain matrix
     manipulations; thus, Maxima sometimes generates an error message
     concerning `fullmap' even though `fullmap' was not explicitly
     called by the user.

     Examples:

          (%i1) a + b * c;
          (%o1)                        b c + a
          (%i2) fullmap (g, %);
          (%o2)                   g(b) g(c) + g(a)
          (%i3) map (g, %th(2));
          (%o3)                     g(b c) + g(a)

Function: fullmapl (<f>, <list_1>, ...)

     Similar to `fullmap', but `fullmapl' only maps onto lists and
     matrices.

     Example:

          (%i1) fullmapl ("+", [3, [4, 5]], [[a, 1], [0, -1.5]]);
          (%o1)                [[a + 3, 4], [4, 3.5]]

System variable: functions

     Default value: `[]'

     `functions' is the list of ordinary Maxima functions in the
     current session.  An ordinary function is a function constructed by
     `define' or `:=' and called with parentheses `()'.  A function may
     be defined at the Maxima prompt or in a Maxima file loaded by
     `load' or `batch'.

     Array functions (called with square brackets, e.g., `F[x]') and
     subscripted functions (called with square brackets and
     parentheses, e.g., `F[x](y)') are listed by the global variable
     `arrays', and not by `functions'.

     Lisp functions are not kept on any list.

     Examples:

          (%i1) F_1 (x) := x - 100;
          (%o1)                   F_1(x) := x - 100
          (%i2) F_2 (x, y) := x / y;
                                                x
          (%o2)                    F_2(x, y) := -
                                                y
          (%i3) define (F_3 (x), sqrt (x));
          (%o3)                   F_3(x) := sqrt(x)
          (%i4) G_1 [x] := x - 100;
          (%o4)                    G_1  := x - 100
                                      x
          (%i5) G_2 [x, y] := x / y;
                                               x
          (%o5)                     G_2     := -
                                       x, y    y
          (%i6) define (G_3 [x], sqrt (x));
          (%o6)                    G_3  := sqrt(x)
                                      x
          (%i7) H_1 [x] (y) := x^y;
                                                y
          (%o7)                     H_1 (y) := x
                                       x
          (%i8) functions;
          (%o8)              [F_1(x), F_2(x, y), F_3(x)]
          (%i9) arrays;
          (%o9)                 [G_1, G_2, G_3, H_1]

Function: fundef (<f>)

     Returns the definition of the function <f>.

     The argument may be the name of a macro (defined with `::='), an
     ordinary function (defined with `:=' or `define'), an array
     function (defined with `:=' or `define', but enclosing arguments
     in square brackets `[ ]'), a subscripted function, (defined with
     `:=' or `define', but enclosing some arguments in square brackets
     and others in parentheses `( )') one of a family of subscripted
     functions selected by a particular subscript value, or a
     subscripted function defined with a constant subscript.

     `fundef' quotes its argument; the quote-quote operator `'''
     defeats quotation.

     `fundef (<f>)' returns the definition of <f>.  In contrast,
     `dispfun (<f>)' creates an intermediate expression label and
     assigns the definition to the label.

Function: funmake (<F>, [<arg_1>, ..., <arg_n>])

     Returns an expression `<F>(<arg_1>, ..., <arg_n>)'.  The return
     value is simplified, but not evaluated, so the function <F> is not
     called, even if it exists.

     `funmake' does not attempt to distinguish array functions from
     ordinary functions; when <F> is the name of an array function,
     `funmake' returns `<F>(...)' (that is, a function call with
     parentheses instead of square brackets).  `arraymake' returns a
     function call with square brackets in this case.

     `funmake' evaluates its arguments.

     Examples:

     `funmake' applied to an ordinary Maxima function.

          (%i1) F (x, y) := y^2 - x^2;
                                             2    2
          (%o1)                  F(x, y) := y  - x
          (%i2) funmake (F, [a + 1, b + 1]);
          (%o2)                    F(a + 1, b + 1)
          (%i3) ''%;
                                        2          2
          (%o3)                  (b + 1)  - (a + 1)

     `funmake' applied to a macro.

          (%i1) G (x) ::= (x - 1)/2;
                                            x - 1
          (%o1)                    G(x) ::= -----
                                              2
          (%i2) funmake (G, [u]);
          (%o2)                         G(u)
          (%i3) ''%;
                                        u - 1
          (%o3)                         -----
                                          2

     `funmake' applied to a subscripted function.

          (%i1) H [a] (x) := (x - 1)^a;
                                                  a
          (%o1)                   H (x) := (x - 1)
                                   a
          (%i2) funmake (H [n], [%e]);
                                                 n
          (%o2)               lambda([x], (x - 1) )(%e)
          (%i3) ''%;
                                              n
          (%o3)                       (%e - 1)
          (%i4) funmake ('(H [n]), [%e]);
          (%o4)                        H (%e)
                                        n
          (%i5) ''%;
                                              n
          (%o5)                       (%e - 1)

     `funmake' applied to a symbol which is not a defined function of
     any kind.

          (%i1) funmake (A, [u]);
          (%o1)                         A(u)
          (%i2) ''%;
          (%o2)                         A(u)

     `funmake' evaluates its arguments, but not the return value.

          (%i1) det(a,b,c) := b^2 -4*a*c;
                                              2
          (%o1)              det(a, b, c) := b  - 4 a c
          (%i2) (x : 8, y : 10, z : 12);
          (%o2)                          12
          (%i3) f : det;
          (%o3)                          det
          (%i4) funmake (f, [x, y, z]);
          (%o4)                    det(8, 10, 12)
          (%i5) ''%;
          (%o5)                         - 284

     Maxima simplifies `funmake''s return value.

          (%i1) funmake (sin, [%pi / 2]);
          (%o1)                           1

Function: lambda ([<x_1>, ..., <x_m>], <expr_1>, ..., <expr_n>)
Function: lambda ([[<L>]], <expr_1>, ..., <expr_n>)
Function: lambda ([<x_1>, ..., <x_m>, [<L>]], <expr_1>, ..., <expr_n>)

     Defines and returns a lambda expression (that is, an anonymous
     function).  The function may have required arguments <x_1>, ...,
     <x_m> and/or optional arguments <L>, which appear within the
     function body as a list.  The return value of the function is
     <expr_n>.  A lambda expression can be assigned to a variable and
     evaluated like an ordinary function.  A lambda expression may
     appear in some contexts in which a function name is expected.

     When the function is evaluated, unbound local variables <x_1>, ...,
     <x_m> are created.  `lambda' may appear within `block' or another
     `lambda'; local variables are established each time another
     `block' or `lambda' is evaluated.  Local variables appear to be
     global to any enclosed `block' or `lambda'.  If a variable is not
     local, its value is the value most recently assigned in an
     enclosing `block' or `lambda', if any, otherwise, it is the value
     of the variable in the global environment.  This policy may
     coincide with the usual understanding of "dynamic scope".

     After local variables are established, <expr_1> through <expr_n>
     are evaluated in turn.  The special variable `%%', representing
     the value of the preceding expression, is recognized.  `throw' and
     `catch' may also appear in the list of expressions.

     `return' cannot appear in a lambda expression unless enclosed by
     `block', in which case `return' defines the return value of the
     block and not of the lambda expression, unless the block happens
     to be <expr_n>.  Likewise, `go' cannot appear in a lambda
     expression unless enclosed by `block'.

     `lambda' quotes its arguments; the quote-quote operator `'''
     defeats quotation.

     Examples:

        * A lambda expression can be assigned to a variable and
          evaluated like an ordinary function.

          (%i1) f: lambda ([x], x^2);
                                                2
          (%o1)                    lambda([x], x )
          (%i2) f(a);
                                          2
          (%o2)                          a

        * A lambda expression may appear in contexts in which a
          function evaluation is expected.

          (%i3) lambda ([x], x^2) (a);
                                          2
          (%o3)                          a
          (%i4) apply (lambda ([x], x^2), [a]);
                                          2
          (%o4)                          a
          (%i5) map (lambda ([x], x^2), [a, b, c, d, e]);
                                  2   2   2   2   2
          (%o5)                 [a , b , c , d , e ]

        * Argument variables are local variables.  Other variables
          appear to be global variables.  Global variables are
          evaluated at the time the lambda expression is evaluated,
          unless some special evaluation is forced by some means, such
          as `'''.

          (%i6) a: %pi$
          (%i7) b: %e$
          (%i8) g: lambda ([a], a*b);
          (%o8)                   lambda([a], a b)
          (%i9) b: %gamma$
          (%i10) g(1/2);
                                       %gamma
          (%o10)                       ------
                                         2
          (%i11) g2: lambda ([a], a*''b);
          (%o11)                lambda([a], a %gamma)
          (%i12) b: %e$
          (%i13) g2(1/2);
                                       %gamma
          (%o13)                       ------
                                         2

        * Lambda expressions may be nested.  Local variables within the
          outer lambda expression appear to be global to the inner
          expression unless masked by local variables of the same names.

          (%i14) h: lambda ([a, b], h2: lambda ([a], a*b), h2(1/2));
                                                             1
          (%o14)    lambda([a, b], h2 : lambda([a], a b), h2(-))
                                                             2
          (%i15) h(%pi, %gamma);
                                       %gamma
          (%o15)                       ------
                                         2

        * Since `lambda' quotes its arguments, lambda expression `i'
          below does not define a "multiply by `a'" function.  Such a
          function can be defined via `buildq', as in lambda expression
          `i2' below.

          (%i16) i: lambda ([a], lambda ([x], a*x));
          (%o16)            lambda([a], lambda([x], a x))
          (%i17) i(1/2);
          (%o17)                  lambda([x], a x)
          (%i18) i2: lambda([a], buildq([a: a], lambda([x], a*x)));
          (%o18)    lambda([a], buildq([a : a], lambda([x], a x)))
          (%i19) i2(1/2);
                                               x
          (%o19)                   lambda([x], -)
                                               2
          (%i20) i2(1/2)(%pi);
                                         %pi
          (%o20)                         ---
                                          2

        * A lambda expression may take a variable number of arguments,
          which are indicated by `[<L>]' as the sole or final argument.
          The arguments appear within the function body as a list.

          (%i1) f : lambda ([aa, bb, [cc]], aa * cc + bb);
          (%o1)          lambda([aa, bb, [cc]], aa cc + bb)
          (%i2) f (foo, %i, 17, 29, 256);
          (%o2)       [17 foo + %i, 29 foo + %i, 256 foo + %i]
          (%i3) g : lambda ([[aa]], apply ("+", aa));
          (%o3)             lambda([[aa]], apply(+, aa))
          (%i4) g (17, 29, x, y, z, %e);
          (%o4)                  z + y + x + %e + 46

Function: local (<v_1>, ..., <v_n>)

     Saves the properties associated with the symbols <v_1>, ..., <v_n>,
     removes any properties before evaluating other expressions, and
     restores any saved properties on exit from the block or other
     compound expression in which `local' appears.

     Some declarations are implemented as properties of a symbol,
     including `:=', `array', `dependencies', `atvalue',
     `matchdeclare', `atomgrad', `constant', `nonscalar', `assume', and
     some others.  The effect of `local' is to make such declarations
     effective only within the block or other compound expression in
     which `local' appears; otherwise such declarations are global
     declarations.

     `local' can only appear in `block' or in the body of a function
     definition or `lambda' expression, and only one occurrence is
     permitted in each.

     `local' quotes its arguments.  `local' returns `done'.

     Example:

     A local function definition.

          (%i1) foo (x) := 1 - x;
          (%o1)                    foo(x) := 1 - x
          (%i2) foo (100);
          (%o2)                         - 99
          (%i3) block (local (foo), foo (x) := 2 * x, foo (100));
          (%o3)                          200
          (%i4) foo (100);
          (%o4)                         - 99

Option variable: macroexpansion

     Default value: `false'

     `macroexpansion' controls whether the expansion (that is, the
     return value) of a macro function is substituted for the macro
     function call.  A substitution may speed up subsequent expression
     evaluations, at the cost of storing the expansion.

    `false'
          The expansion of a macro function is not substituted for the
          macro function call.

    `expand'
          The first time a macro function call is evaluated, the
          expansion is stored.  The expansion is not recomputed on
          subsequent calls; any side effects (such as `print' or
          assignment to global variables) happen only when the macro
          function call is first evaluated.  Expansion in an expression
          does not affect other expressions which have the same macro
          function call.

    `displace'
          The first time a macro function call is evaluated, the
          expansion is substituted for the call, thus modifying the
          expression from which the macro function was called.  The
          expansion is not recomputed on subsequent calls; any side
          effects happen only when the macro function call is first
          evaluated.  Expansion in an expression does not affect other
          expressions which have the same macro function call.

     Examples

     When `macroexpansion' is `false', a macro function is called every
     time the calling expression is evaluated, and the calling
     expression is not modified.

          (%i1) f (x) := h (x) / g (x);
                                            h(x)
          (%o1)                     f(x) := ----
                                            g(x)
          (%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                                 return (x + 99));
          (%o2) g(x) ::= block(print("x + 99 is equal to", x),
                                                            return(x + 99))
          (%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                                 return (x - 99));
          (%o3) h(x) ::= block(print("x - 99 is equal to", x),
                                                            return(x - 99))
          (%i4) macroexpansion: false;
          (%o4)                         false
          (%i5) f (a * b);
          x - 99 is equal to x
          x + 99 is equal to x
                                      a b - 99
          (%o5)                       --------
                                      a b + 99
          (%i6) dispfun (f);
                                            h(x)
          (%t6)                     f(x) := ----
                                            g(x)

          (%o6)                         done
          (%i7) f (a * b);
          x - 99 is equal to x
          x + 99 is equal to x
                                      a b - 99
          (%o7)                       --------
                                      a b + 99

     When `macroexpansion' is `expand', a macro function is called once,
     and the calling expression is not modified.

          (%i1) f (x) := h (x) / g (x);
                                            h(x)
          (%o1)                     f(x) := ----
                                            g(x)
          (%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                                 return (x + 99));
          (%o2) g(x) ::= block(print("x + 99 is equal to", x),
                                                            return(x + 99))
          (%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                                 return (x - 99));
          (%o3) h(x) ::= block(print("x - 99 is equal to", x),
                                                            return(x - 99))
          (%i4) macroexpansion: expand;
          (%o4)                        expand
          (%i5) f (a * b);
          x - 99 is equal to x
          x + 99 is equal to x
                                      a b - 99
          (%o5)                       --------
                                      a b + 99
          (%i6) dispfun (f);
                                            h(x)
          (%t6)                     f(x) := ----
                                            g(x)

          (%o6)                         done
          (%i7) f (a * b);
                                      a b - 99
          (%o7)                       --------
                                      a b + 99

     When `macroexpansion' is `expand', a macro function is called once,
     and the calling expression is modified.

          (%i1) f (x) := h (x) / g (x);
                                            h(x)
          (%o1)                     f(x) := ----
                                            g(x)
          (%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                                 return (x + 99));
          (%o2) g(x) ::= block(print("x + 99 is equal to", x),
                                                            return(x + 99))
          (%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                                 return (x - 99));
          (%o3) h(x) ::= block(print("x - 99 is equal to", x),
                                                            return(x - 99))
          (%i4) macroexpansion: displace;
          (%o4)                       displace
          (%i5) f (a * b);
          x - 99 is equal to x
          x + 99 is equal to x
                                      a b - 99
          (%o5)                       --------
                                      a b + 99
          (%i6) dispfun (f);
                                           x - 99
          (%t6)                    f(x) := ------
                                           x + 99

          (%o6)                         done
          (%i7) f (a * b);
                                      a b - 99
          (%o7)                       --------
                                      a b + 99

Option variable: mode_checkp

     Default value: `true'

     When `mode_checkp' is `true', `mode_declare' checks the modes of
     bound variables.

Option variable: mode_check_errorp

     Default value: `false'

     When `mode_check_errorp' is `true', `mode_declare' calls error.

Option variable: mode_check_warnp

     Default value: `true'

     When `mode_check_warnp' is `true', mode errors are described.

Function: mode_declare (<y_1>, <mode_1>, ..., <y_n>, <mode_n>)

     `mode_declare' is used to declare the modes of variables and
     functions for subsequent translation or compilation of functions.
     `mode_declare' is typically placed at the beginning of a function
     definition, at the beginning of a Maxima script, or executed at
     the interactive prompt.

     The arguments of `mode_declare' are pairs consisting of a variable
     and a mode which is one of `boolean', `fixnum', `number',
     `rational', or `float'.  Each variable may also be a list of
     variables all of which are declared to have the same mode.

     If a variable is an array, and if every element of the array which
     is referenced has a value then `array (yi, complete, dim1, dim2,
     ...)' rather than
          array(yi, dim1, dim2, ...)
     should be used when first declaring the bounds of the array.  If
     all the elements of the array are of mode `fixnum' (`float'), use
     `fixnum' (`float') instead of `complete'.  Also if every element
     of the array is of the same mode, say `m', then

          mode_declare (completearray (yi), m))

     should be used for efficient translation.

     Numeric code using arrays might run faster by declaring the
     expected size of the array, as in:

          mode_declare (completearray (a [10, 10]), float)

     for a floating point number array which is 10 x 10.

     One may declare the mode of the result of a function by using
     `function (f_1, f_2, ...)' as an argument; here `f_1', `f_2', ...
     are the names of functions.  For example the expression,

          mode_declare ([function (f_1, f_2, ...)], fixnum)

     declares that the values returned by `f_1', `f_2', ... are
     single-word integers.

     `modedeclare' is a synonym for `mode_declare'.

Function: mode_identity (<arg_1>, <arg_2>)

     A special form used with `mode_declare' and `macros' to declare,
     e.g., a list of lists of flonums, or other compound data object.
     The first argument to `mode_identity' is a primitive value mode
     name as given to `mode_declare' (i.e., one of `float', `fixnum',
     `number', `list', or `any'), and the second argument is an
     expression which is evaluated and returned as the value of
     `mode_identity'.  However, if the return value is not allowed by
     the mode declared in the first argument, an error or warning is
     signalled.  The important thing is that the mode of the expression
     as determined by the Maxima to Lisp translator, will be that given
     as the first argument, independent of anything that goes on in the
     second argument.  E.g., `x: 3.3; mode_identity (fixnum, x);'
     yields an error.  `mode_identity (flonum, x)' returns 3.3 .  This
     has a number of uses, e.g., if you knew that `first (l)' returned a
     number then you might write `mode_identity (number, first (l))'.
     However, a more efficient way to do it would be to define a new
     primitive,

          firstnumb (x) ::= buildq ([x], mode_identity (number, x));

     and use `firstnumb' every time you take the first of a list of
     numbers.

Function: remfunction (<f_1>, ..., <f_n>)
Function: remfunction (all)

     Unbinds the function definitions of the symbols <f_1>, ..., <f_n>.
     The arguments may be the names of ordinary functions (created by
     `:='  or `define' ) or macro functions (created by `::=' ).

     `remfunction (all)' unbinds all function definitions.

     `remfunction' quotes its arguments.

     `remfunction' returns a list of the symbols for which the function
     definition was unbound.  `false' is returned in place of any
     symbol for which there is no function definition.

     `remfunction' does not apply to array functions or subscripted
     functions.  `remarray'  applies to those types of functions.

Option variable: savedef

     Default value: `true'

     When `savedef' is `true', the Maxima version of a user function is
     preserved when the function is translated.  This permits the
     definition to be displayed by `dispfun' and allows the function to
     be edited.

     When `savedef' is `false', the names of translated functions are
     removed from the `functions' list.

Option variable: transcompile

     Default value: `true'

     When `transcompile' is `true', `translate' and `translate_file'
     generate declarations to make the translated code more suitable
     for compilation.

     `compfile' sets `transcompile: true' for the duration.

Function: translate (<f_1>, ..., <f_n>)
Function: translate (functions)
Function: translate (all)

     Translates the user-defined functions <f_1>, ..., <f_n> from the
     Maxima language into Lisp and evaluates the Lisp translations.
     Typically the translated functions run faster than the originals.

     `translate (all)' or `translate (functions)' translates all
     user-defined functions.

     Functions to be translated should include a call to `mode_declare'
     at the beginning when possible in order to produce more efficient
     code.  For example:

          f (x_1, x_2, ...) := block ([v_1, v_2, ...],
              mode_declare (v_1, mode_1, v_2, mode_2, ...), ...)

     where the <x_1>, <x_2>, ...  are the parameters to the function and
     the <v_1>, <v_2>, ... are the local variables.

     The names of translated functions are removed from the `functions'
     list if `savedef' is `false' (see below) and are added to the
     `props' lists.

     Functions should not be translated unless they are fully debugged.

     Expressions are assumed simplified; if they are not, correct but
     non-optimal code gets generated.  Thus, the user should not set
     the `simp' switch to `false' which inhibits simplification of the
     expressions to be translated.

     The switch `translate', if `true', causes automatic translation of
     a user's function to Lisp.

     Note that translated functions may not run identically to the way
     they did before translation as certain incompatabilities may exist
     between the Lisp and Maxima versions.  Principally, the `rat'
     function with more than one argument and the `ratvars' function
     should not be used if any variables are `mode_declare''d canonical
     rational expressions (CRE).  Also the `prederror: false' setting
     will not translate.

     `savedef' - if `true' will cause the Maxima version of a user
     function to remain when the function is `translate''d.  This
     permits the definition to be displayed by `dispfun' and allows the
     function to be edited.

     `transrun' - if `false' will cause the interpreted version of all
     functions to be run (provided they are still around) rather than
     the translated version.

     The result returned by `translate' is a list of the names of the
     functions translated.

Function: translate_file (<maxima_filename>)
Function: translate_file (<maxima_filename>, <lisp_filename>)

     Translates a file of Maxima code into a file of Lisp code.
     `translate_file' returns a list of three filenames: the name of
     the Maxima file, the name of the Lisp file, and the name of file
     containing additional information about the translation.
     `translate_file' evaluates its arguments.

     `translate_file ("foo.mac"); load("foo.LISP")' is the same as the
     command `batch ("foo.mac")' except for certain restrictions, the
     use of `''' and `%', for example.

     `translate_file (<maxima_filename>)' translates a Maxima file
     <maxima_filename> into a similarly-named Lisp file.  For example,
     `foo.mac' is translated into `foo.LISP'.  The Maxima filename may
     include a directory name or names, in which case the Lisp output
     file is written to the same directory from which the Maxima input
     comes.

     `translate_file (<maxima_filename>, <lisp_filename>)' translates a
     Maxima file <maxima_filename> into a Lisp file <lisp_filename>.
     `translate_file' ignores the filename extension, if any, of
     `lisp_filename'; the filename extension of the Lisp output file is
     always `LISP'.  The Lisp filename may include a directory name or
     names, in which case the Lisp output file is written to the
     specified directory.

     `translate_file' also writes a file of translator warning messages
     of various degrees of severity.  The filename extension of this
     file is `UNLISP'.  This file may contain valuable information,
     though possibly obscure, for tracking down bugs in translated code.
     The `UNLISP' file is always written to the same directory from
     which the Maxima input comes.

     `translate_file' emits Lisp code which causes some declarations
     and definitions to take effect as soon as the Lisp code is
     compiled.  See `compile_file' for more on this topic.

     See also
     `tr_array_as_ref',
     `tr_bound_function_applyp',
     `tr_exponent',
     `tr_file_tty_messagesp',
     `tr_float_can_branch_complex',
     `tr_function_call_default',
     `tr_numer',
     `tr_optimize_max_loop',
     `tr_semicompile',
     `tr_state_vars',
     `tr_warnings_get',
     `tr_warn_bad_function_calls',
     `tr_warn_fexpr',
     `tr_warn_meval',
     `tr_warn_mode',
     `tr_warn_undeclared',
     and `tr_warn_undefined_variable'.

Option variable: transrun

     Default value: `true'

     When `transrun' is `false' will cause the interpreted version of
     all functions to be run (provided they are still around) rather
     than the translated version.

Option variable: tr_array_as_ref

     Default value: `true'

     If `translate_fast_arrays' is `false', array references in Lisp
     code emitted by `translate_file' are affected by `tr_array_as_ref'.
     When `tr_array_as_ref' is `true', array names are evaluated,
     otherwise array names appear as literal symbols in translated code.

     `tr_array_as_ref' has no effect if `translate_fast_arrays' is
     `true'.

Option variable: tr_bound_function_applyp

     Default value: `true'

     When `tr_bound_function_applyp' is `true', Maxima gives a warning
     if a bound variable (such as a function argument) is found being
     used as a function.  `tr_bound_function_applyp' does not affect
     the code generated in such cases.

     For example, an expression such as `g (f, x) := f (x+1)' will
     trigger the warning message.

Option variable: tr_file_tty_messagesp

     Default value: `false'

     When `tr_file_tty_messagesp' is `true', messages generated by
     `translate_file' during translation of a file are displayed on the
     console and inserted into the UNLISP file.  When `false', messages
     about translation of the file are only inserted into the UNLISP
     file.

Option variable: tr_float_can_branch_complex

     Default value: `true'

     Tells the Maxima-to-Lisp translator to assume that the functions
     `acos', `asin', `asec', and `acsc' can return complex results.

     The ostensible effect of `tr_float_can_branch_complex' is the
     following.  However, it appears that this flag has no effect on
     the translator output.

     When it is `true' then `acos(x)' is of mode `any' even if `x' is
     of mode `float' (as set by `mode_declare').  When `false' then
     `acos(x)' is of mode `float' if and only if `x' is of mode `float'.

Option variable: tr_function_call_default

     Default value: `general'

     `false' means give up and call `meval', `expr' means assume Lisp
     fixed arg function.  `general', the default gives code good for
     `mexprs' and `mlexprs' but not `macros'.  `general' assures
     variable bindings are correct in compiled code.  In `general'
     mode, when translating F(X), if F is a bound variable, then it
     assumes that `apply (f, [x])' is meant, and translates a such,
     with appropriate warning.  There is no need to turn this off.
     With the default settings, no warning messages implies full
     compatibility of translated and compiled code with the Maxima
     interpreter.

Option variable: tr_numer

     Default value: `false'

     When `tr_numer' is `true', `numer' properties are used for atoms
     which have them, e.g. `%pi'.

Option variable: tr_optimize_max_loop

     Default value: 100

     `tr_optimize_max_loop' is the maximum number of times the
     macro-expansion and optimization pass of the translator will loop
     in considering a form.  This is to catch macro expansion errors,
     and non-terminating optimization properties.

Option variable: tr_semicompile

     Default value: `false'

     When `tr_semicompile' is `true', `translate_file' and `compfile'
     output forms which will be macroexpanded but not compiled into
     machine code by the Lisp compiler.

System variable: tr_state_vars

     Default value:
          [transcompile, tr_semicompile, tr_warn_undeclared, tr_warn_meval,
          tr_warn_fexpr, tr_warn_mode, tr_warn_undefined_variable,
          tr_function_call_default, tr_array_as_ref,tr_numer]

     The list of the switches that affect the form of the translated
     output.  This information is useful to system people when trying
     to debug the translator.  By comparing the translated product to
     what should have been produced for a given state, it is possible to
     track down bugs.

Function: tr_warnings_get ()

     Prints a list of warnings which have been given by the translator
     during the current translation.

Option variable: tr_warn_bad_function_calls

     Default value: `true'

     - Gives a warning when when function calls are being made which
     may not be correct due to improper declarations that were made at
     translate time.

Option variable: tr_warn_fexpr

     Default value: `compfile'

     - Gives a warning if any FEXPRs are encountered.  FEXPRs should
     not normally be output in translated code, all legitimate special
     program forms are translated.

Option variable: tr_warn_meval

     Default value: `compfile'

     - Gives a warning if the function `meval' gets called.  If `meval'
     is called that indicates problems in the translation.

Option variable: tr_warn_mode

     Default value: `all'

     - Gives a warning when variables are assigned values inappropriate
     for their mode.

Option variable: tr_warn_undeclared

     Default value: `compile'

     - Determines when to send warnings about undeclared variables to
     the TTY.

Option variable: tr_warn_undefined_variable

     Default value: `all'

     - Gives a warning when undefined global variables are seen.

Function: compile_file (<filename>)
Function: compile_file (<filename>, <compiled_filename>)
Function: compile_file (<filename>, <compiled_filename>, <lisp_filename>)

     Translates the Maxima file <filename> into Lisp, executes the Lisp
     compiler, and, if the translation and compilation succeed, loads
     the compiled code into Maxima.

     `compile_file' returns a list of the names of four files: the
     original Maxima file, the Lisp translation, notes on translation,
     and the compiled code.  If the compilation fails, the fourth item
     is `false'.

     Some declarations and definitions take effect as soon as the Lisp
     code is compiled (without loading the compiled code).  These
     include functions defined with the `:=' operator, macros define
     with the `::=' operator, `alias', `declare', `define_variable',
     `mode_declare', and `infix', `matchfix', `nofix', `postfix',
     `prefix', and `compfile'.

     Assignments and function calls are not evaluated until the
     compiled code is loaded.  In particular, within the Maxima file,
     assignments to the translation flags (`tr_numer', etc.) have no
     effect on the translation.

     <filename> may not contain `:lisp' statements.

     `compile_file' evaluates its arguments.

Function: declare_translated (<f_1>, <f_2>, ...)

     When translating a file of Maxima code to Lisp, it is important
     for the translator to know which functions it sees in the file are
     to be called as translated or compiled functions, and which ones
     are just Maxima functions or undefined.  Putting this declaration
     at the top of the file, lets it know that although a symbol does
     which does not yet have a Lisp function value, will have one at
     call time.  `(MFUNCTION-CALL fn arg1 arg2 ...)' is generated when
     the translator does not know `fn' is going to be a Lisp function.

37 Program Flow


Lisp and Maxima
Garbage Collection
Introduction to Program Flow
Functions and Variables for Program Flow

37.1 Lisp and Maxima

Maxima is written in Lisp, and it is easy to access Lisp functions and
variables from Maxima and vice versa.  Lisp and Maxima symbols are
distinguished by a naming convention.  A Lisp symbol which begins with
a dollar sign `$' corresponds to a Maxima symbol without the dollar
sign.

   A Maxima symbol which begins with a question mark `?' corresponds to
a Lisp symbol without the question mark.  For example, the Maxima
symbol `foo' corresponds to the Lisp symbol `$FOO', while the Maxima
symbol `?foo' corresponds to the Lisp symbol `FOO'.  Note that `?foo'
is written without a space between `?' and `foo'; otherwise it might be
mistaken for `describe ("foo")'.

   Hyphen `-', asterisk `*', or other special characters in Lisp symbols
must be escaped by backslash `\' where they appear in Maxima code.  For
example, the Lisp identifier `*foo-bar*' is written `?\*foo\-bar\*' in
Maxima.

   Lisp code may be executed from within a Maxima session.  A single
line of Lisp (containing one or more forms) may be executed by the
special command `:lisp'.  For example,

     (%i1) :lisp (foo $x $y)

calls the Lisp function `foo' with Maxima variables `x' and `y' as
arguments.  The `:lisp' construct can appear at the interactive prompt
or in a file processed by `batch'  or `demo',  but not in a file
processed by `load',  `batchload',

   `translate_file',  or `compile_file'.

   The function `to_lisp'  opens an interactive Lisp session.  Entering
`(to-maxima)' closes the Lisp session and returns to Maxima.

   Lisp functions and variables which are to be visible in Maxima as
functions and variables with ordinary names (no special punctuation)
must have Lisp names beginning with the dollar sign `$'.

   Maxima is case-sensitive, distinguishing between lowercase and
uppercase letters in identifiers.  There are some rules governing the
translation of names between Lisp and Maxima.

  1. A Lisp identifier not enclosed in vertical bars corresponds to a
     Maxima identifier in lowercase.  Whether the Lisp identifier is
     uppercase, lowercase, or mixed case, is ignored.  E.g., Lisp
     `$foo', `$FOO', and `$Foo' all correspond to Maxima `foo'.  But
     this is because `$foo', `$FOO' and `$Foo' are converted by the
     Lisp reader by default to the Lisp symbol `$FOO'.

  2. A Lisp identifier which is all uppercase or all lowercase and
     enclosed in vertical bars corresponds to a Maxima identifier with
     case reversed.  That is, uppercase is changed to lowercase and
     lowercase to uppercase.  E.g., Lisp `|$FOO|' and `|$foo|'
     correspond to Maxima `foo' and `FOO', respectively.

  3. A Lisp identifier which is mixed uppercase and lowercase and
     enclosed in vertical bars corresponds to a Maxima identifier with
     the same case.  E.g., Lisp `|$Foo|' corresponds to Maxima `Foo'.

   The `#$' Lisp macro allows the use of Maxima expressions in Lisp
code.  `#$<expr>$' expands to a Lisp expression equivalent to the Maxima
expression <expr>.

     (msetq $foo #$[x, y]$)

This has the same effect as entering

     (%i1) foo: [x, y];

The Lisp function `displa' prints an expression in Maxima format.

     (%i1) :lisp #$[x, y, z]$
     ((MLIST SIMP) $X $Y $Z)
     (%i1) :lisp (displa '((MLIST SIMP) $X $Y $Z))
     [x, y, z]
     NIL

   Functions defined in Maxima are not ordinary Lisp functions.  The
Lisp function `mfuncall' calls a Maxima function.  For example:

     (%i1) foo(x,y) := x*y$
     (%i2) :lisp (mfuncall '$foo 'a 'b)
     ((MTIMES SIMP) A B)

   Some Lisp functions are shadowed in the Maxima package, namely the
following.

   complement     continue      //
   float          functionp     array
   exp            listen        signum
   atan           asin          acos
   asinh          acosh         atanh
   tanh           cosh          sinh
   tan            break         gcd

37.2 Garbage Collection

Symbolic computation tends to create a good deal of garbage (temporary
or intermediate results that are eventually not used), and effective
handling of this can be crucial to successful completion of some
programs.

   Under GCL, on UNIX systems where the mprotect system call is
available (including SUN OS 4.0 and some variants of BSD) a stratified
garbage collection is available.  This limits the collection to pages
which have been recently written to.  See the GCL documentation under
ALLOCATE and GBC.  At the Lisp level doing (setq si::*notify-gbc* t)
will help you determine which areas might need more space.

   For other Lisps that run Maxima, we refer the reader to the
documentation for that Lisp on how to control GC.

37.3 Introduction to Program Flow

Maxima provides a `do' loop for iteration, as well as more primitive
constructs such as `go'.

37.4 Functions and Variables for Program Flow

Function: backtrace ()
Function: backtrace (<n>)

     Prints the call stack, that is, the list of functions which called
     the currently active function.

     `backtrace()' prints the entire call stack.

     `backtrace (<n>)' prints the <n> most recent functions, including
     the currently active function.

     `backtrace' can be called from a script, a function, or the
     interactive prompt (not only in a debugging context).

     Examples:

        * `backtrace()' prints the entire call stack.

               (%i1) h(x) := g(x/7)$
               (%i2) g(x) := f(x-11)$
               (%i3) f(x) := e(x^2)$
               (%i4) e(x) := (backtrace(), 2*x + 13)$
               (%i5) h(10);
               #0: e(x=4489/49)
               #1: f(x=-67/7)
               #2: g(x=10/7)
               #3: h(x=10)
                                             9615
               (%o5)                         ----
                                              49

        * `backtrace (<n>)' prints the <n> most recent functions,
          including the currently active function.

               (%i1) h(x) := (backtrace(1), g(x/7))$
               (%i2) g(x) := (backtrace(1), f(x-11))$
               (%i3) f(x) := (backtrace(1), e(x^2))$
               (%i4) e(x) := (backtrace(1), 2*x + 13)$
               (%i5) h(10);
               #0: h(x=10)
               #0: g(x=10/7)
               #0: f(x=-67/7)
               #0: e(x=4489/49)
                                             9615
               (%o5)                         ----
                                              49

Special operator: do

     The `do' statement is used for performing iteration.  Due to its
     great generality the `do' statement will be described in two parts.
     First the usual form will be given which is analogous to that used
     in several other programming languages (Fortran, Algol, PL/I,
     etc.); then the other features will be mentioned.

     There are three variants of this form that differ only in their
     terminating conditions.  They are:

        * `for <variable>: <initial_value> step <increment>       thru
          <limit> do <body>'

        * `for <variable>: <initial_value> step <increment>       while
          <condition> do <body>'

        * `for <variable>: <initial_value> step <increment>
          unless <condition> do <body>'

     (Alternatively, the `step' may be given after the termination
     condition or limit.)

     <initial_value>, <increment>, <limit>, and <body> can be any
     expressions.  If the increment is 1 then "`step 1'" may be omitted.

     The execution of the `do' statement proceeds by first assigning
     the <initial_value> to the <variable> (henceforth called the
     control-variable).  Then: (1) If the control-variable has exceeded
     the limit of a `thru' specification, or if the condition of the
     `unless' is `true', or if the condition of the `while' is `false'
     then the `do' terminates.  (2) The <body> is evaluated.  (3) The
     increment is added to the control-variable.  The process from (1)
     to (3) is performed repeatedly until the termination condition is
     satisfied.  One may also give several termination conditions in
     which case the `do' terminates when any of them is satisfied.

     In general the `thru' test is satisfied when the control-variable
     is greater than the <limit> if the <increment> was non-negative,
     or when the control-variable is less than the <limit> if the
     <increment> was negative.  The <increment> and <limit> may be
     non-numeric expressions as long as this inequality can be
     determined.  However, unless the <increment> is syntactically
     negative (e.g. is a negative number) at the time the `do'
     statement is input, Maxima assumes it will be positive when the
     `do' is executed.  If it is not positive, then the `do' may not
     terminate properly.

     Note that the <limit>, <increment>, and termination condition are
     evaluated each time through the loop.  Thus if any of these involve
     much computation, and yield a result that does not change during
     all the executions of the <body>, then it is more efficient to set
     a variable to their value prior to the `do' and use this variable
     in the `do' form.

     The value normally returned by a `do' statement is the atom
     `done'.  However, the function `return' may be used inside the
     <body> to exit the `do' prematurely and give it any desired value.
     Note however that a `return' within a `do' that occurs in a
     `block' will exit only the `do' and not the `block'.  Note also
     that the `go' function may not be used to exit from a `do' into a
     surrounding `block'.

     The control-variable is always local to the `do' and thus any
     variable may be used without affecting the value of a variable with
     the same name outside of the `do'.  The control-variable is unbound
     after the `do' terminates.

          (%i1) for a:-3 thru 26 step 7 do display(a)$
                                       a = - 3

                                        a = 4

                                       a = 11

                                       a = 18

                                       a = 25

          (%i1) s: 0$
          (%i2) for i: 1 while i <= 10 do s: s+i;
          (%o2)                         done
          (%i3) s;
          (%o3)                          55

     Note that the condition `while i <= 10' is equivalent to `unless i
     > 10' and also `thru 10'.

          (%i1) series: 1$
          (%i2) term: exp (sin (x))$
          (%i3) for p: 1 unless p > 7 do
                    (term: diff (term, x)/p,
                     series: series + subst (x=0, term)*x^p)$
          (%i4) series;
                            7    6     5    4    2
                           x    x     x    x    x
          (%o4)            -- - --- - -- - -- + -- + x + 1
                           90   240   15   8    2

     which gives 8 terms of the Taylor series for `e^sin(x)'.

          (%i1) poly: 0$
          (%i2) for i: 1 thru 5 do
                    for j: i step -1 thru 1 do
                        poly: poly + i*x^j$
          (%i3) poly;
                            5      4       3       2
          (%o3)          5 x  + 9 x  + 12 x  + 14 x  + 15 x
          (%i4) guess: -3.0$
          (%i5) for i: 1 thru 10 do
                    (guess: subst (guess, x, 0.5*(x + 10/x)),
                     if abs (guess^2 - 10) < 0.00005 then return (guess));
          (%o5)                  - 3.162280701754386

     This example computes the negative square root of 10 using the
     Newton- Raphson iteration a maximum of 10 times.  Had the
     convergence criterion not been met the value returned would have
     been `done'.

     Instead of always adding a quantity to the control-variable one
     may sometimes wish to change it in some other way for each
     iteration.  In this case one may use `next <expression>' instead of
     `step <increment>'.  This will cause the control-variable to be
     set to the result of evaluating <expression> each time through the
     loop.

          (%i6) for count: 2 next 3*count thru 20 do display (count)$
                                      count = 2

                                      count = 6

                                     count = 18

     As an alternative to `for <variable>: <value> ...do...' the syntax
     `for <variable> from <value> ...do...'  may be used.  This permits
     the `from <value>' to be placed after the `step' or `next' value
     or after the termination condition.  If `from <value>' is omitted
     then 1 is used as the initial value.

     Sometimes one may be interested in performing an iteration where
     the control-variable is never actually used.  It is thus
     permissible to give only the termination conditions omitting the
     initialization and updating information as in the following
     example to compute the square-root of 5 using a poor initial guess.

          (%i1) x: 1000$
          (%i2) thru 20 do x: 0.5*(x + 5.0/x)$
          (%i3) x;
          (%o3)                   2.23606797749979
          (%i4) sqrt(5), numer;
          (%o4)                   2.23606797749979

     If it is desired one may even omit the termination conditions
     entirely and just give `do <body>' which will continue to evaluate
     the <body> indefinitely.  In this case the function `return'
     should be used to terminate execution of the `do'.

          (%i1) newton (f, x):= ([y, df, dfx], df: diff (f ('x), 'x),
                    do (y: ev(df), x: x - f(x)/y,
                        if abs (f (x)) < 5e-6 then return (x)))$
          (%i2) sqr (x) := x^2 - 5.0$
          (%i3) newton (sqr, 1000);
          (%o3)                   2.236068027062195

     (Note that `return', when executed, causes the current value of
     `x' to be returned as the value of the `do'.  The `block' is
     exited and this value of the `do' is returned as the value of the
     `block' because the `do' is the last statement in the block.)

     One other form of the `do' is available in Maxima.  The syntax is:

          for <variable> in <list> <end_tests> do <body>

     The elements of <list> are any expressions which will successively
     be assigned to the `variable' on each iteration of the <body>.
     The optional termination tests <end_tests> can be used to
     terminate execution of the `do'; otherwise it will terminate when
     the <list> is exhausted or when a `return' is executed in the
     <body>.  (In fact, `list' may be any non-atomic expression, and
     successive parts are taken.)

          (%i1)  for f in [log, rho, atan] do ldisp(f(1))$
          (%t1)                                  0
          (%t2)                                rho(1)
                                               %pi
          (%t3)                                 ---
                                                4
          (%i4) ev(%t3,numer);
          (%o4)                             0.78539816

Function: errcatch (<expr_1>, ..., <expr_n>)

     Evaluates <expr_1>, ..., <expr_n> one by one and returns
     `[<expr_n>]' (a list) if no error occurs.  If an error occurs in
     the evaluation of any argument, `errcatch' prevents the error from
     propagating and returns the empty list `[]' without evaluating any
     more arguments.

     `errcatch' is useful in `batch' files where one suspects an error
     might occur which would terminate the `batch' if the error weren't
     caught.

Function: error (<expr_1>, ..., <expr_n>)
System variable: error

     Evaluates and prints <expr_1>, ..., <expr_n>, and then causes an
     error return to top level Maxima or to the nearest enclosing
     `errcatch'.

     The variable `error' is set to a list describing the error.  The
     first element of `error' is a format string, which merges all the
     strings among the arguments <expr_1>, ..., <expr_n>, and the
     remaining elements are the values of any non-string arguments.

     `errormsg()' formats and prints `error'.  This is effectively
     reprinting the most recent error message.

Option variable: error_size

     Default value: 10

     `error_size' modifies error messages according to the size of
     expressions which appear in them.  If the size of an expression
     (as determined by the Lisp function `ERROR-SIZE') is greater than
     `error_size', the expression is replaced in the message by a
     symbol, and the symbol is assigned the expression.  The symbols
     are taken from the list `error_syms'.

     Otherwise, the expression is smaller than `error_size', and the
     expression is displayed in the message.

     See also `error' and `error_syms'.

     Example:

     The size of `U', as determined by `ERROR-SIZE', is 24.

          (%i1) U: (C^D^E + B + A)/(cos(X-1) + 1)$

          (%i2) error_size: 20$

          (%i3) error ("Example expression is", U);

          Example expression is errexp1
           -- an error.  Quitting.  To debug this try debugmode(true);
          (%i4) errexp1;
                                      E
                                     D
                                    C   + B + A
          (%o4)                    --------------
                                   cos(X - 1) + 1
          (%i5) error_size: 30$

          (%i6) error ("Example expression is", U);

                                   E
                                  D
                                 C   + B + A
          Example expression is --------------
                                cos(X - 1) + 1
           -- an error.  Quitting.  To debug this try debugmode(true);

Option variable: error_syms

     Default value: `[errexp1, errexp2, errexp3]'

     In error messages, expressions larger than `error_size' are
     replaced by symbols, and the symbols are set to the expressions.
     The symbols are taken from the list `error_syms'.  The first
     too-large expression is replaced by `error_syms[1]', the second by
     `error_syms[2]', and so on.

     If there are more too-large expressions than there are elements of
     `error_syms', symbols are constructed automatically, with the
     <n>-th symbol equivalent to `concat ('errexp, <n>)'.

     See also `error' and `error_size'.

Function: errormsg ()

     Reprints the most recent error message.  The variable `error'
     holds the message, and `errormsg' formats and prints it.

Option variable: errormsg

     Default value: `true'

     When `false' the output of error messages is suppressed.

     The option variable `errormsg' can not be set in a block to a local
     value.  The global value of `errormsg' is always present.

          (%i1) errormsg;
          (%o1)                                true
          (%i2) sin(a,b);
          Wrong number of arguments to sin
           -- an error. To debug this try: debugmode(true);
          (%i3) errormsg:false;
          (%o3)                                false
          (%i4) sin(a,b);

           -- an error. To debug this try: debugmode(true);

     The option variable `errormsg' can not be set in a block to a
     local value.

          (%i1) f(bool):=block([errormsg:bool],
                               print ("value of errormsg is",errormsg))$
          (%i2) errormsg:true;
          (%o2)                                true
          (%i3) f(false);
          value of errormsg is true
          (%o3)                                true
          (%i4) errormsg:false;
          (%o4)                                false
          (%i5) f(true);
          value of errormsg is false
          (%o5)                                false

Special operator: for

     Used in iterations.  See `do' for a description of Maxima's
     iteration facilities.

Function: go (<tag>)

     is used within a `block' to transfer control to the statement of
     the block which is tagged with the argument to `go'.  To tag a
     statement, precede it by an atomic argument as another statement in
     the `block'.  For example:

          block ([x], x:1, loop, x+1, ..., go(loop), ...)

     The argument to `go' must be the name of a tag appearing in the
     same `block'.  One cannot use `go' to transfer to tag in a `block'
     other than the one containing the `go'.

Special operator: if

     Represents conditional evaluation.  Various forms of `if'
     expressions are recognized.

     `if <cond_1> then <expr_1> else <expr_0>' evaluates to <expr_1> if
     <cond_1> evaluates to `true', otherwise the expression evaluates
     to <expr_0>.

     The command `if <cond_1> then <expr_1> elseif <cond_2> then
     <expr_2> elseif ... else <expr_0>' evaluates to <expr_k> if
     <cond_k> is `true' and all preceding conditions are `false'.  If
     none of the conditions are `true', the expression evaluates to
     `expr_0'.

     A trailing `else false' is assumed if `else' is missing.  That is,
     the command `if <cond_1> then <expr_1>' is equivalent to `if
     <cond_1> then <expr_1> else false', and the command `if <cond_1>
     then <expr_1> elseif ... elseif <cond_n> then <expr_n>' is
     equivalent to `if <cond_1> then <expr_1> elseif ... elseif
     <cond_n> then <expr_n> else false'.

     The alternatives <expr_0>, ..., <expr_n> may be any Maxima
     expressions, including nested `if' expressions.  The alternatives
     are neither simplified nor evaluated unless the corresponding
     condition is `true'.

     The conditions <cond_1>, ..., <cond_n> are expressions which
     potentially or actually evaluate to `true' or `false'.  When a
     condition does not actually evaluate to `true' or `false', the
     behavior of `if' is governed by the global flag `prederror'.  When
     `prederror' is `true', it is an error if any evaluated condition
     does not evaluate to `true' or `false'.  Otherwise, conditions
     which do not evaluate to `true' or `false' are accepted, and the
     result is a conditional expression.

     Among other elements, conditions may comprise relational and
     logical operators as follows.

          Operation            Symbol      Type

          less than            <           relational infix
          less than            <=
            or equal to                    relational infix
          equality (syntactic) =           relational infix
          negation of =        #           relational infix
          equality (value)     equal       relational function
          negation of equal    notequal    relational function
          greater than         >=
            or equal to                    relational infix
          greater than         >           relational infix
          and                  and         logical infix
          or                   or          logical infix
          not                  not         logical prefix

Function: map (<f>, <expr_1>, ..., <expr_n>)

     Returns an expression whose leading operator is the same as that
     of the expressions <expr_1>, ..., <expr_n> but whose subparts are
     the results of applying <f> to the corresponding subparts of the
     expressions.  <f> is either the name of a function of n arguments
     or is a `lambda' form of n arguments.

     `maperror' - if `false' will cause all of the mapping functions to
     (1) stop when they finish going down the shortest <expr_i> if not
     all of the <expr_i> are of the same length and (2) apply <f> to
     [<expr_1>, <expr_2>, ...]  if the <expr_i> are not all the same
     type of object.  If `maperror' is `true' then an error message
     will be given in the above two instances.

     One of the uses of this function is to `map' a function (e.g.
     `partfrac') onto each term of a very large expression where it
     ordinarily wouldn't be possible to use the function on the entire
     expression due to an exhaustion of list storage space in the
     course of the computation.

          (%i1) map(f,x+a*y+b*z);
          (%o1)                        f(b z) + f(a y) + f(x)
          (%i2) map(lambda([u],partfrac(u,x)),x+1/(x^3+4*x^2+5*x+2));
                                     1       1        1
          (%o2)                     ----- - ----- + -------- + x
                                   x + 2   x + 1          2
                                                   (x + 1)
          (%i3) map(ratsimp, x/(x^2+x)+(y^2+y)/y);
                                                1
          (%o3)                            y + ----- + 1
                                              x + 1
          (%i4) map("=",[a,b],[-0.5,3]);
          (%o4)                          [a = - 0.5, b = 3]

Function: mapatom (<expr>)

     Returns `true' if and only if <expr> is treated by the mapping
     routines as an atom.  "Mapatoms" are atoms, numbers (including
     rational numbers), and subscripted variables.

Option variable: maperror

     Default value: `true'

     When `maperror' is `false', causes all of the mapping functions,
     for example

          map (<f>, <expr_1>, <expr_2>, ...)

     to (1) stop when they finish going down the shortest <expr_i> if
     not all of the <expr_i> are of the same length and (2) apply <f>
     to [<expr_1>, <expr_2>, ...] if the <expr_i> are not all the same
     type of object.

     If `maperror' is `true' then an error message is displayed in the
     above two instances.

Option variable: mapprint

     Default value: `true'

     When `mapprint' is `true', various information messages from
     `map', `mapl', and `fullmap' are produced in certain situations.
     These include situations where `map' would use `apply', or `map'
     is truncating on the shortest list.

     If `mapprint' is `false', these messages are suppressed.

Function: maplist (<f>, <expr_1>, ..., <expr_n>)

     Returns a list of the applications of <f> to the parts of the
     expressions <expr_1>, ..., <expr_n>.  <f> is the name of a
     function, or a lambda expression.

     `maplist' differs from `map(<f>, <expr_1>, ..., <expr_n>)' which
     returns an expression with the same main operator as <expr_i> has
     (except for simplifications and the case where `map' does an
     `apply').

Option variable: prederror

     Default value: `false'

     When `prederror' is `true', an error message is displayed whenever
     the predicate of an `if' statement or an `is' function fails to
     evaluate to either `true' or `false'.

     If `false', `unknown' is returned instead in this case.  The
     `prederror: false' mode is not supported in translated code;
     however, `maybe' is supported in translated code.

     See also `is' and `maybe'.

Function: return (<value>)

     May be used to exit explicitly from a block, bringing its
     argument.  See `block' for more information.

Function: scanmap (<f>, <expr>)
Function: scanmap (<f>, <expr>, bottomup)

     Recursively applies <f> to <expr>, in a top down manner.  This is
     most useful when complete factorization is desired, for example:

          (%i1) exp:(a^2+2*a+1)*y + x^2$
          (%i2) scanmap(factor,exp);
                                              2      2
          (%o2)                         (a + 1)  y + x

     Note the way in which `scanmap' applies the given function
     `factor' to the constituent subexpressions of <expr>; if another
     form of <expr> is presented to `scanmap' then the result may be
     different.  Thus, `%o2' is not recovered when `scanmap' is applied
     to the expanded form of `exp':

          (%i3) scanmap(factor,expand(exp));
                                     2                  2
          (%o3)                      a  y + 2 a y + y + x

     Here is another example of the way in which `scanmap' recursively
     applies a given function to all subexpressions, including
     exponents:

          (%i4) expr : u*v^(a*x+b) + c$
          (%i5) scanmap('f, expr);
                              f(f(f(a) f(x)) + f(b))
          (%o5) f(f(f(u) f(f(v)                      )) + f(c))

     `scanmap (<f>, <expr>, bottomup)' applies <f> to <expr> in a
     bottom-up manner.  E.g., for undefined `f',

          scanmap(f,a*x+b) ->
             f(a*x+b) -> f(f(a*x)+f(b)) -> f(f(f(a)*f(x))+f(b))
          scanmap(f,a*x+b,bottomup) -> f(a)*f(x)+f(b)
              -> f(f(a)*f(x))+f(b) ->
               f(f(f(a)*f(x))+f(b))

     In this case, you get the same answer both ways.

Function: throw (<expr>)

     Evaluates <expr> and throws the value back to the most recent
     `catch'.  `throw' is used with `catch' as a nonlocal return
     mechanism.

Special operator: while
Special operator: unless

     See `do'.

Function: outermap (<f>, <a_1>, ..., <a_n>)

     Applies the function <f> to each one of the elements of the outer
     product <a_1> cross <a_2> ... cross <a_n>.

     <f> is the name of a function of n arguments or a lambda expression
     of n arguments.  Each argument <a_k> may be a list or nested list,
     or a matrix, or any other kind of expression.

     The `outermap' return value is a nested structure.  Let <x> be the
     return value.  Then <x> has the same structure as the first list,
     nested list, or matrix argument, `<x>[i_1]...[i_m]' has the same
     structure as the second list, nested list, or matrix argument,
     `<x>[i_1]...[i_m][j_1]...[j_n]' has the same structure as the third
     list, nested list, or matrix argument, and so on, where <m>, <n>,
     ... are the numbers of indices required to access the elements of
     each argument (one for a list, two for a matrix, one or more for a
     nested list).  Arguments which are not lists or matrices have no
     effect on the structure of the return value.

     Note that the effect of `outermap' is different from that of
     applying <f> to each one of the elements of the outer product
     returned by `cartesian_product'.  `outermap' preserves the
     structure of the arguments in the return value, while
     `cartesian_product' does not.

     `outermap' evaluates its arguments.

     See also `map', `maplist', and `apply'.

     Examples:

     Elementary examples of `outermap'.  To show the argument
     combinations more clearly, `F' is left undefined.

          (%i1) outermap(F, [a, b, c], [1, 2, 3]);
          (%o1) [[F(a, 1), F(a, 2), F(a, 3)], [F(b, 1), F(b, 2), F(b, 3)],
                                               [F(c, 1), F(c, 2), F(c, 3)]]
          (%i2) outermap(F, matrix([a, b],[c, d]), matrix([1, 2],[3, 4]));
                   [ [ F(a, 1)  F(a, 2) ]  [ F(b, 1)  F(b, 2) ] ]
                   [ [                  ]  [                  ] ]
                   [ [ F(a, 3)  F(a, 4) ]  [ F(b, 3)  F(b, 4) ] ]
          (%o2)    [                                            ]
                   [ [ F(c, 1)  F(c, 2) ]  [ F(d, 1)  F(d, 2) ] ]
                   [ [                  ]  [                  ] ]
                   [ [ F(c, 3)  F(c, 4) ]  [ F(d, 3)  F(d, 4) ] ]
          (%i3) outermap (F, [a, b], x, matrix ([1, 2], [3, 4]));
                 [ F(a, x, 1)  F(a, x, 2) ]  [ F(b, x, 1)  F(b, x, 2) ]
          (%o3) [[                        ], [                        ]]
                 [ F(a, x, 3)  F(a, x, 4) ]  [ F(b, x, 3)  F(b, x, 4) ]
          (%i4) outermap (F, [a, b], matrix ([1, 2]), matrix ([x], [y]));
                 [ [ F(a, 1, x) ]  [ F(a, 2, x) ] ]
          (%o4) [[ [            ]  [            ] ],
                 [ [ F(a, 1, y) ]  [ F(a, 2, y) ] ]
                                        [ [ F(b, 1, x) ]  [ F(b, 2, x) ] ]
                                        [ [            ]  [            ] ]]
                                        [ [ F(b, 1, y) ]  [ F(b, 2, y) ] ]
          (%i5) outermap ("+", [a, b, c], [1, 2, 3]);
          (%o5) [[a + 1, a + 2, a + 3], [b + 1, b + 2, b + 3],
                                                     [c + 1, c + 2, c + 3]]

     A closer examination of the `outermap' return value.  The first,
     second, and third arguments are a matrix, a list, and a matrix,
     respectively.  The return value is a matrix.  Each element of that
     matrix is a list, and each element of each list is a matrix.

          (%i1) arg_1 :  matrix ([a, b], [c, d]);
                                      [ a  b ]
          (%o1)                       [      ]
                                      [ c  d ]
          (%i2) arg_2 : [11, 22];
          (%o2)                       [11, 22]
          (%i3) arg_3 : matrix ([xx, yy]);
          (%o3)                      [ xx  yy ]
          (%i4) xx_0 : outermap(lambda([x, y, z], x / y + z), arg_1,
                                                             arg_2, arg_3);
                         [  [      a        a  ]  [      a        a  ]  ]
                         [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                         [  [      11       11 ]  [      22       22 ]  ]
          (%o4)  Col 1 = [                                              ]
                         [  [      c        c  ]  [      c        c  ]  ]
                         [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                         [  [      11       11 ]  [      22       22 ]  ]
                           [  [      b        b  ]  [      b        b  ]  ]
                           [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                           [  [      11       11 ]  [      22       22 ]  ]
                   Col 2 = [                                              ]
                           [  [      d        d  ]  [      d        d  ]  ]
                           [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                           [  [      11       11 ]  [      22       22 ]  ]
          (%i5) xx_1 : xx_0 [1][1];
                     [      a        a  ]  [      a        a  ]
          (%o5)     [[ xx + --  yy + -- ], [ xx + --  yy + -- ]]
                     [      11       11 ]  [      22       22 ]
          (%i6) xx_2 : xx_0 [1][1] [1];
                                [      a        a  ]
          (%o6)                 [ xx + --  yy + -- ]
                                [      11       11 ]
          (%i7) xx_3 : xx_0 [1][1] [1] [1][1];
                                            a
          (%o7)                        xx + --
                                            11
          (%i8) [op (arg_1), op (arg_2), op (arg_3)];
          (%o8)                  [matrix, [, matrix]
          (%i9) [op (xx_0), op (xx_1), op (xx_2)];
          (%o9)                  [matrix, [, matrix]

     `outermap' preserves the structure of the arguments in the return
     value, while `cartesian_product' does not.

          (%i1) outermap (F, [a, b, c], [1, 2, 3]);
          (%o1) [[F(a, 1), F(a, 2), F(a, 3)], [F(b, 1), F(b, 2), F(b, 3)],
                                               [F(c, 1), F(c, 2), F(c, 3)]]
          (%i2) setify (flatten (%));
          (%o2) {F(a, 1), F(a, 2), F(a, 3), F(b, 1), F(b, 2), F(b, 3),
                                                 F(c, 1), F(c, 2), F(c, 3)}
          (%i3) map(lambda([L], apply(F, L)),
                               cartesian_product({a, b, c}, {1, 2, 3}));
          (%o3) {F(a, 1), F(a, 2), F(a, 3), F(b, 1), F(b, 2), F(b, 3),
                                                 F(c, 1), F(c, 2), F(c, 3)}
          (%i4) is (equal (%, %th (2)));
          (%o4)                         true

38 Debugging


Source Level Debugging
Keyword Commands
Functions and Variables for Debugging

38.1 Source Level Debugging

Maxima has a built-in source level debugger.  The user can set a
breakpoint at a function, and then step line by line from there.  The
call stack may be examined, together with the variables bound at that
level.

   The command `:help' or `:h' shows the list of debugger commands.
(In general, commands may be abbreviated if the abbreviation is unique.
If not unique, the alternatives will be listed.)  Within the debugger,
the user can also use any ordinary Maxima functions to examine, define,
and manipulate variables and expressions.

   A breakpoint is set by the `:br' command at the Maxima prompt.
Within the debugger, the user can advance one line at a time using the
`:n' ("next") command.  The `:bt' ("backtrace") command shows a list of
stack frames.  The `:r' ("resume") command exits the debugger and
continues with execution.  These commands are demonstrated in the
example below.

     (%i1) load ("/tmp/foobar.mac");

     (%o1)                           /tmp/foobar.mac

     (%i2) :br foo
     Turning on debugging debugmode(true)
     Bkpt 0 for foo (in /tmp/foobar.mac line 1)

     (%i2) bar (2,3);
     Bkpt 0:(foobar.mac 1)
     /tmp/foobar.mac:1::

     (dbm:1) :bt                        <-- :bt typed here gives a backtrace
     #0: foo(y=5)(foobar.mac line 1)
     #1: bar(x=2,y=3)(foobar.mac line 9)

     (dbm:1) :n                         <-- Here type :n to advance line
     (foobar.mac 2)
     /tmp/foobar.mac:2::

     (dbm:1) :n                         <-- Here type :n to advance line
     (foobar.mac 3)
     /tmp/foobar.mac:3::

     (dbm:1) u;                         <-- Investigate value of u
     28

     (dbm:1) u: 33;                     <-- Change u to be 33
     33

     (dbm:1) :r                         <-- Type :r to resume the computation

     (%o2)                                1094

   The file `/tmp/foobar.mac' is the following:

     foo(y) := block ([u:y^2],
       u: u+3,
       u: u^2,
       u);

     bar(x,y) := (
       x: x+2,
       y: y+2,
       x: foo(y),
       x+y);

   USE OF THE DEBUGGER THROUGH EMACS

   If the user is running the code under GNU emacs in a shell window
(dbl shell), or is running the graphical interface version, Xmaxima,
then if he stops at a break point, he will see his current position in
the source file which will be displayed in the other half of the
window, either highlighted in red, or with a little arrow pointing at
the right line.  He can advance single lines at a time by typing M-n
(Alt-n).

   Under Emacs you should run in a `dbl' shell, which requires the
`dbl.el' file in the elisp directory.  Make sure you install the elisp
files or add the Maxima elisp directory to your path: e.g., add the
following to your `.emacs' file or the `site-init.el'

     (setq load-path (cons "/usr/share/maxima/5.9.1/emacs" load-path))
     (autoload 'dbl "dbl")

   then in emacs

     M-x dbl

   should start a shell window in which you can run programs, for
example Maxima, gcl, gdb etc.   This shell window also knows about
source level debugging, and display of source code in the other window.

   The user may set a break point at a certain line of the file by
typing `C-x space'.  This figures out which function the cursor is in,
and then it sees which line of that function the cursor is on.   If the
cursor is on, say, line 2 of `foo', then it will insert in the other
window the command, "`:br foo 2'", to break `foo' at its second line.
To have this enabled, the user must have maxima-mode.el turned on in
the window in which the file `foobar.mac' is visiting.  There are
additional commands available in that file window, such as evaluating
the function into the Maxima, by typing `Alt-Control-x'.

38.2 Keyword Commands

Keyword commands are special keywords which are not interpreted as
Maxima expressions.  A keyword command can be entered at the Maxima
prompt or the debugger prompt, although not at the break prompt.
Keyword commands start with a colon, '`:''.  For example, to evaluate a
Lisp form you may type `:lisp' followed by the form to be evaluated.

     (%i1) :lisp (+ 2 3)
     5

   The number of arguments taken depends on the particular command.
Also, you need not type the whole command, just enough to be unique
among the break keywords.   Thus `:br' would suffice for `:break'.

   The keyword commands are listed below.

`:break F n'
     Set a breakpoint in function `F' at line offset `n' from the
     beginning of the function.  If `F' is given as a string, then it
     is assumed to be a file, and `n' is the offset from the beginning
     of the file.  The offset is optional. If not given, it is assumed
     to be zero (first line of the function or file).

`:bt'
     Print a backtrace of the stack frames

`:continue'
     Continue the computation

`:delete'
     Delete the specified breakpoints, or all if none are specified

`:disable'
     Disable the specified breakpoints, or all if none are specified

`:enable'
     Enable the specified breakpoints, or all if none are specified

`:frame n'
     Print stack frame `n', or the current frame if none is specified

`:help'
     Print help on a debugger command, or all commands if none is
     specified

`:info'
     Print information about item

`:lisp some-form'
     Evaluate `some-form' as a Lisp form

`:lisp-quiet some-form'
     Evaluate Lisp form `some-form' without any output

`:next'
     Like `:step', except `:next' steps over function calls

`:quit'
     Quit the current debugger level without completing the computation

`:resume'
     Continue the computation

`:step'
     Continue the computation until it reaches a new source line

`:top'
     Return to the Maxima prompt (from any debugger level) without
     completing the computation

38.3 Functions and Variables for Debugging

Option variable: debugmode

     Default value: `false'

     When a Maxima error occurs, Maxima will start the debugger if
     `debugmode' is `true'.  The user may enter commands to examine the
     call stack, set breakpoints, step through Maxima code, and so on.
     See `debugging' for a list of debugger commands.

     Enabling `debugmode' will not catch Lisp errors.

Option variable: refcheck

     Default value: `false'

     When `refcheck' is `true', Maxima prints a message each time a
     bound variable is used for the first time in a computation.

Option variable: setcheck

     Default value: `false'

     If `setcheck' is set to a list of variables (which can be
     subscripted), Maxima prints a message whenever the variables, or
     subscripted occurrences of them, are bound with the ordinary
     assignment operator `:', the `::' assignment operator, or function
     argument binding, but not the function assignment `:=' nor the
     macro assignment `::=' operators.  The message comprises the name
     of the variable and the value it is bound to.

     `setcheck' may be set to `all' or `true' thereby including all
     variables.

     Each new assignment of `setcheck' establishes a new list of
     variables to check, and any variables previously assigned to
     `setcheck' are forgotten.

     The names assigned to `setcheck' must be quoted if they would
     otherwise evaluate to something other than themselves.  For
     example, if `x', `y', and `z' are already bound, then enter

          setcheck: ['x, 'y, 'z]$

     to put them on the list of variables to check.

     No printout is generated when a variable on the `setcheck' list is
     assigned to itself, e.g., `X: 'X'.

Option variable: setcheckbreak

     Default value: `false'

     When `setcheckbreak' is `true', Maxima will present a break prompt
     whenever a variable on the `setcheck' list is assigned a new value.
     The break occurs before the assignment is carried out.  At this
     point, `setval' holds the value to which the variable is about to
     be assigned.  Hence, one may assign a different value by assigning
     to `setval'.

     See also `setcheck' and `setval'.

System variable: setval

     Holds the value to which a variable is about to be set when a
     `setcheckbreak' occurs.  Hence, one may assign a different value
     by assigning to `setval'.

     See also `setcheck' and `setcheckbreak'.

Function: timer (<f_1>, ..., <f_n>)
Function: timer (all)
Function: timer ()

     Given functions <f_1>, ..., <f_n>, `timer' puts each one on the
     list of functions for which timing statistics are collected.
     `timer(f)$ timer(g)$' puts `f' and then `g' onto the list; the
     list accumulates from one call to the next.

     `timer(all)' puts all user-defined functions (as named by the
     global variable `functions') on the list of timed functions.

     With no arguments,  `timer' returns the list of timed functions.

     Maxima records how much time is spent executing each function on
     the list of timed functions.  `timer_info' returns the timing
     statistics, including the average time elapsed per function call,
     the number of calls, and the total time elapsed.  `untimer'
     removes functions from the list of timed functions.

     `timer' quotes its arguments.  `f(x) := x^2$ g:f$ timer(g)$' does
     not put `f' on the timer list.

     If `trace(f)' is in effect, then `timer(f)' has no effect; `trace'
     and `timer' cannot both be in effect at the same time.

     See also `timer_devalue'.

Function: untimer (<f_1>, ..., <f_n>)
Function: untimer ()

     Given functions <f_1>, ..., <f_n>, `untimer' removes each function
     from the timer list.

     With no arguments, `untimer' removes all functions currently on
     the timer list.

     After `untimer (f)' is executed, `timer_info (f)' still returns
     previously collected timing statistics, although `timer_info()'
     (with no arguments) does not return information about any function
     not currently on the timer list.  `timer (f)' resets all timing
     statistics to zero and puts `f' on the timer list again.

Option variable: timer_devalue

     Default value: `false'

     When `timer_devalue' is `true', Maxima subtracts from each timed
     function the time spent in other timed functions.  Otherwise, the
     time reported for each function includes the time spent in other
     functions.  Note that time spent in untimed functions is not
     subtracted from the total time.

     See also `timer' and `timer_info'.

Function: timer_info (<f_1>, ..., <f_n>)
Function: timer_info ()

     Given functions <f_1>, ..., <f_n>, `timer_info' returns a matrix
     containing timing information for each function.  With no
     arguments, `timer_info' returns timing information for all
     functions currently on the timer list.

     The matrix returned by `timer_info' contains the function name,
     time per function call, number of function calls, total time, and
     `gctime', which meant "garbage collection time" in the original
     Macsyma but is now always zero.

     The data from which `timer_info' constructs its return value can
     also be obtained by the `get' function:

          get(f, 'calls);  get(f, 'runtime);  get(f, 'gctime);

     See also `timer'.

Function: trace (<f_1>, ..., <f_n>)
Function: trace (all)
Function: trace ()

     Given functions <f_1>, ..., <f_n>, `trace' instructs Maxima to
     print out debugging information whenever those functions are
     called.  `trace(f)$ trace(g)$' puts `f' and then `g' onto the list
     of functions to be traced; the list accumulates from one call to
     the next.

     `trace(all)' puts all user-defined functions (as named by the
     global variable `functions') on the list of functions to be traced.

     With no arguments, `trace' returns a list of all the functions
     currently being traced.

     The `untrace' function disables tracing.  See also `trace_options'.

     `trace' quotes its arguments.  Thus, `f(x) := x^2$ g:f$ trace(g)$'
     does not put `f' on the trace list.

     When a function is redefined, it is removed from the timer list.
     Thus after `timer(f)$ f(x) := x^2$', function `f' is no longer on
     the timer list.

     If `timer (f)' is in effect, then `trace (f)' has no effect;
     `trace' and `timer' can't both be in effect for the same function.

Function: trace_options (<f>, <option_1>, ..., <option_n>)
Function: trace_options (<f>)

     Sets the trace options for function <f>.  Any previous options are
     superseded.  `trace_options (<f>, ...)' has no effect unless
     `trace (<f>)' is also called (either before or after
     `trace_options').

     `trace_options (<f>)' resets all options to their default values.

     The option keywords are:

        * `noprint' Do not print a message at function entry and exit.

        * `break' Put a breakpoint before the function is entered, and
          after the function is exited.  See `break'.

        * `lisp_print' Display arguments and return values as Lisp
          objects.

        * `info' Print `-> true' at function entry and exit.

        * `errorcatch' Catch errors, giving the option to signal an
          error, retry the function call, or specify a return value.

     Trace options are specified in two forms.  The presence of the
     option keyword alone puts the option into effect unconditionally.
     (Note that option <foo> is not put into effect by specifying
     `<foo>: true' or a similar form; note also that keywords need not
     be quoted.) Specifying the option keyword with a predicate
     function makes the option conditional on the predicate.

     The argument list to the predicate function is always `[level,
     direction, function, item]' where `level' is the recursion level
     for the function,  `direction' is either `enter' or `exit',
     `function' is the name of the function, and `item' is the argument
     list (on entering) or the return value (on exiting).

     Here is an example of unconditional trace options:

          (%i1) ff(n) := if equal(n, 0) then 1 else n * ff(n - 1)$

          (%i2) trace (ff)$

          (%i3) trace_options (ff, lisp_print, break)$

          (%i4) ff(3);

     Here is the same function, with the `break' option conditional on
     a predicate:

          (%i5) trace_options (ff, break(pp))$

          (%i6) pp (level, direction, function, item) := block (print (item),
              return (function = 'ff and level = 3 and direction = exit))$

          (%i7) ff(6);

Function: untrace (<f_1>, ..., <f_n>)
Function: untrace ()

     Given functions <f_1>, ..., <f_n>, `untrace' disables tracing
     enabled by the `trace' function.  With no arguments, `untrace'
     disables tracing for all functions.

     `untrace' returns a list of the functions for which it disabled
     tracing.

39 asympa


Introduction to asympa
Functions and variables for asympa

39.1 Introduction to asympa

Function: asympa

     `asympa' is a package for asymptotic analysis. The package contains
     simplification functions for asymptotic analysis, including the
     "big O" and "little o" functions that are widely used in
     complexity analysis and numerical analysis.

     `load ("asympa")' loads this package.

39.2 Functions and variables for asympa

40 augmented_lagrangian


Functions and Variables for augmented_lagrangian

40.1 Functions and Variables for augmented_lagrangian

Function: augmented_lagrangian_method (<FOM>, <xx>, <C>, <yy>)
Function: augmented_lagrangian_method (<FOM>, <xx>, <C>, <yy>, optional_args)
Function: augmented_lagrangian_method ([<FOM>, <grad>], <xx>, <C>, <yy>)
Function: augmented_lagrangian_method ([<FOM>, <grad>], <xx>, <C>, <yy>, optional_args)

     Returns an approximate minimum of the expression <FOM> with
     respect to the variables <xx>, holding the constraints <C> equal
     to zero.  <yy> is a list of initial guesses for <xx>.  The method
     employed is the augmented Lagrangian method (see Refs [1] and [2]).

     <grad>, if present, is the gradient of <FOM> with respect to <xx>,
     represented as a list of expressions, one for each variable in
     <xx>.  If not present, the gradient is constructed automatically.

     <FOM> and each element of <grad>, if present, must be ordinary
     expressions, not names of functions or lambda expressions.

     `optional_args' represents additional arguments, specified as
     `<symbol> = <value>'.  The optional arguments recognized are:

    `niter'
          Number of iterations of the augmented Lagrangian algorithm

    `lbfgs_tolerance'
          Tolerance supplied to LBFGS

    `iprint'
          IPRINT parameter (a list of two integers which controls
          verbosity) supplied to LBFGS

    `%lambda'
          Initial value of `%lambda' to be used for calculating the
          augmented Lagrangian

     This implementation minimizes the augmented Lagrangian by applying
     the limited-memory BFGS (LBFGS) algorithm, which is a quasi-Newton
     algorithm.

     `load(augmented_lagrangian)' loads this function.

     See also `lbfgs'.

     References:

     [1]
     `http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/constrained/nonlinearcon/auglag.html'

     [2] `http://www.cs.ubc.ca/spider/ascher/542/chap10.pdf'

     Examples:

          (%i1) load (lbfgs);
          (%o1)     /maxima/share/lbfgs/lbfgs.mac
          (%i2) load (augmented_lagrangian);
          (%o2)
             /maxima/share/contrib/augmented_lagrangian.mac
          (%i3) FOM: x^2 + 2*y^2;
                                         2    2
          (%o3)                       2 y  + x
          (%i4) xx: [x, y];
          (%o4)                        [x, y]
          (%i5) C: [x + y - 1];
          (%o5)                      [y + x - 1]
          (%i6) yy: [1, 1];
          (%o6)                        [1, 1]
          (%i7) augmented_lagrangian_method(FOM, xx, C, yy, iprint=[-1,0]);
          (%o7) [[x = 0.66665984108002, y = 0.33334027245545],
                                           %lambda = [- 1.333337940892525]]

     Same example as before, but this time the gradient is supplied as
     an argument.

          (%i1) load (lbfgs)$
          (%i2) load (augmented_lagrangian)$
          (%i3) FOM: x^2 + 2*y^2;
                                         2    2
          (%o3)                       2 y  + x
          (%i4) FOM: x^2 + 2*y^2;
                                         2    2
          (%o4)                       2 y  + x
          (%i5) xx: [x, y];
          (%o5)                        [x, y]
          (%i6) grad : [2*x, 4*y];
          (%o6)                      [2 x, 4 y]
          (%i7) C: [x + y - 1];
          (%o7)                      [y + x - 1]
          (%i8) yy: [1, 1];
          (%o8)                        [1, 1]
          (%i9) augmented_lagrangian_method ([FOM, grad], xx, C, yy,
                                             iprint = [-1, 0]);
          (%o9) [[x = 0.666659841080025, y = .3333402724554462],
                                           %lambda = [- 1.333337940892543]]

41 Bernstein


Functions and Variables for Bernstein

41.1 Functions and Variables for Bernstein

Function: bernstein_poly (<k>, <n>, <x>)

     Provided `k' is not a negative integer, the Bernstein polynomials
     are defined by `bernstein_poly(k,n,x) = binomial(n,k) x^k
     (1-x)^(n-k)'; for a negative integer `k', the Bernstein polynomial
     `bernstein_poly(k,n,x)' vanishes.  When either `k' or `n' are non
     integers, the option variable `bernstein_explicit' controls the
     expansion of the Bernstein polynomials into its explicit form;
     example:

          (%i1) load(bernstein)$

          (%i2) bernstein_poly(k,n,x);
          (%o2)                bernstein_poly(k, n, x)
          (%i3) bernstein_poly(k,n,x), bernstein_explicit : true;
                                                 n - k  k
          (%o3)            binomial(n, k) (1 - x)      x

     The Bernstein polynomials have both a gradef property and an
     integrate property:

          (%i4) diff(bernstein_poly(k,n,x),x);
          (%o4) (bernstein_poly(k - 1, n - 1, x)
                                           - bernstein_poly(k, n - 1, x)) n
          (%i5) integrate(bernstein_poly(k,n,x),x);
          (%o5)
                                                                      k + 1
           hypergeometric([k + 1, k - n], [k + 2], x) binomial(n, k) x
           ----------------------------------------------------------------
                                        k + 1

     For numeric inputs, both real and complex, the Bernstein
     polynomials evaluate to a numeric result:

          (%i6) bernstein_poly(5,9, 1/2 + %i);
                                  39375 %i   39375
          (%o6)                   -------- + -----
                                    128       256
          (%i7) bernstein_poly(5,9, 0.5b0 + %i);
          (%o7)           3.076171875b2 %i + 1.5380859375b2

     To use `bernstein_poly', first `load("bernstein")'.

Variable: bernstein_explicit

     Default value: `false'

     When either `k' or `n' are non integers, the option variable
     `bernstein_explicit' controls the expansion of `bernstein(k,n,x)'
     into its explicit form; example:

          (%i1) bernstein_poly(k,n,x);
          (%o1)                bernstein_poly(k, n, x)
          (%i2) bernstein_poly(k,n,x), bernstein_explicit : true;
                                                 n - k  k
          (%o2)            binomial(n, k) (1 - x)      x
     When both `k' and `n' are explicitly integers, `bernstein(k,n,x)'
     _always_ expands to its explicit form.

Function: multibernstein_poly (<[k1,k2,..., kp]>, <[n1,n2,..., np]>, <[x1,x2,..., xp]>)

     The multibernstein polynomial `multibernstein_poly (<[k1, k2, ...,
     kp]>, <[n1, n2, ..., np]>, <[x1, x2, ..., xp]>)' is the product of
     bernstein polynomials `bernstein_poly(k1, n1, x1)
     bernstein_poly(k2, n2, x2) ... bernstein_poly(kp, np, xp)'.

     To use `multibernstein_poly', first `load("bernstein")'.

Function: bernstein_approx (<f>, <[x1, x1, ..., xn]>, n)

     Return the `n'-th order uniform Bernstein polynomial approximation
     for the function `(x1, x2, ..., xn) |--> f'.  Examples

          (%i1) bernstein_approx(f(x),[x], 2);
                           2       1                          2
          (%o1)      f(1) x  + 2 f(-) (1 - x) x + f(0) (1 - x)
                                   2
          (%i2) bernstein_approx(f(x,y),[x,y], 2);
                         2  2       1                2
          (%o2) f(1, 1) x  y  + 2 f(-, 1) (1 - x) x y
                                    2
                            2  2          1   2
           + f(0, 1) (1 - x)  y  + 2 f(1, -) x  (1 - y) y
                                          2
                 1  1                               1         2
           + 4 f(-, -) (1 - x) x (1 - y) y + 2 f(0, -) (1 - x)  (1 - y) y
                 2  2                               2
                      2        2       1                      2
           + f(1, 0) x  (1 - y)  + 2 f(-, 0) (1 - x) x (1 - y)
                                       2
                            2        2
           + f(0, 0) (1 - x)  (1 - y)

     To use `bernstein_approx', first `load("bernstein")'.

Function: bernstein_expand (<e>, <[x1, x1, ..., xn]>)

     Express the _polynomial_ `e' exactly as a linear combination of
     multi-variable Bernstein polynomials.

          (%i1) bernstein_expand(x*y+1,[x,y]);
          (%o1)    2 x y + (1 - x) y + x (1 - y) + (1 - x) (1 - y)
          (%i2) expand(%);
          (%o2)                        x y + 1

     Maxima signals an error when the first argument isn't a polynomial.

     To use `bernstein_expand', first `load("bernstein")'.

42 bode


Functions and Variables for bode

42.1 Functions and Variables for bode

Function: bode_gain (<H>, <range>, ...<plot_opts>...)

     Function to draw Bode gain plots.

     Examples (1 through 7 from
          `http://www.swarthmore.edu/NatSci/echeeve1/Ref/Bode/BodeHow.html',
     8 from Ron Crummett):
          (%i1) load("bode")$

          (%i2) H1 (s) := 100 * (1 + s) / ((s + 10) * (s + 100))$

          (%i3) bode_gain (H1 (s), [w, 1/1000, 1000])$

          (%i4) H2 (s) := 1 / (1 + s/omega0)$

          (%i5) bode_gain (H2 (s), [w, 1/1000, 1000]), omega0 = 10$

          (%i6) H3 (s) := 1 / (1 + s/omega0)^2$

          (%i7) bode_gain (H3 (s), [w, 1/1000, 1000]), omega0 = 10$

          (%i8) H4 (s) := 1 + s/omega0$

          (%i9) bode_gain (H4 (s), [w, 1/1000, 1000]), omega0 = 10$

          (%i10) H5 (s) := 1/s$

          (%i11) bode_gain (H5 (s), [w, 1/1000, 1000])$

          (%i12) H6 (s) := 1/((s/omega0)^2 + 2 * zeta * (s/omega0) + 1)$

          (%i13) bode_gain (H6 (s), [w, 1/1000, 1000]),
                            omega0 = 10, zeta = 1/10$

          (%i14) H7 (s) := (s/omega0)^2 + 2 * zeta * (s/omega0) + 1$

          (%i15) bode_gain (H7 (s), [w, 1/1000, 1000]),
                            omega0 = 10, zeta = 1/10$

          (%i16) H8 (s) := 0.5 / (0.0001 * s^3 + 0.002 * s^2 + 0.01 * s)$

          (%i17) bode_gain (H8 (s), [w, 1/1000, 1000])$

     To use this function write first `load("bode")'. See also
     `bode_phase'

Function: bode_phase (<H>, <range>, ...<plot_opts>...)

     Function to draw Bode phase plots.

     Examples (1 through 7 from
          `http://www.swarthmore.edu/NatSci/echeeve1/Ref/Bode/BodeHow.html',
     8 from Ron Crummett):
          (%i1) load("bode")$

          (%i2) H1 (s) := 100 * (1 + s) / ((s + 10) * (s + 100))$

          (%i3) bode_phase (H1 (s), [w, 1/1000, 1000])$

          (%i4) H2 (s) := 1 / (1 + s/omega0)$

          (%i5) bode_phase (H2 (s), [w, 1/1000, 1000]), omega0 = 10$

          (%i6) H3 (s) := 1 / (1 + s/omega0)^2$

          (%i7) bode_phase (H3 (s), [w, 1/1000, 1000]), omega0 = 10$

          (%i8) H4 (s) := 1 + s/omega0$

          (%i9) bode_phase (H4 (s), [w, 1/1000, 1000]), omega0 = 10$

          (%i10) H5 (s) := 1/s$

          (%i11) bode_phase (H5 (s), [w, 1/1000, 1000])$

          (%i12) H6 (s) := 1/((s/omega0)^2 + 2 * zeta * (s/omega0) + 1)$

          (%i13) bode_phase (H6 (s), [w, 1/1000, 1000]),
                             omega0 = 10, zeta = 1/10$

          (%i14) H7 (s) := (s/omega0)^2 + 2 * zeta * (s/omega0) + 1$

          (%i15) bode_phase (H7 (s), [w, 1/1000, 1000]),
                             omega0 = 10, zeta = 1/10$

          (%i16) H8 (s) := 0.5 / (0.0001 * s^3 + 0.002 * s^2 + 0.01 * s)$

          (%i17) bode_phase (H8 (s), [w, 1/1000, 1000])$

          (%i18) block ([bode_phase_unwrap : false],
                        bode_phase (H8 (s), [w, 1/1000, 1000]));

          (%i19) block ([bode_phase_unwrap : true],
                        bode_phase (H8 (s), [w, 1/1000, 1000]));

     To use this function write first `load("bode")'. See also
     `bode_gain'

43 cobyla


Introduction to cobyla
Functions and Variables for cobyla
Examples for cobyla

43.1 Introduction to cobyla

`fmin_cobyla' is a Common Lisp translation (via `f2cl') of the Fortran
constrained optimization routine COBYLA by Powell[1][2][3].

   COBYLA minimizes an objective function F(X) subject to M inequality
constraints of the form g(X) >= 0 on X, where X is a vector of
variables that has N components.

   Equality constraints g(X)=0 can often be implemented by a pair of
inequality constraints g(X)>=0 and -g(X)>= 0.  Maxima's interface to
COBYLA allows equality constraints and internally converts the equality
constraints to a pair of inequality constraints.

   The algorithm employs linear approximations to the objective and
constraint functions, the approximations being formed by linear
interpolation at N+1 points in the space of the variables.  The
interpolation points are regarded as vertices of a simplex. The
parameter RHO controls the size of the simplex and it is reduced
automatically from RHOBEG to RHOEND. For each RHO the subroutine tries
to achieve a good vector of variables for the current size, and then
RHO is reduced until the value RHOEND is reached. Therefore RHOBEG and
RHOEND should be set to reasonable initial changes to and the required
accuracy in the variables respectively, but this accuracy should be
viewed as a subject for experimentation because it is not guaranteed.
The routine treats each constraint individually when calculating a
change to the variables, rather than lumping the constraints together
into a single penalty function. The name of the subroutine is derived
from the phrase Constrained Optimization BY Linear Approximations.

   References:

   [1] Fortran Code is from
`http://plato.asu.edu/sub/nlores.html#general'

   [2] M. J. D. Powell, "A direct search optimization method that
models the objective and constraint functions by linear interpolation,"
in Advances in Optimization and Numerical Analysis, eds. S. Gomez and
J.-P. Hennart (Kluwer Academic: Dordrecht, 1994), p. 51-67.

   [3] M. J. D. Powell, "Direct search algorithms for optimization
calculations," Acta Numerica 7, 287-336 (1998).  Also available as
University of Cambridge, Department of Applied Mathematics and
Theoretical Physics,  Numerical Analysis Group, Report NA1998/04 from
`http://www.damtp.cam.ac.uk/user/na/reports.html'

43.2 Functions and Variables for cobyla

Function: fmin_cobyla (<F>, <X>, <Y>)
Function: fmin_cobyla (<F>, <X>, <Y>, optional_args)

     Returns an approximate minimum of the expression <F> with respect
     to the variables <X>, subject to an optional set of constraints.
     <Y> is a list of initial guesses for <X>.

     <F> must be an ordinary expressions, not names of functions or
     lambda expressions.

     `optional_args' represents additional arguments, specified as
     `<symbol> = <value>'.  The optional arguments recognized are:

    `constraints'
          List of inequality and equality constraints that must be
          satisfied by <X>.  The inequality constraints must be actual
          inequalities of the form `g(<X>) >= h(<X>)' or `g(<X>) <=
          h(<X>)'.  The equality constraints must be of the form
          `g(<X>) = h(<X>)'.

    `rhobeg'
          Initial value of the internal RHO variable which controls the
          size of simplex.  (Defaults to 1.0)

    `rhoend'
          The desired final value rho parameter.  It is approximately
          the accuracy in the variables. (Defaults to 1d-6.)

    `iprint'
          Verbose output level.  (Defaults to 0)
             * 0 - No output

             * 1 - Summary at the end of the calculation

             * 2 - Each new value of RHO and SIGMA is printed, including
               the vector of variables, some function information when
               RHO is reduced.

             * 3 - Like 2, but information is printed when F(X) is
               computed.

    `maxfun'
          The maximum number of function evaluations.  (Defaults to
          1000).

     On return, a vector is given:
       1. The value of the variables giving the minimum.  This is a
          list of elements of the form `<var> = <value>' for each of the
          variables listed in <X>.

       2. The minimized function value

       3. The number of function evaluations.

       4. Return code with the following meanings
            1.  0 - No errors.

            2.  1 - Limit on maximum number of function evaluations
               reached.

            3.  2 - Rounding errors inhibiting progress.

     `load(fmin_cobyla)' loads this function.

Function: bf_fmin_cobyla (<F>, <X>, <Y>)
Function: bf_fmin_cobyla (<F>, <X>, <Y>, optional_args)

     This function is identical to `fmin_cobyla', except that bigfloat
     operations are used, and the default value for <rhoend> is
     `10^(fpprec/2)'.

     See `fmin_cobyla' for more information.

     `load(fmin_cobyla)' loads this function.

43.3 Examples for cobyla

Minimize x1*x2 with 1-x1^2-x2^2 >= 0.  The theoretical solution is x1 =

44 contrib_ode

Introduction to contrib_ode
Functions and Variables for contrib_ode
Possible improvements to contrib_ode
Test cases for contrib_ode
References for contrib_ode

44.1 Introduction to contrib_ode

Maxima's ordinary differential equation (ODE) solver `ode2' solves
elementary linear ODEs of first and second order.  The function
`contrib_ode' extends `ode2' with additional methods for linear and
non-linear first order ODEs and linear homogeneous second order ODEs.
The code is still under development and the calling sequence may change
in future releases.  Once the code has stabilized it may be moved from
the contrib directory and integrated into Maxima.

   This package must be loaded with the command `load('contrib_ode)'
before use.

   The calling convention for `contrib_ode' is identical to `ode2'.  It
takes three arguments: an ODE (only the left hand side need be given if
the right hand side is 0), the dependent variable, and the independent
variable.  When successful, it returns a list of solutions.

   The form of the solution differs from `ode2'.  As non-linear
equations can have multiple solutions, `contrib_ode' returns a list of
solutions.  Each  solution can have a number of forms:
   * an explicit solution for the dependent variable,

   * an implicit solution for the dependent variable,

   * a parametric solution in terms of variable `%t', or

   * a tranformation into another ODE in variable `%u'.

   `%c' is used to represent the constant of integration for first
order equations.  `%k1' and `%k2' are the constants for second order
equations.  If `contrib_ode' cannot obtain a solution for whatever
reason, it returns `false', after perhaps printing out an error message.

   It is necessary to return a list of solutions, as even first order
non-linear ODEs can have multiple solutions.  For example:

     (%i1) load('contrib_ode)$

     (%i2) eqn:x*'diff(y,x)^2-(1+x*y)*'diff(y,x)+y=0;

                         dy 2             dy
     (%o2)            x (--)  - (x y + 1) -- + y = 0
                         dx               dx
     (%i3) contrib_ode(eqn,y,x);

                                                  x
     (%o3)             [y = log(x) + %c, y = %c %e ]
     (%i4) method;

     (%o4)                        factor

   Nonlinear ODEs can have singular solutions without constants of
integration, as in the second solution of the following example:

     (%i1) load('contrib_ode)$

     (%i2) eqn:'diff(y,x)^2+x*'diff(y,x)-y=0;

                            dy 2     dy
     (%o2)                 (--)  + x -- - y = 0
                            dx       dx
     (%i3) contrib_ode(eqn,y,x);

                                                2
                                      2        x
     (%o3)              [y = %c x + %c , y = - --]
                                               4
     (%i4) method;

     (%o4)                       clairault

   The following ODE has two parametric solutions in terms of the dummy
variable `%t'.  In this case the parametric solutions can be manipulated
to give explicit solutions.

     (%i1) load('contrib_ode)$

     (%i2) eqn:'diff(y,x)=(x+y)^2;

                               dy          2
     (%o2)                     -- = (y + x)
                               dx
     (%i3) contrib_ode(eqn,y,x);

     (%o3) [[x = %c - atan(sqrt(%t)), y = - x - sqrt(%t)],
                          [x = atan(sqrt(%t)) + %c, y = sqrt(%t) - x]]
     (%i4) method;

     (%o4)                       lagrange

   The following example (Kamke 1.112) demonstrates an implicit
solution.

     (%i1) load('contrib_ode)$

     (%i2) assume(x>0,y>0);

     (%o2)                    [x > 0, y > 0]
     (%i3) eqn:x*'diff(y,x)-x*sqrt(y^2+x^2)-y;

                          dy           2    2
     (%o3)              x -- - x sqrt(y  + x ) - y
                          dx
     (%i4) contrib_ode(eqn,y,x);

                                       y
     (%o4)                  [x - asinh(-) = %c]
                                       x
     (%i5) method;

     (%o5)                          lie

   The following Riccati equation is transformed into a linear second
order ODE in the variable `%u'.  Maxima is unable to solve the new ODE,
so it is returned unevaluated.
     (%i1) load('contrib_ode)$

     (%i2) eqn:x^2*'diff(y,x)=a+b*x^n+c*x^2*y^2;

                         2 dy      2  2      n
     (%o2)              x  -- = c x  y  + b x  + a
                           dx
     (%i3) contrib_ode(eqn,y,x);

                    d%u
                    ---                            2
                    dx        2     n - 2   a     d %u
     (%o3)  [[y = - ----, %u c  (b x      + --) + ---- c = 0]]
                    %u c                     2      2
                                            x     dx
     (%i4) method;

     (%o4)                        riccati

   For first order ODEs `contrib_ode' calls `ode2'.  It then tries the
following methods: factorization, Clairault, Lagrange, Riccati, Abel
and Lie symmetry methods.  The Lie method is not attempted on Abel
equations if the Abel method fails, but it is tried if the Riccati
method returns an unsolved second order ODE.

   For second order ODEs `contrib_ode' calls `ode2' then `odelin'.

   Extensive debugging traces and messages are displayed if the command
`put('contrib_ode,true,'verbose)' is executed.

44.2 Functions and Variables for contrib_ode

Function: contrib_ode (<eqn>, <y>, <x>)

     Returns a list of solutions of the ODE <eqn> with independent
     variable <x> and dependent variable <y>.

Function: odelin (<eqn>, <y>, <x>)

     `odelin' solves linear homogeneous ODEs of first and second order
     with independent variable <x> and dependent variable <y>.  It
     returns a fundamental solution set of the ODE.

     For second order ODEs, `odelin' uses a method, due to Bronstein
     and Lafaille, that searches for solutions in terms of given
     special functions.

          (%i1) load('contrib_ode);

          (%i2) odelin(x*(x+1)*'diff(y,x,2)+(x+5)*'diff(y,x,1)+(-4)*y,y,x);
          ...trying factor method
          ...solving 7 equations in 4 variables
          ...trying the Bessel solver
          ...solving 1 equations in 2 variables
          ...trying the F01 solver
          ...solving 1 equations in 3 variables
          ...trying the spherodial wave solver
          ...solving 1 equations in 4 variables
          ...trying the square root Bessel solver
          ...solving 1 equations in 2 variables
          ...trying the 2F1 solver
          ...solving 9 equations in 5 variables
                 gauss_a(- 6, - 2, - 3, - x)  gauss_b(- 6, - 2, - 3, - x)
          (%o2) {---------------------------, ---------------------------}
                              4                            4
                             x                            x

Function: ode_check (<eqn>, <soln>)

     Returns the value of ODE <eqn> after substituting a possible
     solution <soln>.  The value is equivalent to zero if <soln> is a
     solution of <eqn>.

          (%i1) load('contrib_ode)$

          (%i2) eqn:'diff(y,x,2)+(a*x+b)*y;

                                   2
                                  d y
          (%o2)                   --- + (a x + b) y
                                    2
                                  dx
          (%i3) ans:[y = bessel_y(1/3,2*(a*x+b)^(3/2)/(3*a))*%k2*sqrt(a*x+b)
                   +bessel_j(1/3,2*(a*x+b)^(3/2)/(3*a))*%k1*sqrt(a*x+b)];

                                            3/2
                              1  2 (a x + b)
          (%o3) [y = bessel_y(-, --------------) %k2 sqrt(a x + b)
                              3       3 a
                                                    3/2
                                      1  2 (a x + b)
                           + bessel_j(-, --------------) %k1 sqrt(a x + b)]
                                      3       3 a
          (%i4) ode_check(eqn,ans[1]);

          (%o4)                           0

System variable: method

     The variable `method' is set to the successful solution method.

Variable: %c

     `%c' is the integration constant for first order ODEs.

Variable: %k1

     `%k1' is the first integration constant for second order ODEs.

Variable: %k2

     `%k2' is the second integration constant for second order ODEs.

Function: gauss_a (<a>, <b>, <c>, <x>)

     `gauss_a(a,b,c,x)' and `gauss_b(a,b,c,x)' are 2F1 geometric
     functions.  They represent any two independent solutions of the
     hypergeometric differential equation `x(1-x) diff(y,x,2) +
     [c-(a+b+1)x diff(y,x) - aby = 0' (A&S 15.5.1).

     The only use of these functions is in solutions of ODEs returned by
     `odelin' and `contrib_ode'.  The definition and use of these
     functions may change in future releases of Maxima.

     See also `gauss_b', `dgauss_a' and `gauss_b'.

Function: gauss_b (<a>, <b>, <c>, <x>)

     See `gauss_a'.

Function: dgauss_a (<a>, <b>, <c>, <x>)

     The derivative with respect to <x> of `gauss_a(<a>, <b>, <c>,
     <x>)'.

Function: dgauss_b (<a>, <b>, <c>, <x>)

     The derivative with respect to <x> of `gauss_b(<a>, <b>, <c>,
     <x>)'.

Function: kummer_m (<a>, <b>, <x>)

     Kummer's M function, as defined in Abramowitz and Stegun, Handbook
     of Mathematical Functions, Section 13.1.2.

     The only use of this function is in solutions of ODEs returned by
     `odelin' and `contrib_ode'.  The definition and use of this
     function may change in future releases of Maxima.

     See also `kummer_u', `dkummer_m' and `dkummer_u'.

Function: kummer_u (<a>, <b>, <x>)

     Kummer's U function, as defined in Abramowitz and Stegun, Handbook
     of Mathematical Functions, Section 13.1.3.

     See `kummer_m'.

Function: dkummer_m (<a>, <b>, <x>)

     The derivative with respect to <x> of `kummer_m(<a>, <b>, <x>)'.

Function: dkummer_u (<a>, <b>, <x>)

     The derivative with respect to <x> of `kummer_u(<a>, <b>, <x>)'.

44.3 Possible improvements to contrib_ode

These routines are work in progress.  I still need to:

   * Extend the FACTOR method `ode1_factor' to work for multiple roots.

   * Extend the FACTOR method `ode1_factor' to attempt to solve higher
     order factors.  At present it only attemps to solve linear factors.

   * Fix the LAGRANGE routine `ode1_lagrange' to prefer real roots over
      complex roots.

   * Add additional methods for Riccati equations.

   * Improve the detection of Abel equations of second kind.  The
     exisiting   pattern matching is weak.

   * Work on the Lie symmetry group routine `ode1_lie'.  There are
     quite a   few problems with it: some parts are unimplemented; some
     test cases   seem to run forever; other test cases crash; yet
     others return very   complex "solutions".  I wonder if it really
     ready for release yet.

   * Add more test cases.

44.4 Test cases for contrib_ode

The routines have been tested on a approximately one thousand  test
cases from Murphy, Kamke, Zwillinger and elsewhere.  These are included
in the tests subdirectory.

   * The Clairault routine `ode1_clairault' finds all known solutions,
     including singular solutions, of the Clairault equations in Murphy
     and   Kamke.

   * The other routines often return a single solution when multiple
     solutions exist.

   * Some of the "solutions" from `ode1_lie' are overly complex and
     impossible to check.

   * There are some crashes.

44.5 References for contrib_ode

  1. E. Kamke, Differentialgleichungen Losungsmethoden und Losungen,
     Vol 1,     Geest & Portig, Leipzig, 1961

  2. G. M. Murphy, Ordinary Differential Equations and Their Solutions,
        Van Nostrand, New York, 1960

  3. D. Zwillinger, Handbook of Differential Equations, 3rd edition,
     Academic Press, 1998

  4. F. Schwarz, Symmetry Analysis of Abel's Equation, Studies in
     Applied Mathematics, 100:269-294 (1998)

  5. F. Schwarz, Algorithmic Solution of Abel's Equation,     Computing
     61, 39-49 (1998)

  6. E. S. Cheb-Terrab, A. D. Roche, Symmetries and First Order     ODE
     Patterns, Computer Physics Communications 113 (1998), p 239.
     (`http://lie.uwaterloo.ca/papers/ode_vii.pdf')

  7. E. S. Cheb-Terrab, T. Kolokolnikov,  First Order ODEs,
     Symmetries and Linear Transformations, European Journal of
     Applied Mathematics, Vol. 14, No. 2, pp. 231-246 (2003).
     (`http://arxiv.org/abs/math-ph/0007023',
     `http://lie.uwaterloo.ca/papers/ode_iv.pdf')

  8. G. W. Bluman, S. C. Anco, Symmetry and Integration Methods for
     Differential Equations, Springer, (2002)

  9. M. Bronstein, S. Lafaille, Solutions of linear ordinary
     differential equations in terms of special functions, Proceedings
     of ISSAC 2002, Lille, ACM Press, 23-28.
     (`http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf')

45 descriptive


Introduction to descriptive
Functions and Variables for data manipulation
Functions and Variables for descriptive statistics
Functions and Variables for statistical graphs

45.1 Introduction to descriptive

Package `descriptive' contains a set of functions for making
descriptive statistical computations and graphing.  Together with the
source code there are three data sets in your Maxima tree:
`pidigits.data', `wind.data' and `biomed.data'.

   Any statistics manual can be used as a reference to the functions in
package `descriptive'.

   For comments, bugs or suggestions, please contact me at <'mario AT
edu DOT xunta DOT es'>.

   Here is a simple example on how the descriptive functions in
`descriptive' do they work, depending on the nature of their arguments,
lists or matrices,

     (%i1) load (descriptive)$
     (%i2) /* univariate sample */   mean ([a, b, c]);
                                 c + b + a
     (%o2)                       ---------
                                     3
     (%i3) matrix ([a, b], [c, d], [e, f]);
                                 [ a  b ]
                                 [      ]
     (%o3)                       [ c  d ]
                                 [      ]
                                 [ e  f ]
     (%i4) /* multivariate sample */ mean (%);
                           e + c + a  f + d + b
     (%o4)                [---------, ---------]
                               3          3

   Note that in multivariate samples the mean is calculated for each
column.

   In case of several samples with possible different sizes, the Maxima
function `map' can be used to get the desired results for each sample,

     (%i1) load (descriptive)$
     (%i2) map (mean, [[a, b, c], [d, e]]);
                             c + b + a  e + d
     (%o2)                  [---------, -----]
                                 3        2

   In this case, two samples of sizes 3 and 2 were stored into a list.

   Univariate samples must be stored in lists like

     (%i1) s1 : [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5];
     (%o1)           [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5]

   and multivariate samples in matrices as in

     (%i1) s2 : matrix ([13.17, 9.29], [14.71, 16.88], [18.50, 16.88],
                  [10.58, 6.63], [13.33, 13.25], [13.21,  8.12]);
                             [ 13.17  9.29  ]
                             [              ]
                             [ 14.71  16.88 ]
                             [              ]
                             [ 18.5   16.88 ]
     (%o1)                   [              ]
                             [ 10.58  6.63  ]
                             [              ]
                             [ 13.33  13.25 ]
                             [              ]
                             [ 13.21  8.12  ]

   In this case, the number of columns equals the random variable
dimension and the number of rows is the sample size.

   Data can be introduced by hand, but big samples are usually stored
in plain text files. For example, file `pidigits.data' contains the
first 100 digits of number `%pi':
           3
           1
           4
           1
           5
           9
           2
           6
           5
           3 ...

   In order to load these digits in Maxima,

     (%i1) s1 : read_list (file_search ("pidigits.data"))$
     (%i2) length (s1);
     (%o2)                          100

   On the other hand, file `wind.data' contains daily average wind
speeds at 5 meteorological stations in the Republic of Ireland (This is
part of a data set taken at 12 meteorological stations. The original
file is freely downloadable from the StatLib Data Repository and its
analysis is discused in Haslett, J., Raftery, A. E. (1989) <Space-time
Modelling with Long-memory Dependence: Assessing Ireland's Wind Power
Resource, with Discussion>. Applied Statistics 38, 1-50). This loads
the data:

     (%i1) s2 : read_matrix (file_search ("wind.data"))$
     (%i2) length (s2);
     (%o2)                          100
     (%i3) s2 [%]; /* last record */
     (%o3)            [3.58, 6.0, 4.58, 7.62, 11.25]

   Some samples contain non numeric data. As an example, file
`biomed.data' (which is part of another bigger one downloaded from the
StatLib Data Repository) contains four blood measures taken from two
groups of patients, `A' and `B', of different ages,

     (%i1) s3 : read_matrix (file_search ("biomed.data"))$
     (%i2) length (s3);
     (%o2)                          100
     (%i3) s3 [1]; /* first record */
     (%o3)            [A, 30, 167.0, 89.0, 25.6, 364]

   The first individual belongs to group `A', is 30 years old and
his/her blood measures were 167.0, 89.0, 25.6 and 364.

   One must take care when working with categorical data. In the next
example, symbol `a' is asigned a value in some previous moment and then
a sample with categorical value `a' is taken,

     (%i1) a : 1$
     (%i2) matrix ([a, 3], [b, 5]);
                                 [ 1  3 ]
     (%o2)                       [      ]
                                 [ b  5 ]

45.2 Functions and Variables for data manipulation

Function: continuous_freq (<list>)
Function: continuous_freq (<list>, <m>)

     The argument of `continuous_freq' must be a list of numbers.
     Divides the range in intervals and counts how many values are
     inside them. The second argument is optional and either equals the
     number of classes we want, 10 by default, or equals a list
     containing the class limits and the number of classes we want, or
     a list containing only the limits.  Argument <list> must be a list
     of (2 or 3) real numbers.  If sample values are all equal, this
     function returns only one class of amplitude 2.

     Examples:

     Optional argument indicates the number of classes we want.  The
     first list in the output contains the interval limits, and the
     second the corresponding counts: there are 16 digits inside the
     interval `[0, 1.8]', 24 digits in `(1.8, 3.6]', and so on.

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) continuous_freq (s1, 5);
          (%o3) [[0, 1.8, 3.6, 5.4, 7.2, 9.0], [16, 24, 18, 17, 25]]

     Optional argument indicates we want 7 classes with limits -2 and
     12:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) continuous_freq (s1, [-2,12,7]);
          (%o3) [[- 2, 0, 2, 4, 6, 8, 10, 12], [8, 20, 22, 17, 20, 13, 0]]

     Optional argument indicates we want the default number of classes
     with limits -2 and 12:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) continuous_freq (s1, [-2,12]);
                          3  4  11  18     32  39  46  53
          (%o3)  [[- 2, - -, -, --, --, 5, --, --, --, --, 12],
                          5  5  5   5      5   5   5   5
                         [0, 8, 20, 12, 18, 9, 8, 25, 0, 0]]

Function: discrete_freq (<list>)

     Counts absolute frequencies in discrete samples, both numeric and
     categorical. Its unique argument is a list,

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) discrete_freq (s1);
          (%o3) [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9],
                                       [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]

     The first list gives the sample values and the second their
     absolute frequencies. Commands `? col' and `? transpose' should
     help you to understand the last input.

Function: subsample (<data_matrix>, <predicate_function>)
Function: subsample (<data_matrix>, <predicate_function>, <col_num1>, <col_num2>, ...)

     This is a sort of variant of the Maxima `submatrix' function.  The
     first argument is the data matrix, the second is a predicate
     function and optional additional arguments are the numbers of the
     columns to be taken.  Its behaviour is better understood with
     examples.

     These are multivariate records in which the wind speed in the
     first meteorological station were greater than 18.  See that in
     the lambda expression the <i>-th component is refered to as `v[i]'.
          (%i1) load (descriptive)$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) subsample (s2, lambda([v], v[1] > 18));
                        [ 19.38  15.37  15.12  23.09  25.25 ]
                        [                                   ]
                        [ 18.29  18.66  19.08  26.08  27.63 ]
          (%o3)         [                                   ]
                        [ 20.25  21.46  19.95  27.71  23.38 ]
                        [                                   ]
                        [ 18.79  18.96  14.46  26.38  21.84 ]

     In the following example, we request only the first, second and
     fifth components of those records with wind speeds greater or
     equal than 16 in station number 1 and less than 25 knots in
     station number 4. The sample contains only data from stations 1, 2
     and 5. In this case, the predicate function is defined as an
     ordinary Maxima function.
          (%i1) load (descriptive)$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) g(x):= x[1] >= 16 and x[4] < 25$
          (%i4) subsample (s2, g, 1, 2, 5);
                               [ 19.38  15.37  25.25 ]
                               [                     ]
                               [ 17.33  14.67  19.58 ]
          (%o4)                [                     ]
                               [ 16.92  13.21  21.21 ]
                               [                     ]
                               [ 17.25  18.46  23.87 ]

     Here is an example with the categorical variables of `biomed.data'.
     We want the records corresponding to those patients in group `B'
     who are older than 38 years.
          (%i1) load (descriptive)$
          (%i2) s3 : read_matrix (file_search ("biomed.data"))$
          (%i3) h(u):= u[1] = B and u[2] > 38 $
          (%i4) subsample (s3, h);
                          [ B  39  28.0  102.3  17.1  146 ]
                          [                               ]
                          [ B  39  21.0  92.4   10.3  197 ]
                          [                               ]
                          [ B  39  23.0  111.5  10.0  133 ]
                          [                               ]
                          [ B  39  26.0  92.6   12.3  196 ]
          (%o4)           [                               ]
                          [ B  39  25.0  98.7   10.0  174 ]
                          [                               ]
                          [ B  39  21.0  93.2   5.9   181 ]
                          [                               ]
                          [ B  39  18.0  95.0   11.3  66  ]
                          [                               ]
                          [ B  39  39.0  88.5   7.6   168 ]

     Probably, the statistical analysis will involve only the blood
     measures,
          (%i1) load (descriptive)$
          (%i2) s3 : read_matrix (file_search ("biomed.data"))$
          (%i3) subsample (s3, lambda([v], v[1] = B and v[2] > 38),
                           3, 4, 5, 6);
                             [ 28.0  102.3  17.1  146 ]
                             [                        ]
                             [ 21.0  92.4   10.3  197 ]
                             [                        ]
                             [ 23.0  111.5  10.0  133 ]
                             [                        ]
                             [ 26.0  92.6   12.3  196 ]
          (%o3)              [                        ]
                             [ 25.0  98.7   10.0  174 ]
                             [                        ]
                             [ 21.0  93.2   5.9   181 ]
                             [                        ]
                             [ 18.0  95.0   11.3  66  ]
                             [                        ]
                             [ 39.0  88.5   7.6   168 ]

     This is the multivariate mean of `s3',
          (%i1) load (descriptive)$
          (%i2) s3 : read_matrix (file_search ("biomed.data"))$
          (%i3) mean (s3);
                 65 B + 35 A  317          6 NA + 8144.999999999999
          (%o3) [-----------, ---, 87.178, ------------------------,
                     100      10                     100
                                                              3 NA + 19587
                                                      18.123, ------------]
                                                                  100

     Here, the first component is meaningless, since `A' and `B' are
     categorical, the second component is the mean age of individuals
     in rational form, and the fourth and last values exhibit some
     strange behaviour. This is because symbol `NA' is used here to
     indicate <non available> data, and the two means are nonsense. A
     possible solution would be to take out from the matrix those rows
     with `NA' symbols, although this deserves some loss of information.
          (%i1) load (descriptive)$
          (%i2) s3 : read_matrix (file_search ("biomed.data"))$
          (%i3) g(v):= v[4] # NA and v[6] # NA $
          (%i4) mean (subsample (s3, g, 3, 4, 5, 6));
          (%o4) [79.4923076923077, 86.2032967032967, 16.93186813186813,
                                                                      2514
                                                                      ----]
                                                                       13

45.3 Functions and Variables for descriptive statistics

Function: mean (<list>)
Function: mean (<matrix>)

     This is the sample mean, defined as
                                 n
                               ====
                       _   1   \
                       x = -    >    x
                           n   /      i
                               ====
                               i = 1

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) mean (s1);
                                         471
          (%o3)                          ---
                                         100
          (%i4) %, numer;
          (%o4)                         4.71
          (%i5) s2 : read_matrix (file_search ("wind.data"))$
          (%i6) mean (s2);
          (%o6)     [9.9485, 10.1607, 10.8685, 15.7166, 14.8441]

Function: var (<list>)
Function: var (<matrix>)

     This is the sample variance, defined as
                               n
                             ====
                     2   1   \          _ 2
                    s  = -    >    (x - x)
                         n   /       i
                             ====
                             i = 1

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) var (s1), numer;
          (%o3)                   8.425899999999999

     See also function `var1'.

Function: var1 (<list>)
Function: var1 (<matrix>)

     This is the sample variance, defined as
                               n
                             ====
                         1   \          _ 2
                        ---   >    (x - x)
                        n-1  /       i
                             ====
                             i = 1

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) var1 (s1), numer;
          (%o3)                    8.5110101010101
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) var1 (s2);
          (%o5) [17.39586540404041, 15.13912778787879, 15.63204924242424,
                                      32.50152569696971, 24.66977392929294]

     See also function `var'.

Function: std (<list>)
Function: std (<matrix>)

     This is the the square root of function `var', the variance with
     denominator n.

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) std (s1), numer;
          (%o3)                   2.902740084816414
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) std (s2);
          (%o5) [4.149928523480858, 3.871399812729241, 3.933920277534866,
                                      5.672434260526957, 4.941970881136392]

     See also functions `var' and `std1'.

Function: std1 (<list>)
Function: std1 (<matrix>)

     This is the the square root of function `var1', the variance with
     denominator n-1.

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) std1 (s1), numer;
          (%o3)                   2.917363553109228
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) std1 (s2);
          (%o5) [4.170835096721089, 3.89090320978032, 3.953738641137555,
                                      5.701010936401517, 4.966867617451963]

     See also functions `var1' and `std'.

Function: noncentral_moment (<list>, <k>)
Function: noncentral_moment (<matrix>, <k>)

     The non central moment of order k, defined as
                                 n
                               ====
                           1   \      k
                           -    >    x
                           n   /      i
                               ====
                               i = 1

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) noncentral_moment (s1, 1), numer; /* the mean */
          (%o3)                         4.71
          (%i5) s2 : read_matrix (file_search ("wind.data"))$
          (%i6) noncentral_moment (s2, 5);
          (%o6) [319793.8724761505, 320532.1923892463,
                391249.5621381556, 2502278.205988911, 1691881.797742255]

     See also function `central_moment'.

Function: central_moment (<list>, <k>)
Function: central_moment (<matrix>, <k>)

     The central moment of order k, defined as
                              n
                            ====
                        1   \          _ k
                        -    >    (x - x)
                        n   /       i
                            ====
                            i = 1

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) central_moment (s1, 2), numer; /* the variance */
          (%o3)                   8.425899999999999
          (%i5) s2 : read_matrix (file_search ("wind.data"))$
          (%i6) central_moment (s2, 3);
          (%o6) [11.29584771375004, 16.97988248298583, 5.626661952750102,
                                       37.5986572057918, 25.85981904394192]

     See also functions `central_moment' and `mean'.

Function: cv (<list>)
Function: cv (<matrix>)

     The variation coefficient is the quotient between the sample
     standard deviation (`std') and the `mean',

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) cv (s1), numer;
          (%o3)                   .6193977819764815
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) cv (s2);
          (%o5) [.4192426091090204, .3829365309260502, 0.363779605385983,
                                      .3627381836021478, .3346021393989506]

     See also functions `std' and `mean'.

Function: smin (<list>)
Function: smin (<matrix>)

     This is the minimum value of the sample <list>.  When the argument
     is a matrix, `smin' returns a list containing the minimum values
     of the columns, which are associated to statistical variables.

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) smin (s1);
          (%o3)                           0
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) smin (s2);
          (%o5)             [0.58, 0.5, 2.67, 5.25, 5.17]

     See also function `smax'.

Function: smax (<list>)
Function: smax (<matrix>)

     This is the maximum value of the sample <list>.  When the argument
     is a matrix, `smax' returns a list containing the maximum values
     of the columns, which are associated to statistical variables.

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) smax (s1);
          (%o3)                           9
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) smax (s2);
          (%o5)          [20.25, 21.46, 20.04, 29.63, 27.63]

     See also function `smin'.

Function: range (<list>)
Function: range (<matrix>)

     The range is the difference between the extreme values.

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) range (s1);
          (%o3)                           9
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) range (s2);
          (%o5)          [19.67, 20.96, 17.37, 24.38, 22.46]

Function: quantile (<list>, <p>)
Function: quantile (<matrix>, <p>)

     This is the <p>-quantile, with <p> a number in [0, 1], of the
     sample <list>.  Although there are several definitions for the
     sample quantile (Hyndman, R. J., Fan, Y. (1996) <Sample quantiles
     in statistical packages>. American Statistician, 50, 361-365), the
     one based on linear interpolation is implemented in package
     `descriptive'.

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) /* 1st and 3rd quartiles */
                   [quantile (s1, 1/4), quantile (s1, 3/4)], numer;
          (%o3)                      [2.0, 7.25]
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) quantile (s2, 1/4);
          (%o5)    [7.2575, 7.477500000000001, 7.82, 11.28, 11.48]

Function: median (<list>)
Function: median (<matrix>)

     Once the sample is ordered, if the sample size is odd the median
     is the central value, otherwise it is the mean of the two central
     values.

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) median (s1);
                                          9
          (%o3)                           -
                                          2
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) median (s2);
          (%o5)         [10.06, 9.855, 10.73, 15.48, 14.105]

     The median is the 1/2-quantile.

     See also function `quantile'.

Function: qrange (<list>)
Function: qrange (<matrix>)

     The interquartilic range is the difference between the third and
     first quartiles, `quantile(<list>,3/4) - quantile(<list>,1/4)',

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) qrange (s1);
                                         21
          (%o3)                          --
                                         4
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) qrange (s2);
          (%o5) [5.385, 5.572499999999998, 6.022500000000001,
                                      8.729999999999999, 6.649999999999999]

     See also function `quantile'.

Function: mean_deviation (<list>)
Function: mean_deviation (<matrix>)

     The mean deviation, defined as
                               n
                             ====
                         1   \          _
                         -    >    |x - x|
                         n   /       i
                             ====
                             i = 1

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) mean_deviation (s1);
                                         51
          (%o3)                          --
                                         20
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) mean_deviation (s2);
          (%o5) [3.287959999999999, 3.075342, 3.23907, 4.715664000000001,
                                                         4.028546000000002]

     See also function `mean'.

Function: median_deviation (<list>)
Function: median_deviation (<matrix>)

     The median deviation, defined as
                           n
                         ====
                     1   \
                     -    >    |x - med|
                     n   /       i
                         ====
                         i = 1
     where `med' is the median of <list>.

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) median_deviation (s1);
                                          5
          (%o3)                           -
                                          2
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) median_deviation (s2);
          (%o5)           [2.75, 2.755, 3.08, 4.315, 3.31]

     See also function `mean'.

Function: harmonic_mean (<list>)
Function: harmonic_mean (<matrix>)

     The harmonic mean, defined as
                            n
                         --------
                          n
                         ====
                         \     1
                          >    --
                         /     x
                         ====   i
                         i = 1

     Example:

          (%i1) load (descriptive)$
          (%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
          (%i3) harmonic_mean (y), numer;
          (%o3)                   3.901858027632205
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) harmonic_mean (s2);
          (%o5) [6.948015590052786, 7.391967752360356, 9.055658197151745,
                                      13.44199028193692, 13.01439145898509]

     See also functions `mean' and `geometric_mean'.

Function: geometric_mean (<list>)
Function: geometric_mean (<matrix>)

     The geometric mean, defined as
                           /  n      \ 1/n
                           | /===\   |
                           |  ! !    |
                           |  ! !  x |
                           |  ! !   i|
                           | i = 1   |
                           \         /

     Example:

          (%i1) load (descriptive)$
          (%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
          (%i3) geometric_mean (y), numer;
          (%o3)                   4.454845412337012
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) geometric_mean (s2);
          (%o5) [8.82476274347979, 9.22652604739361, 10.0442675714889,
                                      14.61274126349021, 13.96184163444275]

     See also functions `mean' and `harmonic_mean'.

Function: kurtosis (<list>)
Function: kurtosis (<matrix>)

     The kurtosis coefficient, defined as
                              n
                            ====
                      1     \          _ 4
                     ----    >    (x - x)  - 3
                        4   /       i
                     n s    ====
                            i = 1

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) kurtosis (s1), numer;
          (%o3)                  - 1.273247946514421
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) kurtosis (s2);
          (%o5) [- .2715445622195385, 0.119998784429451,
               - .4275233490482861, - .6405361979019522, - .4952382132352935]

     See also functions `mean', `var' and `skewness'.

Function: skewness (<list>)
Function: skewness (<matrix>)

     The skewness coefficient, defined as
                              n
                            ====
                      1     \          _ 3
                     ----    >    (x - x)
                        3   /       i
                     n s    ====
                            i = 1

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) skewness (s1), numer;
          (%o3)                  .009196180476450424
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) skewness (s2);
          (%o5) [.1580509020000978, .2926379232061854, .09242174416107717,
                                      .2059984348148687, .2142520248890831]

     See also functions `mean', `var' and `kurtosis'.

Function: pearson_skewness (<list>)
Function: pearson_skewness (<matrix>)

     Pearson's skewness coefficient, defined as
                          _
                       3 (x - med)
                       -----------
                            s
     where <med> is the median of <list>.

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) pearson_skewness (s1), numer;
          (%o3)                   .2159484029093895
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) pearson_skewness (s2);
          (%o5) [- .08019976629211892, .2357036272952649,
                   .1050904062491204, .1245042340592368, .4464181795804519]

     See also functions `mean', `var' and `median'.

Function: quartile_skewness (<list>)
Function: quartile_skewness (<matrix>)

     The quartile skewness coefficient, defined as
                         c    - 2 c    + c
                          3/4      1/2    1/4
                         --------------------
                             c    - c
                              3/4    1/4
     where c_p is the <p>-quantile of sample <list>.

     Example:

          (%i1) load (descriptive)$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) quartile_skewness (s1), numer;
          (%o3)                  .04761904761904762
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) quartile_skewness (s2);
          (%o5) [- 0.0408542246982353, .1467025572005382,
                 0.0336239103362392, .03780068728522298, .2105263157894735]

     See also function `quantile'.

Function: cov (<matrix>)

     The covariance matrix of the multivariate sample, defined as
                        n
                       ====
                    1  \           _        _
                S = -   >    (X  - X) (X  - X)'
                    n  /       j        j
                       ====
                       j = 1
     where X_j is the j-th row of the sample matrix.

     Example:

          (%i1) load (descriptive)$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) fpprintprec : 7$  /* change precision for pretty output */
          (%i4) cov (s2);
                [ 17.22191  13.61811  14.37217  19.39624  15.42162 ]
                [                                                  ]
                [ 13.61811  14.98774  13.30448  15.15834  14.9711  ]
                [                                                  ]
          (%o4) [ 14.37217  13.30448  15.47573  17.32544  16.18171 ]
                [                                                  ]
                [ 19.39624  15.15834  17.32544  32.17651  20.44685 ]
                [                                                  ]
                [ 15.42162  14.9711   16.18171  20.44685  24.42308 ]

     See also function `cov1'.

Function: cov1 (<matrix>)

     The covariance matrix of the multivariate sample, defined as
                        n
                       ====
                   1   \           _        _
             S  = ---   >    (X  - X) (X  - X)'
              1   n-1  /       j        j
                       ====
                       j = 1
     where X_j is the j-th row of the sample matrix.

     Example:

          (%i1) load (descriptive)$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) fpprintprec : 7$ /* change precision for pretty output */
          (%i4) cov1 (s2);
                [ 17.39587  13.75567  14.51734  19.59216  15.5774  ]
                [                                                  ]
                [ 13.75567  15.13913  13.43887  15.31145  15.12232 ]
                [                                                  ]
          (%o4) [ 14.51734  13.43887  15.63205  17.50044  16.34516 ]
                [                                                  ]
                [ 19.59216  15.31145  17.50044  32.50153  20.65338 ]
                [                                                  ]
                [ 15.5774   15.12232  16.34516  20.65338  24.66977 ]

     See also function `cov'.

Function: global_variances (<matrix>)
Function: global_variances (<matrix>, <logical_value>)

     Function `global_variances' returns a list of global variance
     measures:

        * <total variance>: `trace(S_1)',

        * <mean variance>: `trace(S_1)/p',

        * <generalized variance>: `determinant(S_1)',

        * <generalized standard deviation>: `sqrt(determinant(S_1))',

        * <efective variance> `determinant(S_1)^(1/p)', (defined in:
          Pen~a, D. (2002) <Ana'lisis de datos multivariantes>;
          McGraw-Hill, Madrid.)

        * <efective standard deviation>: `determinant(S_1)^(1/(2*p))'.
     where <p> is the dimension of the multivariate random variable and
     S_1 the covariance matrix returned by `cov1'.

     Example:

          (%i1) load (descriptive)$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) global_variances (s2);
          (%o3) [105.338342060606, 21.06766841212119, 12874.34690469686,
                   113.4651792608501, 6.636590811800795, 2.576158149609762]

     The function `global_variances' has an optional logical argument:
     `global_variances(x, true)' tells Maxima that `x' is the data
     matrix, making the same as `global_variances(x)'. On the other
     hand, `global_variances(x, false)' means that `x' is not the data
     matrix, but the covariance matrix, avoiding its recalculation,

          (%i1) load (descriptive)$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) s : cov1 (s2)$
          (%i4) global_variances (s, false);
          (%o4) [105.338342060606, 21.06766841212119, 12874.34690469686,
                   113.4651792608501, 6.636590811800795, 2.576158149609762]

     See also `cov' and `cov1'.

Function: cor (<matrix>)
Function: cor (<matrix>, <logical_value>)

     The correlation matrix of the multivariate sample.

     Example:

          (%i1) load (descriptive)$
          (%i2) fpprintprec : 7 $
          (%i3) s2 : read_matrix (file_search ("wind.data"))$
          (%i4) cor (s2);
                [   1.0     .8476339  .8803515  .8239624  .7519506 ]
                [                                                  ]
                [ .8476339    1.0     .8735834  .6902622  0.782502 ]
                [                                                  ]
          (%o4) [ .8803515  .8735834    1.0     .7764065  .8323358 ]
                [                                                  ]
                [ .8239624  .6902622  .7764065    1.0     .7293848 ]
                [                                                  ]
                [ .7519506  0.782502  .8323358  .7293848    1.0    ]

     Function `cor' has an optional logical argument: `cor(x,true)'
     tells Maxima that `x' is the data matrix, making the same as
     `cor(x)'. On the other hand, `cor(x,false)' means that `x' is not
     the data matrix, but the covariance matrix, avoiding its
     recalculation,

          (%i1) load (descriptive)$
          (%i2) fpprintprec : 7 $
          (%i3) s2 : read_matrix (file_search ("wind.data"))$
          (%i4) s : cov1 (s2)$
          (%i5) cor (s, false); /* this is faster */
                [   1.0     .8476339  .8803515  .8239624  .7519506 ]
                [                                                  ]
                [ .8476339    1.0     .8735834  .6902622  0.782502 ]
                [                                                  ]
          (%o5) [ .8803515  .8735834    1.0     .7764065  .8323358 ]
                [                                                  ]
                [ .8239624  .6902622  .7764065    1.0     .7293848 ]
                [                                                  ]
                [ .7519506  0.782502  .8323358  .7293848    1.0    ]

     See also `cov' and `cov1'.

Function: list_correlations (<matrix>)
Function: list_correlations (<matrix>, <logical_value>)

     Function `list_correlations' returns a list of correlation
     measures:

        * <precision matrix>: the inverse of the covariance matrix S_1,
                      -1     ij
                     S   = (s  )
                      1         i,j = 1,2,...,p

        * <multiple correlation vector>:  (R_1^2, R_2^2, ..., R_p^2),
          with
                      2          1
                     R  = 1 - -------
                      i        ii
                              s   s
                                   ii
          being an indicator of the goodness of fit of the linear
          multivariate regression model on X_i when the rest of
          variables are used as regressors.

        * <partial correlation matrix>: with element (i, j) being
                                        ij
                                       s
                     r        = - ------------
                      ij.rest     / ii  jj\ 1/2
                                  |s   s  |
                                  \       /

45.4 Functions and Variables for statistical graphs

Function: barsplot (<data1>, <data2>, ..., <option_1>, <option_2>, ...)
Function: barsplot_description (...)

     Plots bars diagrams for discrete statistical variables, both for
     one or multiple samples.

     <data> can be a list of outcomes representing one sample, or a
     matrix of <m> rows and <n> columns, representing <n> samples of
     size <m> each.

     Available options are:

        * <box_width> (default, `3/4'): relative width of rectangles.
          This value must be in the range `[0,1]'.

        * <grouping> (default, `clustered'): indicates how multiple
          samples are shown. Valid values are: `clustered' and
          `stacked'.

        * <groups_gap> (default, `1'): a positive integer number
          representing the gap between two consecutive groups of bars.

        * <bars_colors> (default, `[]'): a list of colors for multiple
          samples.  When there are more samples than specified colors,
          the extra necesary colors are chosen at random. See `color'
          to learn more about them.

        * <frequency> (default, `absolute'): indicates the scale of the
          ordinates. Possible values are:  `absolute', `relative', and
          `percent'.

        * <ordering> (default, `orderlessp'): possible values are
          `orderlessp' or `ordergreatp', indicating how statistical
          outcomes should be ordered on the <x>-axis.

        * <sample_keys> (default, `[]'): a list with the strings to be
          used in the legend.  When the list length is other than 0 or
          the number of samples, an error message is returned.

        * <start_at> (default, `0'): indicates where the plot begins to
          be plotted on the x axis.

        * All global `draw' options, except `xtics', which is
          internally assigned by `barsplot'.  If you want to set your
          own values for this option or want to build complex scenes,
          make use of `barsplot_description'. See example below.

        * The following local `draw' options: `key', `color',
          `fill_color', `fill_density' and `line_width'. See also
          `bars'.

Function: boxplot (<data>)
Function: boxplot (<data>, <option_1>, <option_2>, ...)
Function: boxplot_description (...)

     This function plots box-and-whishker diagrams. Argument <data> can
     be a list, which is not of great interest, since these diagrams
     are mainly used for comparing different samples, or a matrix, so
     it is possible to compare two or more components of a multivariate
     statistical variable.  But it is also allowed <data> to be a list
     of samples with possible different sample sizes, in fact this is
     the only function in package `descriptive' that admits this type
     of data structure.

     Available options are:

        * <box_width> (default, `3/4'): relative width of boxes.  This
          value must be in the range `[0,1]'.

        * <box_orientation> (default, `vertical'): possible values:
          `vertical' and `horizontal'.

        * All `draw' options, except `points_joined', `point_size',
          `point_type', `xtics', `ytics', `xrange', and `yrange', which
          are internally assigned by `boxplot'.  If you want to set
          your own values for this options or want to build complex
          scenes, make use of `boxplot_description'.

        * The following local `draw' options: `key', `color', and
          `line_width'.

Function: histogram (<list>)
Function: histogram (<list>, <option_1>, <option_2>, ...)
Function: histogram (<one_column_matrix>)
Function: histogram (<one_column_matrix>, <option_1>, <option_2>, ...)
Function: histogram (<one_row_matrix>)
Function: histogram (<one_row_matrix>, <option_1>, <option_2>, ...)
Function: histogram_description (...)

     This function plots an histogram from a continuous sample.  Sample
     data must be stored in a list of numbers or a one dimensional
     matrix.

     Available options are:

        * <nclasses> (default, `10'): number of classes of the
          histogram, or a list indicating the limits of the classes and
          the number of them, or only the limits.

        * <frequency> (default, `absolute'): indicates the scale of the
          ordinates. Possible values are:  `absolute', `relative', and
          `percent'.

        * <htics> (default, `auto'): format of the histogram tics.
          Possible values are: `auto', `endpoints', `intervals', or a
          list of labels.

        * All global `draw' options, except `xrange', `yrange', and
          `xtics', which are internally assigned by `histogram'.  If
          you want to set your own values for these options, make use of
          `histogram_description'. See examples bellow.

        * The following local `draw' options: `key', `color',
          `fill_color', `fill_density' and `line_width'. See also
          `bars'.

Function: piechart (<list>)
Function: piechart (<list>, <option_1>, <option_2>, ...)
Function: piechart (<one_column_matrix>)
Function: piechart (<one_column_matrix>, <option_1>, <option_2>, ...)
Function: piechart (<one_row_matrix>)
Function: piechart (<one_row_matrix>, <option_1>, <option_2>, ...)
Function: piechart_description (...)

     Similar to `barsplot', but plots sectors instead of rectangles.

     Available options are:

        * <sector_colors> (default, `[]'): a list of colors for sectors.
          When there are more sectors than specified colors, the extra
          necesary colors are chosen at random. See `color' to learn
          more about them.

        * <pie_center> (default, `[0,0]'): diagram's center.

        * <pie_radius> (default, `1'): diagram's radius.

        * All global `draw' options, except `key', which is internally
          assigned by `piechart'.  If you want to set your own values
          for this option or want to build complex scenes, make use of
          `piechart_description'.

        * The following local `draw' options: `key', `color',
          `fill_density' and `line_width'. See also `ellipse'.

Function: scatterplot (<list>)
Function: scatterplot (<list>, <option_1>, <option_2>, ...)
Function: scatterplot (<matrix>)
Function: scatterplot (<matrix>, <option_1>, <option_2>, ...)
Function: scatterplot_description (...)

     Plots scatter diagrams both for univariate (<list>) and
     multivariate (<matrix>) samples.

     Available options are the same admitted by `histogram'.

     Function `scatterplot_description' creates a graphic object
     suitable for creating complex scenes, together with other graphic
     objects. There is also a function `wxscatterplot' for creating
     embedded histograms in interfaces wxMaxima and iMaxima.

     Examples:

     Univariate scatter diagram from a simulated Gaussian sample.

          (%i1) load (descriptive)$
          (%i2) load (distrib)$
          (%i3) scatterplot(
                  random_normal(0,1,200),
                  xaxis      = true,
                  point_size = 2,
                  dimensions = [600,150])$

     Two dimensional scatter plot.

          (%i1) load (descriptive)$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) scatterplot(
                 submatrix(s2, 1,2,3),
                 title      = "Data from stations #4 and #5",
                 point_type = diamant,
                 point_size = 2,
                 color      = blue)$

     Three dimensional scatter plot.

          (%i1) load (descriptive)$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) scatterplot(submatrix (s2, 1,2), nclasses=4)$

     Five dimensional scatter plot, with five classes histograms.

          (%i1) load (descriptive)$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) scatterplot(
                  s2,
                  nclasses     = 5,
                  frequency    = relative,
                  fill_color   = blue,
                  fill_density = 0.3,
                  xtics        = 5)$

     For plotting isolated or line-joined points in two and three
     dimensions, see `points'. See also `histogram'.

Function: starplot (<data1>, <data2>, ..., <option_1>, <option_2>, ...)
Function: starplot_description (...)

     Plots star diagrams for discrete statistical variables, both for
     one or multiple samples.

     <data> can be a list of outcomes representing one sample, or a
     matrix of <m> rows and <n> columns, representing <n> samples of
     size <m> each.

     Available options are:

        * <stars_colors> (default, `[]'): a list of colors for multiple
          samples.  When there are more samples than specified colors,
          the extra necesary colors are chosen at random. See `color'
          to learn more about them.

        * <frequency> (default, `absolute'): indicates the scale of the
          radii. Possible values are:  `absolute' and `relative'.

        * <ordering> (default, `orderlessp'): possible values are
          `orderlessp' or `ordergreatp', indicating how statistical
          outcomes should be ordered.

        * <sample_keys> (default, `[]'): a list with the strings to be
          used in the legend.  When the list length is other than 0 or
          the number of samples, an error message is returned.

        * <star_center> (default, `[0,0]'): diagram's center.

        * <star_radius> (default, `1'): diagram's radius.

        * All global `draw' options, except `points_joined',
          `point_type', and `key', which are internally assigned by
          `starplot'.  If you want to set your own values for this
          options or want to build complex scenes, make use of
          `starplot_description'.

        * The following local `draw' option: `line_width'.

Function: stemplot (<data>)
Function: stemplot (<data>, <option>)

     Plots stem and leaf diagrams.

     Unique available option is:

        * <leaf_unit> (default, `1'): indicates the unit of the leaves;
          must be a power of 10.

46 diag


Functions and Variables for diag

46.1 Functions and Variables for diag

Function: diag (<lm>)

     Constructs a square matrix with the matrices of <lm> in the
     diagonal. <lm> is a list of matrices or scalars.

     Example:
          (%i1) load("diag")$

          (%i2) a1:matrix([1,2,3],[0,4,5],[0,0,6])$

          (%i3) a2:matrix([1,1],[1,0])$

          (%i4) diag([a1,x,a2]);
                             [ 1  2  3  0  0  0 ]
                             [                  ]
                             [ 0  4  5  0  0  0 ]
                             [                  ]
                             [ 0  0  6  0  0  0 ]
          (%o4)              [                  ]
                             [ 0  0  0  x  0  0 ]
                             [                  ]
                             [ 0  0  0  0  1  1 ]
                             [                  ]
                             [ 0  0  0  0  1  0 ]

     To use this function write first `load("diag")'.

Function: JF (<lambda>,<n>)

     Returns the Jordan cell of order <n> with eigenvalue <lambda>.

     Example:
          (%i1) load("diag")$

          (%i2) JF(2,5);
                              [ 2  1  0  0  0 ]
                              [               ]
                              [ 0  2  1  0  0 ]
                              [               ]
          (%o2)               [ 0  0  2  1  0 ]
                              [               ]
                              [ 0  0  0  2  1 ]
                              [               ]
                              [ 0  0  0  0  2 ]
          (%i3) JF(3,2);
                                   [ 3  1 ]
          (%o3)                    [      ]
                                   [ 0  3 ]

     To use this function write first `load("diag")'.

Function: jordan (<mat>)

     Returns the Jordan form of matrix <mat>, but codified in a Maxima
     list.  To get the corresponding matrix, call function `dispJordan'
     using as argument the output of `jordan'.

     Example:
          (%i1) load("diag")$

          (%i3) a:matrix([2,0,0,0,0,0,0,0],
                         [1,2,0,0,0,0,0,0],
                         [-4,1,2,0,0,0,0,0],
                         [2,0,0,2,0,0,0,0],
                         [-7,2,0,0,2,0,0,0],
                         [9,0,-2,0,1,2,0,0],
                         [-34,7,1,-2,-1,1,2,0],
                         [145,-17,-16,3,9,-2,0,3])$

          (%i34) jordan(a);
          (%o4)             [[2, 3, 3, 1], [3, 1]]
          (%i5) dispJordan(%);
                          [ 2  1  0  0  0  0  0  0 ]
                          [                        ]
                          [ 0  2  1  0  0  0  0  0 ]
                          [                        ]
                          [ 0  0  2  0  0  0  0  0 ]
                          [                        ]
                          [ 0  0  0  2  1  0  0  0 ]
          (%o5)           [                        ]
                          [ 0  0  0  0  2  1  0  0 ]
                          [                        ]
                          [ 0  0  0  0  0  2  0  0 ]
                          [                        ]
                          [ 0  0  0  0  0  0  2  0 ]
                          [                        ]
                          [ 0  0  0  0  0  0  0  3 ]

     To use this function write first `load("diag")'. See also
     `dispJordan' and `minimalPoly'.

Function: dispJordan (<l>)

     Returns the Jordan matrix associated to the codification given by
     the Maxima list <l>, which is the output given by function
     `jordan'.

     Example:
          (%i1) load("diag")$

          (%i2) b1:matrix([0,0,1,1,1],
                          [0,0,0,1,1],
                          [0,0,0,0,1],
                          [0,0,0,0,0],
                          [0,0,0,0,0])$

          (%i3) jordan(b1);
          (%o3)                  [[0, 3, 2]]
          (%i4) dispJordan(%);
                              [ 0  1  0  0  0 ]
                              [               ]
                              [ 0  0  1  0  0 ]
                              [               ]
          (%o4)               [ 0  0  0  0  0 ]
                              [               ]
                              [ 0  0  0  0  1 ]
                              [               ]
                              [ 0  0  0  0  0 ]

     To use this function write first `load("diag")'. See also `jordan'
     and `minimalPoly'.

Function: minimalPoly (<l>)

     Returns the minimal polynomial associated to the codification
     given by the Maxima list <l>, which is the output given by
     function `jordan'.

     Example:
          (%i1) load("diag")$

          (%i2) a:matrix([2,1,2,0],
                         [-2,2,1,2],
                         [-2,-1,-1,1],
                         [3,1,2,-1])$

          (%i3) jordan(a);
          (%o3)               [[- 1, 1], [1, 3]]
          (%i4) minimalPoly(%);
                                      3
          (%o4)                (x - 1)  (x + 1)

     To use this function write first `load("diag")'. See also `jordan'
     and `dispJordan'.

Function: ModeMatrix (<A>,<l>)

     Returns the matrix <M> such that (M^^-1).A.M=J, where <J> is the
     Jordan form of <A>. The Maxima list <l> is the codified form of
     the Jordan form as returned by function `jordan'.

     Example:
          (%i1) load("diag")$

          (%i2) a:matrix([2,1,2,0],
                    [-2,2,1,2],
                    [-2,-1,-1,1],
                    [3,1,2,-1])$

          (%i3) jordan(a);
          (%o3)               [[- 1, 1], [1, 3]]
          (%i4) M: ModeMatrix(a,%);
                            [  1    - 1   1   1 ]
                            [                   ]
                            [   1               ]
                            [ - -   - 1   0   0 ]
                            [   9               ]
                            [                   ]
          (%o4)             [   13              ]
                            [ - --   1   - 1  0 ]
                            [   9               ]
                            [                   ]
                            [  17               ]
                            [  --   - 1   1   1 ]
                            [  9                ]
          (%i5) is(  (M^^-1).a.M = dispJordan(%o3)  );
          (%o5)                      true
     Note that `dispJordan(%o3)' is the Jordan form of matrix `a'.

     To use this function write first `load("diag")'. See also `jordan'
     and `dispJordan'.

Function: mat_function (<f>,<mat>)

     Returns f(mat), where <f> is an analytic function and <mat> a
     matrix. This computation is based on Cauchy's integral formula,
     which states that if `f(x)' is analytic and

          mat = diag([JF(m1,n1),...,JF(mk,nk)]),

     then

          f(mat) = ModeMatrix*diag([f(JF(m1,n1)), ..., f(JF(mk,nk))])
                                                        *ModeMatrix^^(-1)

     Note that there are about 6 or 8 other methods for this
     calculation.

     Some examples follow.

     Example 1:
          (%i1) load("diag")$

          (%i2) b2:matrix([0,1,0], [0,0,1], [-1,-3,-3])$

          (%i3) mat_function(exp,t*b2);
                         2   - t
                        t  %e          - t     - t
          (%o3) matrix([-------- + t %e    + %e   ,
                           2
                  - t     - t                           - t
           2    %e      %e        - t           - t   %e
          t  (- ----- - ----- + %e   ) + t (2 %e    - -----)
                  t       2                             t
                         t
                                   - t          - t     - t
                 - t       - t   %e        2  %e      %e
           + 2 %e   , t (%e    - -----) + t  (----- - -----)
                                   t            2       t
                         2   - t            - t     - t
               - t      t  %e        2    %e      %e        - t
           + %e   ], [- --------, - t  (- ----- - ----- + %e   ),
                           2                t       2
                                                   t
                  - t     - t      2   - t
             2  %e      %e        t  %e          - t
          - t  (----- - -----)], [-------- - t %e   ,
                  2       t          2
                  - t     - t                           - t
           2    %e      %e        - t           - t   %e
          t  (- ----- - ----- + %e   ) - t (2 %e    - -----),
                  t       2                             t
                         t
                - t     - t                 - t
           2  %e      %e            - t   %e
          t  (----- - -----) - t (%e    - -----)])
                2       t                   t
          (%i4) ratsimp(%);
                         [   2              - t ]
                         [ (t  + 2 t + 2) %e    ]
                         [ -------------------- ]
                         [          2           ]
                         [                      ]
                         [         2   - t      ]
          (%o4)  Col 1 = [        t  %e         ]
                         [      - --------      ]
                         [           2          ]
                         [                      ]
                         [     2          - t   ]
                         [   (t  - 2 t) %e      ]
                         [   ----------------   ]
                         [          2           ]
                   [      2        - t    ]
                   [    (t  + t) %e       ]
                   [                      ]
           Col 2 = [     2            - t ]
                   [ - (t  - t - 1) %e    ]
                   [                      ]
                   [     2          - t   ]
                   [   (t  - 3 t) %e      ]
                   [        2   - t       ]
                   [       t  %e          ]
                   [       --------       ]
                   [          2           ]
                   [                      ]
                   [      2          - t  ]
           Col 3 = [    (t  - 2 t) %e     ]
                   [  - ----------------  ]
                   [           2          ]
                   [                      ]
                   [   2              - t ]
                   [ (t  - 4 t + 2) %e    ]
                   [ -------------------- ]
                   [          2           ]

     Example 2:
          (%i5) b1:matrix([0,0,1,1,1],
                          [0,0,0,1,1],
                          [0,0,0,0,1],
                          [0,0,0,0,0],
                          [0,0,0,0,0])$

          (%i6) mat_function(exp,t*b1);
                            [              2     ]
                            [             t      ]
                            [ 1  0  t  t  -- + t ]
                            [             2      ]
                            [                    ]
          (%o6)             [ 0  1  0  t    t    ]
                            [                    ]
                            [ 0  0  1  0    t    ]
                            [                    ]
                            [ 0  0  0  1    0    ]
                            [                    ]
                            [ 0  0  0  0    1    ]
          (%i7) minimalPoly(jordan(b1));
                                       3
          (%o7)                       x
          (%i8) ident(5)+t*b1+1/2*(t^2)*b1^^2;
                            [              2     ]
                            [             t      ]
                            [ 1  0  t  t  -- + t ]
                            [             2      ]
                            [                    ]
          (%o8)             [ 0  1  0  t    t    ]
                            [                    ]
                            [ 0  0  1  0    t    ]
                            [                    ]
                            [ 0  0  0  1    0    ]
                            [                    ]
                            [ 0  0  0  0    1    ]
          (%i9) mat_function(exp,%i*t*b1);
                       [                           2 ]
                       [                          t  ]
                       [ 1  0  %i t  %i t  %i t - -- ]
                       [                          2  ]
                       [                             ]
          (%o9)        [ 0  1   0    %i t    %i t    ]
                       [                             ]
                       [ 0  0   1     0      %i t    ]
                       [                             ]
                       [ 0  0   0     1        0     ]
                       [                             ]
                       [ 0  0   0     0        1     ]
          (%i10) mat_function(cos,t*b1)+%i*mat_function(sin,t*b1);
                        [                           2 ]
                        [                          t  ]
                        [ 1  0  %i t  %i t  %i t - -- ]
                        [                          2  ]
                        [                             ]
          (%o10)        [ 0  1   0    %i t    %i t    ]
                        [                             ]
                        [ 0  0   1     0      %i t    ]
                        [                             ]
                        [ 0  0   0     1        0     ]
                        [                             ]
                        [ 0  0   0     0        1     ]

     Example 3:
          (%i11) a1:matrix([2,1,0,0,0,0],
                           [-1,4,0,0,0,0],
                           [-1,1,2,1,0,0],
                           [-1,1,-1,4,0,0],
                           [-1,1,-1,1,3,0],
                           [-1,1,-1,1,1,2])$

          (%i12) fpow(x):=block([k],declare(k,integer),x^k)$

          (%i13) mat_function(fpow,a1);
                          [  k      k - 1 ]         [      k - 1    ]
                          [ 3  - k 3      ]         [   k 3         ]
                          [               ]         [               ]
                          [       k - 1   ]         [  k      k - 1 ]
                          [  - k 3        ]         [ 3  + k 3      ]
                          [               ]         [               ]
                          [       k - 1   ]         [      k - 1    ]
                          [  - k 3        ]         [   k 3         ]
          (%o13)  Col 1 = [               ] Col 2 = [               ]
                          [       k - 1   ]         [      k - 1    ]
                          [  - k 3        ]         [   k 3         ]
                          [               ]         [               ]
                          [       k - 1   ]         [      k - 1    ]
                          [  - k 3        ]         [   k 3         ]
                          [               ]         [               ]
                          [       k - 1   ]         [      k - 1    ]
                          [  - k 3        ]         [   k 3         ]
                   [       0       ]         [       0       ]
                   [               ]         [               ]
                   [       0       ]         [       0       ]
                   [               ]         [               ]
                   [  k      k - 1 ]         [      k - 1    ]
                   [ 3  - k 3      ]         [   k 3         ]
                   [               ]         [               ]
           Col 3 = [       k - 1   ] Col 4 = [  k      k - 1 ]
                   [  - k 3        ]         [ 3  + k 3      ]
                   [               ]         [               ]
                   [       k - 1   ]         [      k - 1    ]
                   [  - k 3        ]         [   k 3         ]
                   [               ]         [               ]
                   [       k - 1   ]         [      k - 1    ]
                   [  - k 3        ]         [   k 3         ]
                   [    0    ]
                   [         ]         [ 0  ]
                   [    0    ]         [    ]
                   [         ]         [ 0  ]
                   [    0    ]         [    ]
                   [         ]         [ 0  ]
           Col 5 = [    0    ] Col 6 = [    ]
                   [         ]         [ 0  ]
                   [    k    ]         [    ]
                   [   3     ]         [ 0  ]
                   [         ]         [    ]
                   [  k    k ]         [  k ]
                   [ 3  - 2  ]         [ 2  ]

     To use this function write first `load("diag")'.

47 distrib


Introduction to distrib
Functions and Variables for continuous distributions
Functions and Variables for discrete distributions

47.1 Introduction to distrib

Package `distrib' contains a set of functions for making probability
computations on both discrete and continuous univariate models.

   What follows is a short reminder of basic probabilistic related
definitions.

   Let f(x) be the <density function> of an absolute continuous random
variable X. The <distribution function> is defined as
                            x
                           /
                           [
                    F(x) = I     f(u) du
                           ]
                           /
                            minf
   which equals the probability <Pr(X <= x)>.

   The <mean> value is a localization parameter and is defined as
                          inf
                         /
                         [
                E[X]  =  I   x f(x) dx
                         ]
                         /
                          minf

   The <variance> is a measure of variation,
                      inf
                     /
                     [                    2
              V[X] = I     f(x) (x - E[X])  dx
                     ]
                     /
                      minf
   which is a positive real number. The square root of the variance is
the <standard deviation>, D[X]=sqrt(V[X]), and it is another measure of
variation.

   The <skewness coefficient> is a measure of non-symmetry,
                      inf
                     /
                 1   [                    3
       SK[X] = ----- I     f(x) (x - E[X])  dx
                   3 ]
               D[X]  /
                      minf

   And the <kurtosis coefficient> measures the peakedness of the
distribution,
                      inf
                     /
                 1   [                    4
       KU[X] = ----- I     f(x) (x - E[X])  dx - 3
                   4 ]
               D[X]  /
                      minf
   If X is gaussian, KU[X]=0. In fact, both skewness and kurtosis are
shape parameters used to measure the non-gaussianity of a distribution.

   If the random variable X is discrete, the density, or <probability>,
function f(x) takes positive values within certain countable set of
numbers x_i, and zero elsewhere. In this case, the distribution
function is
                            ====
                            \
                     F(x) =  >    f(x )
                            /        i
                            ====
                           x <= x
                            i

   The mean, variance, standard deviation, skewness coefficient and
kurtosis coefficient take the form
                            ====
                            \
                     E[X] =  >  x  f(x ) ,
                            /    i    i
                            ====
                             x
                              i

                     ====
                     \                     2
             V[X] =   >    f(x ) (x - E[X])  ,
                     /        i    i
                     ====
                      x
                       i

                    D[X] = sqrt(V[X]),

                          ====
                   1      \                     3
       SK[X] =  -------    >    f(x ) (x - E[X])
                D[X]^3    /        i    i
                          ====
                           x
                            i
   and
                          ====
                   1      \                     4
       KU[X] =  -------    >    f(x ) (x - E[X])   - 3 ,
                D[X]^4    /        i    i
                          ====
                           x
                            i
   respectively.

   There is a naming convention in package `distrib'. Every function
name has two parts, the first one makes reference to the function or
parameter we want to calculate,
     Functions:
        Density function            (pdf_*)
        Distribution function       (cdf_*)
        Quantile                    (quantile_*)
        Mean                        (mean_*)
        Variance                    (var_*)
        Standard deviation          (std_*)
        Skewness coefficient        (skewness_*)
        Kurtosis coefficient        (kurtosis_*)
        Random variate              (random_*)

   The second part is an explicit reference to the probabilistic model,
     Continuous distributions:
        Normal              (*normal)
        Student             (*student_t)
        Chi^2               (*chi2)
        Noncentral Chi^2    (*noncentral_chi2)
        F                   (*f)
        Exponential         (*exp)
        Lognormal           (*lognormal)
        Gamma               (*gamma)
        Beta                (*beta)
        Continuous uniform  (*continuous_uniform)
        Logistic            (*logistic)
        Pareto              (*pareto)
        Weibull             (*weibull)
        Rayleigh            (*rayleigh)
        Laplace             (*laplace)
        Cauchy              (*cauchy)
        Gumbel              (*gumbel)

     Discrete distributions:
        Binomial             (*binomial)
        Poisson              (*poisson)
        Bernoulli            (*bernoulli)
        Geometric            (*geometric)
        Discrete uniform     (*discrete_uniform)
        hypergeometric       (*hypergeometric)
        Negative binomial    (*negative_binomial)
        Finite discrete      (*general_finite_discrete)

   For example, `pdf_student_t(x,n)' is the density function of the
Student distribution with <n> degrees of freedom, `std_pareto(a,b)' is
the standard deviation of the Pareto distribution with parameters <a>
and <b> and `kurtosis_poisson(m)' is the kurtosis coefficient of the
Poisson distribution with mean <m>.

   In order to make use of package `distrib' you need first to load it
by typing
     (%i1) load(distrib)$

   For comments, bugs or suggestions, please contact the author at
<'mario AT edu DOT xunta DOT es'>.

47.2 Functions and Variables for continuous distributions

Function: pdf_normal (<x>,<m>,<s>)

     Returns the value at <x> of the density function of a Normal(m,s)
     random variable, with s>0. To make use of this function, write
     first `load(distrib)'.

Function: cdf_normal (<x>,<m>,<s>)

     Returns the value at <x> of the distribution function of a
     Normal(m,s) random variable, with s>0. This function is defined in
     terms of Maxima's built-in error function `erf'.

          (%i1) load (distrib)$
          (%i2) assume(s>0)$ cdf_normal(x,m,s);
                                       x - m
                                 erf(---------)
                                     sqrt(2) s    1
          (%o3)                  -------------- + -
                                       2          2

     See also `erf'.

Function: quantile_normal (<q>,<m>,<s>)

     Returns the <q>-quantile of a Normal(m,s) random variable, with
     s>0; in other words, this is the inverse of `cdf_normal'. Argument
     <q> must be an element of [0,1]. To make use of this function,
     write first `load(distrib)'.

          (%i1) load (distrib)$
          (%i2) quantile_normal(95/100,0,1);
                                                9
          (%o2)             sqrt(2) inverse_erf(--)
                                                10
          (%i3) float(%);
          (%o3)               1.644853626951472

Function: mean_normal (<m>,<s>)

     Returns the mean of a Normal(m,s) random variable, with s>0,
     namely <m>. To make use of this function, write first
     `load(distrib)'.

Function: var_normal (<m>,<s>)

     Returns the variance of a Normal(m,s) random variable, with s>0,
     namely <s^2>. To make use of this function, write first
     `load(distrib)'.

Function: std_normal (<m>,<s>)

     Returns the standard deviation of a Normal(m,s) random variable,
     with s>0, namely <s>. To make use of this function, write first
     `load(distrib)'.

Function: skewness_normal (<m>,<s>)

     Returns the skewness coefficient of a Normal(m,s) random variable,
     with s>0, which is always equal to 0. To make use of this
     function, write first `load(distrib)'.

Function: kurtosis_normal (<m>,<s>)

     Returns the kurtosis coefficient of a Normal(m,s) random variable,
     with s>0, which is always equal to 0. To make use of this
     function, write first `load(distrib)'.

Function: random_normal (<m>,<s>)
Function: random_normal (<m>,<s>,<n>)

     Returns a Normal(m,s) random variate, with s>0. Calling
     `random_normal' with a third argument <n>, a random sample of size
     <n> will be simulated.

     This is an implementation of the Box-Mueller algorithm, as
     described in Knuth, D.E. (1981) <Seminumerical Algorithms. The Art
     of Computer Programming.> Addison-Wesley.

     To make use of this function, write first `load(distrib)'.

Function: pdf_student_t (<x>,<n>)

     Returns the value at <x> of the density function of a Student
     random variable t(n), with n>0 degrees of freedom. To make use of
     this function, write first `load(distrib)'.

Function: cdf_student_t (<x>,<n>)

     Returns the value at <x> of the distribution function of a Student
     random variable t(n), with n>0 degrees of freedom.

          (%i1) load (distrib)$
          (%i2) cdf_student_t(1/2, 7/3);
                                                   7  1  28
                       beta_incomplete_regularized(-, -, --)
                                                   6  2  31
          (%o2)    1 - -------------------------------------
                                         2
          (%i3) float(%);
          (%o3)                .6698450596140415

Function: quantile_student_t (<q>,<n>)

     Returns the <q>-quantile of a Student random variable t(n), with
     n>0; in other words, this is the inverse of `cdf_student_t'.
     Argument <q> must be an element of [0,1]. To make use of this
     function, write first `load(distrib)'.

Function: mean_student_t (<n>)

     Returns the mean of a Student random variable t(n), with n>0,
     which is always equal to 0. To make use of this function, write
     first `load(distrib)'.

Function: var_student_t (<n>)

     Returns the variance of a Student random variable t(n), with n>2.

          (%i1) load (distrib)$
          (%i2) assume(n>2)$  var_student_t(n);
                                          n
          (%o3)                         -----
                                        n - 2

Function: std_student_t (<n>)

     Returns the standard deviation of a Student random variable t(n),
     with n>2. To make use of this function, write first
     `load(distrib)'.

Function: skewness_student_t (<n>)

     Returns the skewness coefficient of a Student random variable
     t(n), with n>3, which is always equal to 0. To make use of this
     function, write first `load(distrib)'.

Function: kurtosis_student_t (<n>)

     Returns the kurtosis coefficient of a Student random variable
     t(n), with n>4. To make use of this function, write first
     `load(distrib)'.

Function: random_student_t (<n>)
Function: random_student_t (<n>,<m>)

     Returns a Student random variate t(n), with n>0. Calling
     `random_student_t' with a second argument <m>, a random sample of
     size <m> will be simulated.

     The implemented algorithm is based on the fact that if <Z> is a
     normal random variable N(0,1) and S^2 is a chi square random
     variable with <n> degrees of freedom, Chi^2(n), then
                                     Z
                           X = -------------
                               /   2  \ 1/2
                               |  S   |
                               | ---  |
                               \  n   /
     is a Student random variable with <n> degrees of freedom, t(n).

     To make use of this function, write first `load(distrib)'.

Function: pdf_noncentral_student_t (<x>,<n>,<ncp>)

     Returns the value at <x> of the density function of a noncentral
     Student random variable nc_t(n,ncp), with n>0 degrees of freedom
     and noncentrality parameter ncp. To make use of this function,
     write first `load(distrib)'.

     Sometimes an extra work is necessary to get the final result.

          (%i1) load (distrib)$
          (%i2) expand(pdf_noncentral_student_t(3,5,0.1));
                 .01370030107589574 sqrt(5)
          (%o2)  --------------------------
                 sqrt(2) sqrt(14) sqrt(%pi)
             1.654562884111515E-4 sqrt(5)
           + ----------------------------
                      sqrt(%pi)
             .02434921505438663 sqrt(5)
           + --------------------------
                        %pi
          (%i3) float(%);
          (%o3)          .02080593159405669

Function: cdf_noncentral_student_t (<x>,<n>,<ncp>)

     Returns the value at <x> of the distribution function of a
     noncentral Student random variable nc_t(n,ncp), with n>0 degrees
     of freedom and noncentrality parameter ncp. This function has no
     closed form and it is numerically computed if the global variable
     `numer' equals `true' or at least one of the arguments is a float,
     otherwise it returns a nominal expression.

          (%i1) load (distrib)$
          (%i2) cdf_noncentral_student_t(-2,5,-5);
          (%o2) cdf_noncentral_student_t(- 2, 5, - 5)
          (%i3) cdf_noncentral_student_t(-2.0,5,-5);
          (%o3)          .9952030093319743

Function: quantile_noncentral_student_t (<q>,<n>,<ncp>)

     Returns the <q>-quantile of a noncentral Student random variable
     nc_t(n,ncp), with n>0 degrees of freedom and noncentrality
     parameter ncp; in other words, this is the inverse of
     `cdf_noncentral_student_t'. Argument <q> must be an element of
     [0,1]. To make use of this function, write first `load(distrib)'.

Function: mean_noncentral_student_t (<n>,<ncp>)

     Returns the mean of a noncentral Student random variable
     nc_t(n,ncp), with n>1 degrees of freedom and noncentrality
     parameter ncp. To make use of this function, write first
     `load(distrib)'.

          (%i1) load (distrib)$
          (%i2) (assume(df>1), mean_noncentral_student_t(df,k));
                             df - 1
                       gamma(------) sqrt(df) k
                               2
          (%o2)        ------------------------
                                        df
                          sqrt(2) gamma(--)
                                        2

Function: var_noncentral_student_t (<n>,<ncp>)

     Returns the variance of a noncentral Student random variable
     nc_t(n,ncp), with n>2 degrees of freedom and noncentrality
     parameter ncp. To make use of this function, write first
     `load(distrib)'.

Function: std_noncentral_student_t (<n>,<ncp>)

     Returns the standard deviation of a noncentral Student random
     variable nc_t(n,ncp), with n>2 degrees of freedom and
     noncentrality parameter ncp. To make use of this function, write
     first `load(distrib)'.

Function: skewness_noncentral_student_t (<n>,<ncp>)

     Returns the skewness coefficient of a noncentral Student random
     variable nc_t(n,ncp), with n>3 degrees of freedom and
     noncentrality parameter ncp. To make use of this function, write
     first `load(distrib)'.

Function: kurtosis_noncentral_student_t (<n>,<ncp>)

     Returns the kurtosis coefficient of a noncentral Student random
     variable nc_t(n,ncp), with n>4 degrees of freedom and
     noncentrality parameter ncp. To make use of this function, write
     first `load(distrib)'.

Function: random_noncentral_student_t (<n>,<ncp>)
Function: random_noncentral_student_t (<n>,<ncp>,<m>)

     Returns a noncentral Student random variate nc_t(n,ncp), with n>0.
     Calling `random_noncentral_student_t' with a third argument <m>, a
     random sample of size <m> will be simulated.

     The implemented algorithm is based on the fact that if <X> is a
     normal random variable N(ncp,1) and S^2 is a chi square random
     variable with <n> degrees of freedom, Chi^2(n), then
                                     X
                           U = -------------
                               /   2  \ 1/2
                               |  S   |
                               | ---  |
                               \  n   /
     is a noncentral Student random variable with <n> degrees of
     freedom and noncentrality parameter ncp, nc_t(n,ncp).

     To make use of this function, write first `load(distrib)'.

Function: pdf_chi2 (<x>,<n>)

     Returns the value at <x> of the density function of a Chi-square
     random variable Chi^2(n), with n>0.

     The Chi^2(n) random variable is equivalent to the Gamma(n/2,2),
     therefore when Maxima has not enough information to get the
     result, a noun form based on the gamma density is returned.

          (%i1) load (distrib)$
          (%i2) pdf_chi2(x,n);
                                              n
          (%o2)                  pdf_gamma(x, -, 2)
                                              2
          (%i3) assume(x>0, n>0)$  pdf_chi2(x,n);
                                   n/2 - 1   - x/2
                                  x        %e
          (%o4)                   ----------------
                                    n/2       n
                                   2    gamma(-)
                                              2

Function: cdf_chi2 (<x>,<n>)

     Returns the value at <x> of the distribution function of a
     Chi-square random variable Chi^2(n), with n>0.

          (%i1) load (distrib)$
          (%i2) cdf_chi2(3,4);
                                                         3
          (%o2)      1 - gamma_incomplete_regularized(2, -)
                                                         2
          (%i3) float(%);
          (%o3)               .4421745996289256

Function: quantile_chi2 (<q>,<n>)

     Returns the <q>-quantile of a Chi-square random variable Chi^2(n),
     with n>0; in other words, this is the inverse of `cdf_chi2'.
     Argument <q> must be an element of [0,1].

     This function has no closed form and it is numerically computed if
     the global variable `numer' equals `true',  otherwise it returns a
     nominal expression based on the gamma quantile function, since the
     Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

          (%i1) load (distrib)$
          (%i2) quantile_chi2(0.99,9);
          (%o2)                   21.66599433346194
          (%i3) quantile_chi2(0.99,n);
                                                  n
          (%o3)              quantile_gamma(0.99, -, 2)
                                                  2

Function: mean_chi2 (<n>)

     Returns the mean of a Chi-square random variable Chi^2(n), with
     n>0.

     The Chi^2(n) random variable is equivalent to the Gamma(n/2,2),
     therefore when Maxima has not enough information to get the
     result, a noun form based on the gamma mean is returned.

          (%i1) load (distrib)$
          (%i2) mean_chi2(n);
                                             n
          (%o2)                   mean_gamma(-, 2)
                                             2
          (%i3) assume(n>0)$ mean_chi2(n);
          (%o4)                           n

Function: var_chi2 (<n>)

     Returns the variance of a Chi-square random variable Chi^2(n),
     with n>0.

     The Chi^2(n) random variable is equivalent to the Gamma(n/2,2),
     therefore when Maxima has not enough information to get the
     result, a noun form based on the gamma variance is returned.

          (%i1) load (distrib)$
          (%i2) var_chi2(n);
                                             n
          (%o2)                    var_gamma(-, 2)
                                             2
          (%i3) assume(n>0)$ var_chi2(n);
          (%o4)                          2 n

Function: std_chi2 (<n>)

     Returns the standard deviation of a Chi-square random variable
     Chi^2(n), with n>0.

     The Chi^2(n) random variable is equivalent to the Gamma(n/2,2),
     therefore when Maxima has not enough information to get the
     result, a noun form based on the gamma standard deviation is
     returned.

          (%i1) load (distrib)$
          (%i2) std_chi2(n);
                                             n
          (%o2)                    std_gamma(-, 2)
                                             2
          (%i3) assume(n>0)$ std_chi2(n);
          (%o4)                    sqrt(2) sqrt(n)

Function: skewness_chi2 (<n>)

     Returns the skewness coefficient of a Chi-square random variable
     Chi^2(n), with n>0.

     The Chi^2(n) random variable is equivalent to the Gamma(n/2,2),
     therefore when Maxima has not enough information to get the
     result, a noun form based on the gamma skewness coefficient is
     returned.

          (%i1) load (distrib)$
          (%i2) skewness_chi2(n);
                                               n
          (%o2)                 skewness_gamma(-, 2)
                                               2
          (%i3) assume(n>0)$ skewness_chi2(n);
                                      2 sqrt(2)
          (%o4)                       ---------
                                       sqrt(n)

Function: kurtosis_chi2 (<n>)

     Returns the kurtosis coefficient of a Chi-square random variable
     Chi^2(n), with n>0.

     The Chi^2(n) random variable is equivalent to the Gamma(n/2,2),
     therefore when Maxima has not enough information to get the
     result, a noun form based on the gamma kurtosis coefficient is
     returned.

          (%i1) load (distrib)$
          (%i2) kurtosis_chi2(n);
                                               n
          (%o2)                 kurtosis_gamma(-, 2)
                                               2
          (%i3) assume(n>0)$ kurtosis_chi2(n);
                                         12
          (%o4)                          --
                                         n

Function: random_chi2 (<n>)
Function: random_chi2 (<n>,<m>)

     Returns a Chi-square random variate Chi^2(n), with n>0. Calling
     `random_chi2' with a second argument <m>, a random sample of size
     <m> will be simulated.

     The simulation is based on the Ahrens-Cheng algorithm. See
     `random_gamma' for details.

     To make use of this function, write first `load(distrib)'.

Function: pdf_noncentral_chi2 (<x>,<n>,<ncp>)

     Returns the value at <x> of the density function of a noncentral
     Chi-square random variable nc_Chi^2(n,ncp), with n>0 and
     noncentrality parameter ncp>=0. To make use of this function,
     write first `load(distrib)'.

Function: cdf_noncentral_chi2 (<x>,<n>,<ncp>)

     Returns the value at <x> of the distribution function of a
     noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0
     and noncentrality parameter ncp>=0. To make use of this function,
     write first `load(distrib)'.

Function: quantile_noncentral_chi2 (<q>,<n>,<ncp>)

     Returns the <q>-quantile of a noncentral Chi-square random
     variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter
     ncp>=0; in other words, this is the inverse of
     `cdf_noncentral_chi2'. Argument <q> must be an element of [0,1].

     This function has no closed form and it is numerically computed if
     the global variable `numer' equals `true',  otherwise it returns a
     nominal expression.

Function: mean_noncentral_chi2 (<n>,<ncp>)

     Returns the mean of a noncentral Chi-square random variable
     nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

Function: var_noncentral_chi2 (<n>,<ncp>)

     Returns the variance of a noncentral Chi-square random variable
     nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

Function: std_noncentral_chi2 (<n>,<ncp>)

     Returns the standard deviation of a noncentral Chi-square random
     variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter
     ncp>=0.

Function: skewness_noncentral_chi2 (<n>,<ncp>)

     Returns the skewness coefficient of a noncentral Chi-square random
     variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter
     ncp>=0.

Function: kurtosis_noncentral_chi2 (<n>,<ncp>)

     Returns the kurtosis coefficient of a noncentral Chi-square random
     variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter
     ncp>=0.

Function: random_noncentral_chi2 (<n>,<ncp>)
Function: random_noncentral_chi2 (<n>,<ncp>,<m>)

     Returns a noncentral Chi-square random variate nc_Chi^2(n,ncp),
     with n>0 and noncentrality parameter ncp>=0. Calling
     `random_noncentral_chi2' with a third argument <m>, a random
     sample of size <m> will be simulated.

     To make use of this function, write first `load(distrib)'.

Function: pdf_f (<x>,<m>,<n>)

     Returns the value at <x> of the density function of a F random
     variable F(m,n), with m,n>0. To make use of this function, write
     first `load(distrib)'.

Function: cdf_f (<x>,<m>,<n>)

     Returns the value at <x> of the distribution function of a F
     random variable F(m,n), with m,n>0.

          (%i1) load (distrib)$
          (%i2) cdf_f(2,3,9/4);
                                                   9  3  3
          (%o2)    1 - beta_incomplete_regularized(-, -, --)
                                                   8  2  11
          (%i3) float(%);
          (%o3)                 0.66756728179008

Function: quantile_f (<q>,<m>,<n>)

     Returns the <q>-quantile of a F random variable F(m,n), with
     m,n>0; in other words, this is the inverse of `cdf_f'. Argument
     <q> must be an element of [0,1].

     This function has no closed form and it is numerically computed if
     the global variable `numer' equals `true',  otherwise it returns a
     nominal expression.

          (%i1) load (distrib)$
          (%i2) quantile_f(2/5,sqrt(3),5);
                                         2
          (%o2)               quantile_f(-, sqrt(3), 5)
                                         5
          (%i3) %,numer;
          (%o3)                   0.518947838573693

Function: mean_f (<m>,<n>)

     Returns the mean of a F random variable F(m,n), with m>0, n>2. To
     make use of this function, write first `load(distrib)'.

Function: var_f (<m>,<n>)

     Returns the variance of a F random variable F(m,n), with m>0, n>4.
     To make use of this function, write first `load(distrib)'.

Function: std_f (<m>,<n>)

     Returns the standard deviation of a F random variable F(m,n), with
     m>0, n>4. To make use of this function, write first
     `load(distrib)'.

Function: skewness_f (<m>,<n>)

     Returns the skewness coefficient of a F random variable F(m,n),
     with m>0, n>6. To make use of this function, write first
     `load(distrib)'.

Function: kurtosis_f (<m>,<n>)

     Returns the kurtosis coefficient of a F random variable F(m,n),
     with m>0, n>8. To make use of this function, write first
     `load(distrib)'.

Function: random_f (<m>,<n>)
Function: random_f (<m>,<n>,<k>)

     Returns a F random variate F(m,n), with m,n>0. Calling `random_f'
     with a third argument <k>, a random sample of size <k> will be
     simulated.

     The simulation algorithm is based on the fact that if <X> is a
     Chi^2(m) random variable and Y is a Chi^2(n) random variable, then
                                  n X
                              F = ---
                                  m Y
     is a F random variable with <m> and <n> degrees of freedom, F(m,n).

     To make use of this function, write first `load(distrib)'.

Function: pdf_exp (<x>,<m>)

     Returns the value at <x> of the density function of an
     Exponential(m) random variable, with m>0.

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull density is
     returned.

          (%i1) load (distrib)$
          (%i2) pdf_exp(x,m);
                                                  1
          (%o2)                 pdf_weibull(x, 1, -)
                                                  m
          (%i3) assume(x>0,m>0)$  pdf_exp(x,m);
                                          - m x
          (%o4)                       m %e

Function: cdf_exp (<x>,<m>)

     Returns the value at <x> of the distribution function of an
     Exponential(m) random variable, with m>0.

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull distribution
     is returned.

          (%i1) load (distrib)$
          (%i2) cdf_exp(x,m);
                                                  1
          (%o2)                 cdf_weibull(x, 1, -)
                                                  m
          (%i3) assume(x>0,m>0)$  cdf_exp(x,m);
                                           - m x
          (%o4)                      1 - %e

Function: quantile_exp (<q>,<m>)

     Returns the <q>-quantile of an Exponential(m) random variable,
     with m>0; in other words, this is the inverse of `cdf_exp'.
     Argument <q> must be an element of [0,1].

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull quantile is
     returned.

          (%i1) load (distrib)$
          (%i2) quantile_exp(0.56,5);
          (%o2)                   .1641961104139661
          (%i3) quantile_exp(0.56,m);
                                                      1
          (%o3)             quantile_weibull(0.56, 1, -)
                                                      m

Function: mean_exp (<m>)

     Returns the mean of an Exponential(m) random variable, with m>0.

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull mean is
     returned.

          (%i1) load (distrib)$
          (%i2) mean_exp(m);
                                                 1
          (%o2)                  mean_weibull(1, -)
                                                 m
          (%i3) assume(m>0)$  mean_exp(m);
                                          1
          (%o4)                           -
                                          m

Function: var_exp (<m>)

     Returns the variance of an Exponential(m) random variable, with
     m>0.

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull variance is
     returned.

          (%i1) load (distrib)$
          (%i2) var_exp(m);
                                                 1
          (%o2)                   var_weibull(1, -)
                                                 m
          (%i3) assume(m>0)$  var_exp(m);
                                         1
          (%o4)                          --
                                          2
                                         m

Function: std_exp (<m>)

     Returns the standard deviation of an Exponential(m) random
     variable, with m>0.

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull standard
     deviation is returned.

          (%i1) load (distrib)$
          (%i2) std_exp(m);
                                                 1
          (%o2)                   std_weibull(1, -)
                                                 m
          (%i3) assume(m>0)$  std_exp(m);
                                          1
          (%o4)                           -
                                          m

Function: skewness_exp (<m>)

     Returns the skewness coefficient of an Exponential(m) random
     variable, with m>0.

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull skewness
     coefficient is returned.

          (%i1) load (distrib)$
          (%i2) skewness_exp(m);
                                                   1
          (%o2)                skewness_weibull(1, -)
                                                   m
          (%i3) assume(m>0)$  skewness_exp(m);
          (%o4)                           2

Function: kurtosis_exp (<m>)

     Returns the kurtosis coefficient of an Exponential(m) random
     variable, with m>0.

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull kurtosis
     coefficient is returned.

          (%i1) load (distrib)$
          (%i2) kurtosis_exp(m);
                                                   1
          (%o2)                kurtosis_weibull(1, -)
                                                   m
          (%i3) assume(m>0)$  kurtosis_exp(m);
          (%o4)                           6

Function: random_exp (<m>)
Function: random_exp (<m>,<k>)

     Returns an Exponential(m) random variate, with m>0. Calling
     `random_exp' with a second argument <k>, a random sample of size
     <k> will be simulated.

     The simulation algorithm is based on the general inverse method.

     To make use of this function, write first `load(distrib)'.

Function: pdf_lognormal (<x>,<m>,<s>)

     Returns the value at <x> of the density function of a
     Lognormal(m,s) random variable, with s>0. To make use of this
     function, write first `load(distrib)'.

Function: cdf_lognormal (<x>,<m>,<s>)

     Returns the value at <x> of the distribution function of a
     Lognormal(m,s) random variable, with s>0. This function is defined
     in terms of Maxima's built-in error function `erf'.

          (%i1) load (distrib)$
          (%i2) assume(x>0, s>0)$  cdf_lognormal(x,m,s);
                                     log(x) - m
                                 erf(----------)
                                     sqrt(2) s     1
          (%o3)                  --------------- + -
                                        2          2

     See also `erf'.

Function: quantile_lognormal (<q>,<m>,<s>)

     Returns the <q>-quantile of a Lognormal(m,s) random variable, with
     s>0; in other words, this is the inverse of `cdf_lognormal'.
     Argument <q> must be an element of [0,1]. To make use of this
     function, write first `load(distrib)'.

          (%i1) load (distrib)$
          (%i2) quantile_lognormal(95/100,0,1);
                            sqrt(2) inverse_erf(9/10)
          (%o2)           %e
          (%i3) float(%);
          (%o3)               5.180251602233015

Function: mean_lognormal (<m>,<s>)

     Returns the mean of a Lognormal(m,s) random variable, with s>0. To
     make use of this function, write first `load(distrib)'.

Function: var_lognormal (<m>,<s>)

     Returns the variance of a Lognormal(m,s) random variable, with
     s>0. To make use of this function, write first `load(distrib)'.

Function: std_lognormal (<m>,<s>)

     Returns the standard deviation of a Lognormal(m,s) random
     variable, with s>0. To make use of this function, write first
     `load(distrib)'.

Function: skewness_lognormal (<m>,<s>)

     Returns the skewness coefficient of a Lognormal(m,s) random
     variable, with s>0. To make use of this function, write first
     `load(distrib)'.

Function: kurtosis_lognormal (<m>,<s>)

     Returns the kurtosis coefficient of a Lognormal(m,s) random
     variable, with s>0. To make use of this function, write first
     `load(distrib)'.

Function: random_lognormal (<m>,<s>)
Function: random_lognormal (<m>,<s>,<n>)

     Returns a Lognormal(m,s) random variate, with s>0. Calling
     `random_lognormal' with a third argument <n>, a random sample of
     size <n> will be simulated.

     Log-normal variates are simulated by means of random normal
     variates. See `random_normal' for details.

     To make use of this function, write first `load(distrib)'.

Function: pdf_gamma (<x>,<a>,<b>)

     Returns the value at <x> of the density function of a Gamma(a,b)
     random variable, with a,b>0. To make use of this function, write
     first `load(distrib)'.

Function: cdf_gamma (<x>,<a>,<b>)

     Returns the value at <x> of the distribution function of a
     Gamma(a,b) random variable, with a,b>0.

          (%i1) load (distrib)$
          (%i2) cdf_gamma(3,5,21);
                                                        1
          (%o2)     1 - gamma_incomplete_regularized(5, -)
                                                        7
          (%i3) float(%);
          (%o3)              4.402663157376807E-7

Function: quantile_gamma (<q>,<a>,<b>)

     Returns the <q>-quantile of a Gamma(a,b) random variable, with
     a,b>0; in other words, this is the inverse of `cdf_gamma'.
     Argument <q> must be an element of [0,1]. To make use of this
     function, write first `load(distrib)'.

Function: mean_gamma (<a>,<b>)

     Returns the mean of a Gamma(a,b) random variable, with a,b>0. To
     make use of this function, write first `load(distrib)'.

Function: var_gamma (<a>,<b>)

     Returns the variance of a Gamma(a,b) random variable, with a,b>0.
     To make use of this function, write first `load(distrib)'.

Function: std_gamma (<a>,<b>)

     Returns the standard deviation of a Gamma(a,b) random variable,
     with a,b>0. To make use of this function, write first
     `load(distrib)'.

Function: skewness_gamma (<a>,<b>)

     Returns the skewness coefficient of a Gamma(a,b) random variable,
     with a,b>0. To make use of this function, write first
     `load(distrib)'.

Function: kurtosis_gamma (<a>,<b>)

     Returns the kurtosis coefficient of a Gamma(a,b) random variable,
     with a,b>0. To make use of this function, write first
     `load(distrib)'.

Function: random_gamma (<a>,<b>)
Function: random_gamma (<a>,<b>,<n>)

     Returns a Gamma(a,b) random variate, with a,b>0. Calling
     `random_gamma' with a third argument <n>, a random sample of size
     <n> will be simulated.

     The implemented algorithm is a combinantion of two procedures,
     depending on the value of parameter <a>:

     For a>=1, Cheng, R.C.H. and Feast, G.M. (1979). <Some simple gamma
     variate generators>. Appl. Stat., 28, 3, 290-295.

     For 0<a<1, Ahrens, J.H. and Dieter, U. (1974). <Computer methods
     for sampling from gamma, beta, poisson and binomial
     cdf_tributions>. Computing, 12, 223-246.

     To make use of this function, write first `load(distrib)'.

Function: pdf_beta (<x>,<a>,<b>)

     Returns the value at <x> of the density function of a Beta(a,b)
     random variable, with a,b>0. To make use of this function, write
     first `load(distrib)'.

Function: cdf_beta (<x>,<a>,<b>)

     Returns the value at <x> of the distribution function of a
     Beta(a,b) random variable, with a,b>0.

          (%i1) load (distrib)$
          (%i2) cdf_beta(1/3,15,2);
                                       11
          (%o2)                     --------
                                    14348907
          (%i3) float(%);
          (%o3)              7.666089131388195E-7

Function: quantile_beta (<q>,<a>,<b>)

     Returns the <q>-quantile of a Beta(a,b) random variable, with
     a,b>0; in other words, this is the inverse of `cdf_beta'. Argument
     <q> must be an element of [0,1]. To make use of this function,
     write first `load(distrib)'.

Function: mean_beta (<a>,<b>)

     Returns the mean of a Beta(a,b) random variable, with a,b>0. To
     make use of this function, write first `load(distrib)'.

Function: var_beta (<a>,<b>)

     Returns the variance of a Beta(a,b) random variable, with a,b>0.
     To make use of this function, write first `load(distrib)'.

Function: std_beta (<a>,<b>)

     Returns the standard deviation of a Beta(a,b) random variable,
     with a,b>0. To make use of this function, write first
     `load(distrib)'.

Function: skewness_beta (<a>,<b>)

     Returns the skewness coefficient of a Beta(a,b) random variable,
     with a,b>0. To make use of this function, write first
     `load(distrib)'.

Function: kurtosis_beta (<a>,<b>)

     Returns the kurtosis coefficient of a Beta(a,b) random variable,
     with a,b>0. To make use of this function, write first
     `load(distrib)'.

Function: random_beta (<a>,<b>)
Function: random_beta (<a>,<b>,<n>)

     Returns a Beta(a,b) random variate, with a,b>0. Calling
     `random_beta' with a third argument <n>, a random sample of size
     <n> will be simulated.

     The implemented algorithm is defined in Cheng, R.C.H.  (1978).
     <Generating Beta Variates with Nonintegral Shape Parameters>.
     Communications of the ACM, 21:317-322

     To make use of this function, write first `load(distrib)'.

Function: pdf_continuous_uniform (<x>,<a>,<b>)

     Returns the value at <x> of the density function of a Continuous
     Uniform(a,b) random variable, with a<b. To make use of this
     function, write first `load(distrib)'.

Function: cdf_continuous_uniform (<x>,<a>,<b>)

     Returns the value at <x> of the distribution function of a
     Continuous Uniform(a,b) random variable, with a<b. To make use of
     this function, write first `load(distrib)'.

Function: quantile_continuous_uniform (<q>,<a>,<b>)

     Returns the <q>-quantile of a Continuous Uniform(a,b) random
     variable, with a<b; in other words, this is the inverse of
     `cdf_continuous_uniform'. Argument <q> must be an element of
     [0,1]. To make use of this function, write first `load(distrib)'.

Function: mean_continuous_uniform (<a>,<b>)

     Returns the mean of a Continuous Uniform(a,b) random variable,
     with a<b. To make use of this function, write first
     `load(distrib)'.

Function: var_continuous_uniform (<a>,<b>)

     Returns the variance of a Continuous Uniform(a,b) random variable,
     with a<b. To make use of this function, write first
     `load(distrib)'.

Function: std_continuous_uniform (<a>,<b>)

     Returns the standard deviation of a Continuous Uniform(a,b) random
     variable, with a<b. To make use of this function, write first
     `load(distrib)'.

Function: skewness_continuous_uniform (<a>,<b>)

     Returns the skewness coefficient of a Continuous Uniform(a,b)
     random variable, with a<b. To make use of this function, write
     first `load(distrib)'.

Function: kurtosis_continuous_uniform (<a>,<b>)

     Returns the kurtosis coefficient of a Continuous Uniform(a,b)
     random variable, with a<b. To make use of this function, write
     first `load(distrib)'.

Function: random_continuous_uniform (<a>,<b>)
Function: random_continuous_uniform (<a>,<b>,<n>)

     Returns a Continuous Uniform(a,b) random variate, with a<b.
     Calling `random_continuous_uniform' with a third argument <n>, a
     random sample of size <n> will be simulated.

     This is a direct application of the `random' built-in Maxima
     function.

     See also `random'. To make use of this function, write first
     `load(distrib)'.

Function: pdf_logistic (<x>,<a>,<b>)

     Returns the value at <x> of the density function of a
     Logistic(a,b) random variable , with b>0. To make use of this
     function, write first `load(distrib)'.

Function: cdf_logistic (<x>,<a>,<b>)

     Returns the value at <x> of the distribution function of a
     Logistic(a,b) random variable , with b>0. To make use of this
     function, write first `load(distrib)'.

Function: quantile_logistic (<q>,<a>,<b>)

     Returns the <q>-quantile of a Logistic(a,b) random variable , with
     b>0; in other words, this is the inverse of `cdf_logistic'.
     Argument <q> must be an element of [0,1]. To make use of this
     function, write first `load(distrib)'.

Function: mean_logistic (<a>,<b>)

     Returns the mean of a Logistic(a,b) random variable , with b>0. To
     make use of this function, write first `load(distrib)'.

Function: var_logistic (<a>,<b>)

     Returns the variance of a Logistic(a,b) random variable , with
     b>0. To make use of this function, write first `load(distrib)'.

Function: std_logistic (<a>,<b>)

     Returns the standard deviation of a Logistic(a,b) random variable
     , with b>0. To make use of this function, write first
     `load(distrib)'.

Function: skewness_logistic (<a>,<b>)

     Returns the skewness coefficient of a Logistic(a,b) random
     variable , with b>0. To make use of this function, write first
     `load(distrib)'.

Function: kurtosis_logistic (<a>,<b>)

     Returns the kurtosis coefficient of a Logistic(a,b) random
     variable , with b>0. To make use of this function, write first
     `load(distrib)'.

Function: random_logistic (<a>,<b>)
Function: random_logistic (<a>,<b>,<n>)

     Returns a Logistic(a,b) random variate, with b>0. Calling
     `random_logistic' with a third argument <n>, a random sample of
     size <n> will be simulated.

     The implemented algorithm is based on the general inverse method.

     To make use of this function, write first `load(distrib)'.

Function: pdf_pareto (<x>,<a>,<b>)

     Returns the value at <x> of the density function of a Pareto(a,b)
     random variable, with a,b>0. To make use of this function, write
     first `load(distrib)'.

Function: cdf_pareto (<x>,<a>,<b>)

     Returns the value at <x> of the distribution function of a
     Pareto(a,b) random variable, with a,b>0. To make use of this
     function, write first `load(distrib)'.

Function: quantile_pareto (<q>,<a>,<b>)

     Returns the <q>-quantile of a Pareto(a,b) random variable, with
     a,b>0; in other words, this is the inverse of `cdf_pareto'.
     Argument <q> must be an element of [0,1]. To make use of this
     function, write first `load(distrib)'.

Function: mean_pareto (<a>,<b>)

     Returns the mean of a Pareto(a,b) random variable, with a>1,b>0.
     To make use of this function, write first `load(distrib)'.

Function: var_pareto (<a>,<b>)

     Returns the variance of a Pareto(a,b) random variable, with
     a>2,b>0. To make use of this function, write first `load(distrib)'.

Function: std_pareto (<a>,<b>)

     Returns the standard deviation of a Pareto(a,b) random variable,
     with a>2,b>0. To make use of this function, write first
     `load(distrib)'.

Function: skewness_pareto (<a>,<b>)

     Returns the skewness coefficient of a Pareto(a,b) random variable,
     with a>3,b>0. To make use of this function, write first
     `load(distrib)'.

Function: kurtosis_pareto (<a>,<b>)

     Returns the kurtosis coefficient of a Pareto(a,b) random variable,
     with a>4,b>0. To make use of this function, write first
     `load(distrib)'.

Function: random_pareto (<a>,<b>)
Function: random_pareto (<a>,<b>,<n>)

     Returns a Pareto(a,b) random variate, with a>0,b>0. Calling
     `random_pareto' with a third argument <n>, a random sample of size
     <n> will be simulated.

     The implemented algorithm is based on the general inverse method.

     To make use of this function, write first `load(distrib)'.

Function: pdf_weibull (<x>,<a>,<b>)

     Returns the value at <x> of the density function of a Weibull(a,b)
     random variable, with a,b>0. To make use of this function, write
     first `load(distrib)'.

Function: cdf_weibull (<x>,<a>,<b>)

     Returns the value at <x> of the distribution function of a
     Weibull(a,b) random variable, with a,b>0. To make use of this
     function, write first `load(distrib)'.

Function: quantile_weibull (<q>,<a>,<b>)

     Returns the <q>-quantile of a Weibull(a,b) random variable, with
     a,b>0; in other words, this is the inverse of `cdf_weibull'.
     Argument <q> must be an element of [0,1]. To make use of this
     function, write first `load(distrib)'.

Function: mean_weibull (<a>,<b>)

     Returns the mean of a Weibull(a,b) random variable, with a,b>0. To
     make use of this function, write first `load(distrib)'.

Function: var_weibull (<a>,<b>)

     Returns the variance of a Weibull(a,b) random variable, with
     a,b>0. To make use of this function, write first `load(distrib)'.

Function: std_weibull (<a>,<b>)

     Returns the standard deviation of a Weibull(a,b) random variable,
     with a,b>0. To make use of this function, write first
     `load(distrib)'.

Function: skewness_weibull (<a>,<b>)

     Returns the skewness coefficient of a Weibull(a,b) random
     variable, with a,b>0. To make use of this function, write first
     `load(distrib)'.

Function: kurtosis_weibull (<a>,<b>)

     Returns the kurtosis coefficient of a Weibull(a,b) random
     variable, with a,b>0. To make use of this function, write first
     `load(distrib)'.

Function: random_weibull (<a>,<b>)
Function: random_weibull (<a>,<b>,<n>)

     Returns a Weibull(a,b) random variate, with a,b>0. Calling
     `random_weibull' with a third argument <n>, a random sample of
     size <n> will be simulated.

     The implemented algorithm is based on the general inverse method.

     To make use of this function, write first `load(distrib)'.

Function: pdf_rayleigh (<x>,<b>)

     Returns the value at <x> of the density function of a Rayleigh(b)
     random variable, with b>0.

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull density is
     returned.

          (%i1) load (distrib)$
          (%i2) pdf_rayleigh(x,b);
                                                  1
          (%o2)                 pdf_weibull(x, 2, -)
                                                  b
          (%i3) assume(x>0,b>0)$ pdf_rayleigh(x,b);
                                              2  2
                                     2     - b  x
          (%o4)                   2 b  x %e

Function: cdf_rayleigh (<x>,<b>)

     Returns the value at <x> of the distribution function of a
     Rayleigh(b) random variable, with b>0.

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull distribution
     is returned.

          (%i1) load (distrib)$
          (%i2) cdf_rayleigh(x,b);
                                                  1
          (%o2)                 cdf_weibull(x, 2, -)
                                                  b
          (%i3) assume(x>0,b>0)$ cdf_rayleigh(x,b);
                                             2  2
                                          - b  x
          (%o4)                     1 - %e

Function: quantile_rayleigh (<q>,<b>)

     Returns the <q>-quantile of a Rayleigh(b) random variable, with
     b>0; in other words, this is the inverse of `cdf_rayleigh'.
     Argument <q> must be an element of [0,1].

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull quantile is
     returned.

          (%i1) load (distrib)$
          (%i2) quantile_rayleigh(0.99,b);
                                                      1
          (%o2)             quantile_weibull(0.99, 2, -)
                                                      b
          (%i3) assume(x>0,b>0)$ quantile_rayleigh(0.99,b);
                                  2.145966026289347
          (%o4)                   -----------------
                                          b

Function: mean_rayleigh (<b>)

     Returns the mean of a Rayleigh(b) random variable, with b>0.

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull mean is
     returned.

          (%i1) load (distrib)$
          (%i2) mean_rayleigh(b);
                                                 1
          (%o2)                  mean_weibull(2, -)
                                                 b
          (%i3) assume(b>0)$ mean_rayleigh(b);
                                      sqrt(%pi)
          (%o4)                       ---------
                                         2 b

Function: var_rayleigh (<b>)

     Returns the variance of a Rayleigh(b) random variable, with b>0.

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull variance is
     returned.

          (%i1) load (distrib)$
          (%i2) var_rayleigh(b);
                                                 1
          (%o2)                   var_weibull(2, -)
                                                 b
          (%i3) assume(b>0)$ var_rayleigh(b);
                                           %pi
                                       1 - ---
                                            4
          (%o4)                        -------
                                          2
                                         b

Function: std_rayleigh (<b>)

     Returns the standard deviation of a Rayleigh(b) random variable,
     with b>0.

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull standard
     deviation is returned.

          (%i1) load (distrib)$
          (%i2) std_rayleigh(b);
                                                 1
          (%o2)                   std_weibull(2, -)
                                                 b
          (%i3) assume(b>0)$ std_rayleigh(b);
                                             %pi
                                    sqrt(1 - ---)
                                              4
          (%o4)                     -------------
                                          b

Function: skewness_rayleigh (<b>)

     Returns the skewness coefficient of a Rayleigh(b) random variable,
     with b>0.

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull skewness
     coefficient is returned.

          (%i1) load (distrib)$
          (%i2) skewness_rayleigh(b);
                                                   1
          (%o2)                skewness_weibull(2, -)
                                                   b
          (%i3) assume(b>0)$ skewness_rayleigh(b);
                                   3/2
                                %pi      3 sqrt(%pi)
                                ------ - -----------
                                  4           4
          (%o4)                 --------------------
                                         %pi 3/2
                                    (1 - ---)
                                          4

Function: kurtosis_rayleigh (<b>)

     Returns the kurtosis coefficient of a Rayleigh(b) random variable,
     with b>0.

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b), therefore when Maxima has not enough information
     to get the result, a noun form based on the Weibull kurtosis
     coefficient is returned.

          (%i1) load (distrib)$
          (%i2) kurtosis_rayleigh(b);
                                                   1
          (%o2)                kurtosis_weibull(2, -)
                                                   b
          (%i3) assume(b>0)$ kurtosis_rayleigh(b);
                                            2
                                       3 %pi
                                   2 - ------
                                         16
          (%o4)                    ---------- - 3
                                        %pi 2
                                   (1 - ---)
                                         4

Function: random_rayleigh (<b>)
Function: random_rayleigh (<b>,<n>)

     Returns a Rayleigh(b) random variate, with b>0. Calling
     `random_rayleigh' with a second argument <n>, a random sample of
     size <n> will be simulated.

     The implemented algorithm is based on the general inverse method.

     To make use of this function, write first `load(distrib)'.

Function: pdf_laplace (<x>,<a>,<b>)

     Returns the value at <x> of the density function of a Laplace(a,b)
     random variable, with b>0. To make use of this function, write
     first `load(distrib)'.

Function: cdf_laplace (<x>,<a>,<b>)

     Returns the value at <x> of the distribution function of a
     Laplace(a,b) random variable, with b>0. To make use of this
     function, write first `load(distrib)'.

Function: quantile_laplace (<q>,<a>,<b>)

     Returns the <q>-quantile of a Laplace(a,b) random variable, with
     b>0; in other words, this is the inverse of `cdf_laplace'.
     Argument <q> must be an element of [0,1]. To make use of this
     function, write first `load(distrib)'.

Function: mean_laplace (<a>,<b>)

     Returns the mean of a Laplace(a,b) random variable, with b>0. To
     make use of this function, write first `load(distrib)'.

Function: var_laplace (<a>,<b>)

     Returns the variance of a Laplace(a,b) random variable, with b>0.
     To make use of this function, write first `load(distrib)'.

Function: std_laplace (<a>,<b>)

     Returns the standard deviation of a Laplace(a,b) random variable,
     with b>0. To make use of this function, write first
     `load(distrib)'.

Function: skewness_laplace (<a>,<b>)

     Returns the skewness coefficient of a Laplace(a,b) random
     variable, with b>0. To make use of this function, write first
     `load(distrib)'.

Function: kurtosis_laplace (<a>,<b>)

     Returns the kurtosis coefficient of a Laplace(a,b) random
     variable, with b>0. To make use of this function, write first
     `load(distrib)'.

Function: random_laplace (<a>,<b>)
Function: random_laplace (<a>,<b>,<n>)

     Returns a Laplace(a,b) random variate, with b>0. Calling
     `random_laplace' with a third argument <n>, a random sample of
     size <n> will be simulated.

     The implemented algorithm is based on the general inverse method.

     To make use of this function, write first `load(distrib)'.

Function: pdf_cauchy (<x>,<a>,<b>)

     Returns the value at <x> of the density function of a Cauchy(a,b)
     random variable, with b>0. To make use of this function, write
     first `load(distrib)'.

Function: cdf_cauchy (<x>,<a>,<b>)

     Returns the value at <x> of the distribution function of a
     Cauchy(a,b) random variable, with b>0. To make use of this
     function, write first `load(distrib)'.

Function: quantile_cauchy (<q>,<a>,<b>)

     Returns the <q>-quantile of a Cauchy(a,b) random variable, with
     b>0; in other words, this is the inverse of `cdf_cauchy'. Argument
     <q> must be an element of [0,1]. To make use of this function,
     write first `load(distrib)'.

Function: random_cauchy (<a>,<b>)
Function: random_cauchy (<a>,<b>,<n>)

     Returns a Cauchy(a,b) random variate, with b>0. Calling
     `random_cauchy' with a third argument <n>, a random sample of size
     <n> will be simulated.

     The implemented algorithm is based on the general inverse method.

     To make use of this function, write first `load(distrib)'.

Function: pdf_gumbel (<x>,<a>,<b>)

     Returns the value at <x> of the density function of a Gumbel(a,b)
     random variable, with b>0. To make use of this function, write
     first `load(distrib)'.

Function: cdf_gumbel (<x>,<a>,<b>)

     Returns the value at <x> of the distribution function of a
     Gumbel(a,b) random variable, with b>0. To make use of this
     function, write first `load(distrib)'.

Function: quantile_gumbel (<q>,<a>,<b>)

     Returns the <q>-quantile of a Gumbel(a,b) random variable, with
     b>0; in other words, this is the inverse of `cdf_gumbel'. Argument
     <q> must be an element of [0,1]. To make use of this function,
     write first `load(distrib)'.

Function: mean_gumbel (<a>,<b>)

     Returns the mean of a Gumbel(a,b) random variable, with b>0.

          (%i1) load (distrib)$
          (%i2) assume(b>0)$  mean_gumbel(a,b);
          (%o3)                     %gamma b + a
     where symbol `%gamma' stands for the Euler-Mascheroni constant.
     See also `%gamma'.

Function: var_gumbel (<a>,<b>)

     Returns the variance of a Gumbel(a,b) random variable, with b>0.
     To make use of this function, write first `load(distrib)'.

Function: std_gumbel (<a>,<b>)

     Returns the standard deviation of a Gumbel(a,b) random variable,
     with b>0. To make use of this function, write first
     `load(distrib)'.

Function: skewness_gumbel (<a>,<b>)

     Returns the skewness coefficient of a Gumbel(a,b) random variable,
     with b>0.

          (%i1) load (distrib)$
          (%i2) assume(b>0)$ skewness_gumbel(a,b);
                                 12 sqrt(6) zeta(3)
          (%o3)                  ------------------
                                           3
                                        %pi
          (%i4) numer:true$ skewness_gumbel(a,b);
          (%o5)                   1.139547099404649
     where `zeta' stands for the Riemann's zeta function.

Function: kurtosis_gumbel (<a>,<b>)

     Returns the kurtosis coefficient of a Gumbel(a,b) random variable,
     with b>0. To make use of this function, write first
     `load(distrib)'.

Function: random_gumbel (<a>,<b>)
Function: random_gumbel (<a>,<b>,<n>)

     Returns a Gumbel(a,b) random variate, with b>0. Calling
     `random_gumbel' with a third argument <n>, a random sample of size
     <n> will be simulated.

     The implemented algorithm is based on the general inverse method.

     To make use of this function, write first `load(distrib)'.

47.3 Functions and Variables for discrete distributions

Function: pdf_general_finite_discrete (<x>,<v>)

     Returns the value at <x> of the probability function of a general
     finite discrete random variable, with vector probabilities v, such
     that `Pr(X=i) = v_i'. Vector v can be a list of nonnegative
     expressions, whose components will be normalized to get a vector
     of probabilities. To make use of this function, write first
     `load(distrib)'.

          (%i1) load (distrib)$
          (%i2) pdf_general_finite_discrete(2, [1/7, 4/7, 2/7]);
                                          4
          (%o2)                           -
                                          7
          (%i3) pdf_general_finite_discrete(2, [1, 4, 2]);
                                          4
          (%o3)                           -
                                          7

Function: cdf_general_finite_discrete (<x>,<v>)

     Returns the value at <x> of the distribution function of a general
     finite discrete random variable, with vector probabilities v.

     See `pdf_general_finite_discrete' for more details.

          (%i1) load (distrib)$
          (%i2) cdf_general_finite_discrete(2, [1/7, 4/7, 2/7]);
                                          5
          (%o2)                           -
                                          7
          (%i3) cdf_general_finite_discrete(2, [1, 4, 2]);
                                          5
          (%o3)                           -
                                          7
          (%i4) cdf_general_finite_discrete(2+1/2, [1, 4, 2]);
                                          5
          (%o4)                           -
                                          7

Function: quantile_general_finite_discrete (<q>,<v>)

     Returns the <q>-quantile of a general finite discrete random
     variable, with vector probabilities v.

     See `pdf_general_finite_discrete' for more details.

Function: mean_general_finite_discrete (<v>)

     Returns the mean of a general finite discrete random variable,
     with vector probabilities v.

     See `pdf_general_finite_discrete' for more details.

Function: var_general_finite_discrete (<v>)

     Returns the variance of a general finite discrete random variable,
     with vector probabilities v.

     See `pdf_general_finite_discrete' for more details.

Function: std_general_finite_discrete (<v>)

     Returns the standard deviation of a general finite discrete random
     variable, with vector probabilities v.

     See `pdf_general_finite_discrete' for more details.

Function: skewness_general_finite_discrete (<v>)

     Returns the skewness coefficient of a general finite discrete
     random variable, with vector probabilities v.

     See `pdf_general_finite_discrete' for more details.

Function: kurtosis_general_finite_discrete (<v>)

     Returns the kurtosis coefficient of a general finite discrete
     random variable, with vector probabilities v.

     See `pdf_general_finite_discrete' for more details.

Function: random_general_finite_discrete (<v>)
Function: random_general_finite_discrete (<v>,<m>)

     Returns a general finite discrete random variate, with vector
     probabilities v. Calling `random_general_finite_discrete' with a
     second argument <m>, a random sample of size <m> will be simulated.

     See `pdf_general_finite_discrete' for more details.

          (%i1) load (distrib)$
          (%i2) random_general_finite_discrete([1,3,1,5]);
          (%o2)                          4
          (%i3) random_general_finite_discrete([1,3,1,5], 10);
          (%o3)           [4, 2, 2, 3, 2, 4, 4, 1, 2, 2]

Function: pdf_binomial (<x>,<n>,<p>)

     Returns the value at <x> of the probability function of a
     Binomial(n,p) random variable, with 0<p<1 and n a positive
     integer. To make use of this function, write first `load(distrib)'.
                                      4 (%o6)
      -                                   7

Function: cdf_binomial (<x>,<n>,<p>)

     Returns the value at <x> of the distribution function of a
     Binomial(n,p) random variable, with 0<p<1 and n a positive integer.

          (%i1) load (distrib)$
          (%i2) cdf_binomial(5,7,1/6);
                                      7775
          (%o2)                       ----
                                      7776
          (%i3) float(%);
          (%o3)               .9998713991769548

Function: quantile_binomial (<q>,<n>,<p>)

     Returns the <q>-quantile of a Binomial(n,p) random variable, with
     0<p<1 and n a positive integer; in other words, this is the
     inverse of `cdf_binomial'. Argument <q> must be an element of
     [0,1]. To make use of this function, write first `load(distrib)'.

Function: mean_binomial (<n>,<p>)

     Returns the mean of a Binomial(n,p) random variable, with 0<p<1
     and n a positive integer. To make use of this function, write
     first `load(distrib)'.

Function: var_binomial (<n>,<p>)

     Returns the variance of a Binomial(n,p) random variable, with
     0<p<1 and n a positive integer. To make use of this function,
     write first `load(distrib)'.

Function: std_binomial (<n>,<p>)

     Returns the standard deviation of a Binomial(n,p) random variable,
     with 0<p<1 and n a positive integer. To make use of this function,
     write first `load(distrib)'.

Function: skewness_binomial (<n>,<p>)

     Returns the skewness coefficient of a Binomial(n,p) random
     variable, with 0<p<1 and n a positive integer. To make use of this
     function, write first `load(distrib)'.

Function: kurtosis_binomial (<n>,<p>)

     Returns the kurtosis coefficient of a Binomial(n,p) random
     variable, with 0<p<1 and n a positive integer. To make use of this
     function, write first `load(distrib)'.

Function: random_binomial (<n>,<p>)
Function: random_binomial (<n>,<p>,<m>)

     Returns a Binomial(n,p) random variate, with 0<p<1 and n a
     positive integer. Calling `random_binomial' with a third argument
     <m>, a random sample of size <m> will be simulated.

     The implemented algorithm is based on the one described in
     Kachitvichyanukul, V. and Schmeiser, B.W. (1988) <Binomial Random
     Variate Generation>. Communications of the ACM, 31, Feb., 216.

     To make use of this function, write first `load(distrib)'.

Function: pdf_poisson (<x>,<m>)

     Returns the value at <x> of the probability function of a
     Poisson(m) random variable, with m>0. To make use of this
     function, write first `load(distrib)'.

Function: cdf_poisson (<x>,<m>)

     Returns the value at <x> of the distribution function of a
     Poisson(m) random variable, with m>0.

          (%i1) load (distrib)$
          (%i2) cdf_poisson(3,5);
          (%o2)       gamma_incomplete_regularized(4, 5)
          (%i3) float(%);
          (%o3)               .2650259152973623

Function: quantile_poisson (<q>,<m>)

     Returns the <q>-quantile of a Poisson(m) random variable, with
     m>0; in other words, this is the inverse of `cdf_poisson'.
     Argument <q> must be an element of [0,1]. To make use of this
     function, write first `load(distrib)'.

Function: mean_poisson (<m>)

     Returns the mean of a Poisson(m) random variable, with  m>0. To
     make use of this function, write first `load(distrib)'.

Function: var_poisson (<m>)

     Returns the variance of a Poisson(m) random variable, with  m>0.
     To make use of this function, write first `load(distrib)'.

Function: std_poisson (<m>)

     Returns the standard deviation of a Poisson(m) random variable,
     with m>0. To make use of this function, write first
     `load(distrib)'.

Function: skewness_poisson (<m>)

     Returns the skewness coefficient of a Poisson(m) random variable,
     with m>0. To make use of this function, write first
     `load(distrib)'.

Function: kurtosis_poisson (<m>)

     Returns the kurtosis coefficient of a Poisson random variable
     Poi(m), with m>0. To make use of this function, write first
     `load(distrib)'.

Function: random_poisson (<m>)
Function: random_poisson (<m>,<n>)

     Returns a Poisson(m) random variate, with m>0. Calling
     `random_poisson' with a second argument <n>, a random sample of
     size <n> will be simulated.

     The implemented algorithm is the one described in Ahrens, J.H. and
     Dieter, U. (1982) <Computer Generation of Poisson Deviates From
     Modified Normal Distributions>. ACM Trans. Math. Software, 8, 2,
     June,163-179.

     To make use of this function, write first `load(distrib)'.

Function: pdf_bernoulli (<x>,<p>)

     Returns the value at <x> of the probability function of a
     Bernoulli(p) random variable, with 0<p<1.

     The Bernoulli(p) random variable is equivalent to the
     Binomial(1,p), therefore when Maxima has not enough information to
     get the result, a noun form based on the binomial probability
     function is returned.

          (%i1) load (distrib)$
          (%i2) pdf_bernoulli(1,p);
          (%o2)                 pdf_binomial(1, 1, p)
          (%i3) assume(0<p,p<1)$ pdf_bernoulli(1,p);
          (%o4)                           p

Function: cdf_bernoulli (<x>,<p>)

     Returns the value at <x> of the distribution function of a
     Bernoulli(p) random variable, with 0<p<1. To make use of this
     function, write first `load(distrib)'.

Function: quantile_bernoulli (<q>,<p>)

     Returns the <q>-quantile of a Bernoulli(p) random variable, with
     0<p<1; in other words, this is the inverse of `cdf_bernoulli'.
     Argument <q> must be an element of [0,1]. To make use of this
     function, write first `load(distrib)'.

Function: mean_bernoulli (<p>)

     Returns the mean of a Bernoulli(p) random variable, with 0<p<1.

     The Bernoulli(p) random variable is equivalent to the
     Binomial(1,p), therefore when Maxima has not enough information to
     get the result, a noun form based on the binomial mean is returned.

          (%i1) load (distrib)$
          (%i2) mean_bernoulli(p);
          (%o2)                  mean_binomial(1, p)
          (%i3) assume(0<p,p<1)$ mean_bernoulli(p);
          (%o4)                           p

Function: var_bernoulli (<p>)

     Returns the variance of a Bernoulli(p) random variable, with 0<p<1.

     The Bernoulli(p) random variable is equivalent to the
     Binomial(1,p), therefore when Maxima has not enough information to
     get the result, a noun form based on the binomial variance is
     returned.

          (%i1) load (distrib)$
          (%i2) var_bernoulli(p);
          (%o2)                  var_binomial(1, p)
          (%i3) assume(0<p,p<1)$ var_bernoulli(p);
          (%o4)                       (1 - p) p

Function: std_bernoulli (<p>)

     Returns the standard deviation of a Bernoulli(p) random variable,
     with 0<p<1.

     The Bernoulli(p) random variable is equivalent to the
     Binomial(1,p), therefore when Maxima has not enough information to
     get the result, a noun form based on the binomial standard
     deviation is returned.

          (%i1) load (distrib)$
          (%i2) std_bernoulli(p);
          (%o2)                  std_binomial(1, p)
          (%i3) assume(0<p,p<1)$ std_bernoulli(p);
          (%o4)                  sqrt(1 - p) sqrt(p)

Function: skewness_bernoulli (<p>)

     Returns the skewness coefficient of a Bernoulli(p) random
     variable, with 0<p<1.

     The Bernoulli(p) random variable is equivalent to the
     Binomial(1,p), therefore when Maxima has not enough information to
     get the result, a noun form based on the binomial skewness
     coefficient is returned.

          (%i1) load (distrib)$
          (%i2) skewness_bernoulli(p);
          (%o2)                skewness_binomial(1, p)
          (%i3) assume(0<p,p<1)$ skewness_bernoulli(p);
                                       1 - 2 p
          (%o4)                  -------------------
                                 sqrt(1 - p) sqrt(p)

Function: kurtosis_bernoulli (<p>)

     Returns the kurtosis coefficient of a Bernoulli(p) random
     variable, with 0<p<1.

     The Bernoulli(p) random variable is equivalent to the
     Binomial(1,p), therefore when Maxima has not enough information to
     get the result, a noun form based on the binomial kurtosis
     coefficient is returned.

          (%i1) load (distrib)$
          (%i2) kurtosis_bernoulli(p);
          (%o2)                kurtosis_binomial(1, p)
          (%i3) assume(0<p,p<1)$ kurtosis_bernoulli(p);
                                   1 - 6 (1 - p) p
          (%o4)                    ---------------
                                      (1 - p) p

Function: random_bernoulli (<p>)
Function: random_bernoulli (<p>,<n>)

     Returns a Bernoulli(p) random variate, with 0<p<1. Calling
     `random_bernoulli' with a second argument <n>, a random sample of
     size <n> will be simulated.

     This is a direct application of the `random' built-in Maxima
     function.

     See also `random'. To make use of this function, write first
     `load(distrib)'.

Function: pdf_geometric (<x>,<p>)

     Returns the value at <x> of the probability function of a
     Geometric(p) random variable, with 0<p<1. To make use of this
     function, write first `load(distrib)'.

Function: cdf_geometric (<x>,<p>)

     Returns the value at <x> of the distribution function of a
     Geometric(p) random variable, with 0<p<1. To make use of this
     function, write first `load(distrib)'.

Function: quantile_geometric (<q>,<p>)

     Returns the <q>-quantile of a Geometric(p) random variable, with
     0<p<1; in other words, this is the inverse of `cdf_geometric'.
     Argument <q> must be an element of [0,1]. To make use of this
     function, write first `load(distrib)'.

Function: mean_geometric (<p>)

     Returns the mean of a Geometric(p) random variable, with 0<p<1. To
     make use of this function, write first `load(distrib)'.

Function: var_geometric (<p>)

     Returns the variance of a Geometric(p) random variable, with
     0<p<1. To make use of this function, write first `load(distrib)'.

Function: std_geometric (<p>)

     Returns the standard deviation of a Geometric(p) random variable,
     with 0<p<1. To make use of this function, write first
     `load(distrib)'.

Function: skewness_geometric (<p>)

     Returns the skewness coefficient of a Geometric(p) random
     variable, with 0<p<1. To make use of this function, write first
     `load(distrib)'.

Function: kurtosis_geometric (<p>)

     Returns the kurtosis coefficient of a geometric random variable
     Geo(p), with 0<p<1. To make use of this function, write first
     `load(distrib)'.

Function: random_geometric (<p>)
Function: random_geometric (<p>,<n>)

     Returns a Geometric(p) random variate, with 0<p<1. Calling
     `random_geometric' with a second argument <n>, a random sample of
     size <n> will be simulated.

     The algorithm is based on simulation of Bernoulli trials.

     To make use of this function, write first `load(distrib)'.

Function: pdf_discrete_uniform (<x>,<n>)

     Returns the value at <x> of the probability function of a Discrete
     Uniform(n) random variable, with n a strictly positive integer. To
     make use of this function, write first `load(distrib)'.

Function: cdf_discrete_uniform (<x>,<n>)

     Returns the value at <x> of the distribution function of a
     Discrete Uniform(n) random variable, with n a strictly positive
     integer. To make use of this function, write first `load(distrib)'.

Function: quantile_discrete_uniform (<q>,<n>)

     Returns the <q>-quantile of a Discrete Uniform(n) random variable,
     with n a strictly positive integer; in other words, this is the
     inverse of `cdf_discrete_uniform'. Argument <q> must be an element
     of [0,1]. To make use of this function, write first
     `load(distrib)'.

Function: mean_discrete_uniform (<n>)

     Returns the mean of a Discrete Uniform(n) random variable, with n
     a strictly positive integer. To make use of this function, write
     first `load(distrib)'.

Function: var_discrete_uniform (<n>)

     Returns the variance of a Discrete Uniform(n) random variable,
     with n a strictly positive integer. To make use of this function,
     write first `load(distrib)'.

Function: std_discrete_uniform (<n>)

     Returns the standard deviation of a Discrete Uniform(n) random
     variable, with n a strictly positive integer. To make use of this
     function, write first `load(distrib)'.

Function: skewness_discrete_uniform (<n>)

     Returns the skewness coefficient of a Discrete Uniform(n) random
     variable, with n a strictly positive integer. To make use of this
     function, write first `load(distrib)'.

Function: kurtosis_discrete_uniform (<n>)

     Returns the kurtosis coefficient of a Discrete Uniform(n) random
     variable, with n a strictly positive integer. To make use of this
     function, write first `load(distrib)'.

Function: random_discrete_uniform (<n>)
Function: random_discrete_uniform (<n>,<m>)

     Returns a Discrete Uniform(n) random variate, with n a strictly
     positive integer. Calling `random_discrete_uniform' with a second
     argument <m>, a random sample of size <m> will be simulated.

     This is a direct application of the `random' built-in Maxima
     function.

     See also `random'. To make use of this function, write first
     `load(distrib)'.

Function: pdf_hypergeometric (<x>,<n1>,<n2>,<n>)

     Returns the value at <x> of the probability function of a
     Hypergeometric(n1,n2,n) random variable, with <n1>, <n2> and <n>
     non negative integers and n<=n1+n2.  Being <n1> the number of
     objects of class A, <n2> the number of objects of class B, and <n>
     the size of the sample without replacement, this function returns
     the probability of event "exactly <x> objects are of class A".

     To make use of this function, write first `load(distrib)'.

Function: cdf_hypergeometric (<x>,<n1>,<n2>,<n>)

     Returns the value at <x> of the distribution function of a
     Hypergeometric(n1,n2,n) random variable, with <n1>, <n2> and <n>
     non negative integers and n<=n1+n2.  See `pdf_hypergeometric' for
     a more complete description.

     To make use of this function, write first `load(distrib)'.

Function: quantile_hypergeometric (<q>,<n1>,<n2>,<n>)

     Returns the <q>-quantile of a Hypergeometric(n1,n2,n) random
     variable, with <n1>, <n2> and <n> non negative integers and
     n<=n1+n2; in other words, this is the inverse of
     `cdf_hypergeometric'. Argument <q> must be an element of [0,1]. To
     make use of this function, write first `load(distrib)'.

Function: mean_hypergeometric (<n1>,<n2>,<n>)

     Returns the mean of a discrete uniform random variable
     Hyp(n1,n2,n), with <n1>, <n2> and <n> non negative integers and
     n<=n1+n2. To make use of this function, write first
     `load(distrib)'.

Function: var_hypergeometric (<n1>,<n2>,<n>)

     Returns the variance of a hypergeometric  random variable
     Hyp(n1,n2,n), with <n1>, <n2> and <n> non negative integers and
     n<=n1+n2. To make use of this function, write first
     `load(distrib)'.

Function: std_hypergeometric (<n1>,<n2>,<n>)

     Returns the standard deviation of a Hypergeometric(n1,n2,n) random
     variable, with <n1>, <n2> and <n> non negative integers and
     n<=n1+n2. To make use of this function, write first
     `load(distrib)'.

Function: skewness_hypergeometric (<n1>,<n2>,<n>)

     Returns the skewness coefficient of a Hypergeometric(n1,n2,n)
     random variable, with <n1>, <n2> and <n> non negative integers and
     n<=n1+n2. To make use of this function, write first
     `load(distrib)'.

Function: kurtosis_hypergeometric (<n1>,<n2>,<n>)

     Returns the kurtosis coefficient of a Hypergeometric(n1,n2,n)
     random variable, with <n1>, <n2> and <n> non negative integers and
     n<=n1+n2. To make use of this function, write first
     `load(distrib)'.

Function: random_hypergeometric (<n1>,<n2>,<n>)
Function: random_hypergeometric (<n1>,<n2>,<n>,<m>)

     Returns a Hypergeometric(n1,n2,n) random variate, with <n1>, <n2>
     and <n> non negative integers and n<=n1+n2. Calling
     `random_hypergeometric' with a fourth argument <m>, a random
     sample of size <m> will be simulated.

     Algorithm described in Kachitvichyanukul, V., Schmeiser, B.W.
     (1985) <Computer generation of hypergeometric random variates.>
     Journal of Statistical Computation and Simulation 22, 127-145.

     To make use of this function, write first `load(distrib)'.

Function: pdf_negative_binomial (<x>,<n>,<p>)

     Returns the value at <x> of the probability function of a Negative
     Binomial(n,p) random variable, with 0<p<1 and n a positive
     integer. To make use of this function, write first `load(distrib)'.

Function: cdf_negative_binomial (<x>,<n>,<p>)

     Returns the value at <x> of the distribution function of a
     Negative Binomial(n,p) random variable, with 0<p<1 and n a
     positive integer.

          (%i1) load (distrib)$
          (%i2) cdf_negative_binomial(3,4,1/8);
                                      3271
          (%o2)                      ------
                                     524288
          (%i3) float(%);
          (%o3)              .006238937377929687

Function: quantile_negative_binomial (<q>,<n>,<p>)

     Returns the <q>-quantile of a Negative Binomial(n,p) random
     variable, with 0<p<1 and n a positive integer; in other words,
     this is the inverse of `cdf_negative_binomial'. Argument <q> must
     be an element of [0,1]. To make use of this function, write first
     `load(distrib)'.

Function: mean_negative_binomial (<n>,<p>)

     Returns the mean of a Negative Binomial(n,p) random variable, with
     0<p<1 and n a positive integer. To make use of this function,
     write first `load(distrib)'.

Function: var_negative_binomial (<n>,<p>)

     Returns the variance of a Negative Binomial(n,p) random variable,
     with 0<p<1 and n a positive integer. To make use of this function,
     write first `load(distrib)'.

Function: std_negative_binomial (<n>,<p>)

     Returns the standard deviation of a Negative Binomial(n,p) random
     variable, with 0<p<1 and n a positive integer. To make use of this
     function, write first `load(distrib)'.

Function: skewness_negative_binomial (<n>,<p>)

     Returns the skewness coefficient of a Negative Binomial(n,p)
     random variable, with 0<p<1 and n a positive integer. To make use
     of this function, write first `load(distrib)'.

Function: kurtosis_negative_binomial (<n>,<p>)

     Returns the kurtosis coefficient of a Negative Binomial(n,p)
     random variable, with 0<p<1 and n a positive integer. To make use
     of this function, write first `load(distrib)'.

Function: random_negative_binomial (<n>,<p>)
Function: random_negative_binomial (<n>,<p>,<m>)

     Returns a Negative Binomial(n,p) random variate, with 0<p<1 and n
     a positive integer. Calling `random_negative_binomial' with a
     third argument <m>, a random sample of size <m> will be simulated.

     Algorithm described in Devroye, L. (1986) <Non-Uniform Random
     Variate Generation>. Springer Verlag, p. 480.

     To make use of this function, write first `load(distrib)'.

48 draw


Introduction to draw
Functions and Variables for draw
Functions and Variables for pictures
Functions and Variables for worldmap

48.1 Introduction to draw

`draw' is a Maxima-Gnuplot interface.

   There are three main functions to be used at Maxima level: `draw2d',
`draw3d' and `draw'.

   Follow this link for more elaborated examples of this package:

   `http://riotorto.users.sourceforge.net/gnuplot'

   You need Gnuplot 4.2 or newer to run this program.

48.2 Functions and Variables for draw

Scene constructor: gr2d (<graphic option>, ..., <graphic object>, ...)

     Function `gr2d' builds an object describing a 2D scene. Arguments
     are graphic options, graphic objects, or lists containing both
     graphic options and objects.  This scene is interpreted
     sequentially: graphic options affect those graphic objects placed
     on its right. Some graphic options affect the global appearence of
     the scene.

     This is the list of graphic objects available for scenes in two
     dimensions: `bars', `ellipse', `explicit', `image', `implicit',
     `label', `parametric', `points', `polar', `polygon',
     `quadrilateral', `rectangle', `triangle', `vector', and `geomap'
     (this one defined in package `worldmap').

     See also `draw' and `draw2d'.

     To make use of this object, write first `load(draw)'.

Scene constructor: gr3d (<graphic option>, ..., <graphic object>, ...)

     Function `gr3d' builds an object describing a 3d scene. Arguments
     are graphic options, graphic objects, or lists containing both
     graphic options and objects. This scene is interpreted
     sequentially: graphic options affect those graphic objects placed
     on its right. Some graphic options affect the global appearence of
     the scene.

     This is the list of graphic objects available for scenes in three
     dimensions:
     `cylindrical', `elevation_grid', `explicit', `implicit', `label',
     `mesh', `parametric',
     `parametric_surface', `points', `quadrilateral', `spherical',
     `triangle', `tube',
     `vector', and `geomap' (this one defined in package `worldmap').

     See also `draw' and `draw3d'.

     To make use of this object, write first `load(draw)'.

Function: draw (<gr2d>, ..., <gr3d>, ..., <options>, ...)

     Plots a series of scenes; its arguments are `gr2d' and/or `gr3d'
     objects, together with some options, or lists of scenes and
     options.  By default, the scenes are put together in one column.

     Function `draw' accepts the following global options: `terminal',
     `columns', `dimensions', `file_name' and `delay'.

     Functions `draw2d' and `draw3d' are short cuts to be used when
     only one scene is required, in two or three dimensions,
     respectively.

     See also `gr2d' and `gr3d'.

     To make use of this function, write first `load(draw)'.

     Example:

          (%i1) load(draw)$
          (%i2) scene1: gr2d(title="Ellipse",
                             nticks=30,
                             parametric(2*cos(t),5*sin(t),t,0,2*%pi))$
          (%i3) scene2: gr2d(title="Triangle",
                             polygon([4,5,7],[6,4,2]))$
          (%i4) draw(scene1, scene2, columns = 2)$

     The two draw sentences are equivalent:
          (%i1) load(draw)$
          (%i2) draw(gr3d(explicit(x^2+y^2,x,-1,1,y,-1,1)));
          (%o2)                          [gr3d(explicit)]
          (%i3) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1));
          (%o3)                          [gr3d(explicit)]

     An animated gif file:
          (%i1) load(draw)$
          (%i2) draw(
                  delay     = 100,
                  file_name = "zzz",
                  terminal  = 'animated_gif,
                  gr2d(explicit(x^2,x,-1,1)),
                  gr2d(explicit(x^3,x,-1,1)),
                  gr2d(explicit(x^4,x,-1,1)));
          End of animation sequence
          (%o2)          [gr2d(explicit), gr2d(explicit), gr2d(explicit)]

     See also `gr2d', `gr3d', `draw2d' and `draw3d'..

Function: draw2d (<option>, <graphic_object>, ...)

     This function is a short cut for `draw(gr2d(<options>, ...,
     <graphic_object>, ...))'.

     It can be used to plot a unique scene in 2d.

     To make use of this function, write first `load(draw)'.

     See also `draw' and `gr2d'.

Function: draw3d (<option>, <graphic_object>, ...)

     This function is a short cut for `draw(gr3d(<options>, ...,
     <graphic_object>, ...))'.

     It can be used to plot a unique scene in 3d.

     To make use of this function, write first `load(draw)'.

     See also `draw' and `gr3d'.

Function: draw_file (<graphic option>, ..., <graphic object>, ...)

     Saves the current plot into a file. Accepted graphics options are:
     `terminal', `dimensions', `file_name' and `background_color'.

     Example:

          (%i1) load(draw)$
          (%i2) /* screen plot */
                draw(gr3d(explicit(x^2+y^2,x,-1,1,y,-1,1)))$
          (%i3) /* same plot in eps format */
                draw_file(terminal  = eps,
                          dimensions = [5,5]) $

Function: multiplot_mode (<term>)

     This function enables Maxima to work in one-window multiplot mode
     with terminal <term>; accepted arguments for this function are
     `screen', `wxt', `aquaterm' and `none'.

     When multiplot mode is enabled, each call to `draw' sends a new
     plot to the same window, without erasing the previous ones. To
     disable the multiplot mode, write `multiplot_mode(none)'.

     When multiplot mode is enabled, global option `terminal' is
     blocked and you have to disable this working mode before changing
     to another terminal.

     This feature does not work in Windows platforms.

     Example:

          (%i1) load(draw)$
          (%i2) set_draw_defaults(
                   xrange = [-1,1],
                   yrange = [-1,1],
                   grid   = true,
                   title  = "Step by step plot" )$
          (%i3) multiplot_mode(screen)$
          (%i4) draw2d(color=blue,  explicit(x^2,x,-1,1))$
          (%i5) draw2d(color=red,   explicit(x^3,x,-1,1))$
          (%i6) draw2d(color=brown, explicit(x^4,x,-1,1))$
          (%i7) multiplot_mode(none)$

Function: set_draw_defaults (<graphic option>, ..., <graphic

          object>, ...)
     Sets user graphics options. This function is useful for plotting a
     sequence of graphics with common graphics options. Calling this
     function without arguments removes user defaults.

     Example:

          (%i1) load(draw)$
          (%i2) set_draw_defaults(
                   xrange = [-10,10],
                   yrange = [-2, 2],
                   color  = blue,
                   grid   = true)$
          (%i3) /* plot with user defaults */
                draw2d(explicit(((1+x)**2/(1+x*x))-1,x,-10,10))$
          (%i4) set_draw_defaults()$
          (%i5) /* plot with standard defaults */
                draw2d(explicit(((1+x)**2/(1+x*x))-1,x,-10,10))$

     To make use of this function, write first `load(draw)'.

Graphic option: adapt_depth

     Default value: 10

     `adapt_depth' is the maximum number of splittings used by the
     adaptive plotting routine.

     This option is relevant only for 2d `explicit' functions.

Graphic option: axis_3d

     Default value: `true'

     If `axis_3d' is `true', the <x>, <y> and <z> axis are shown in 3d
     scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(axis_3d = false,
                       explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$

     See also `axis_bottom',  `axis_left', `axis_top', and `axis_right'
     for axis in 2d.

Graphic option: axis_bottom

     Default value: `true'

     If `axis_bottom' is `true', the bottom axis is shown in 2d scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(axis_bottom = false,
                       explicit(x^3,x,-1,1))$

     See also `axis_left',  `axis_top', `axis_right', and `axis_3d'.

Graphic option: axis_left

     Default value: `true'

     If `axis_left' is `true', the left axis is shown in 2d scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(axis_left = false,
                       explicit(x^3,x,-1,1))$

     See also `axis_bottom',  `axis_top', `axis_right', and `axis_3d'.

Graphic option: axis_right

     Default value: `true'

     If `axis_right' is `true', the right axis is shown in 2d scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(axis_right = false,
                       explicit(x^3,x,-1,1))$

     See also `axis_bottom',  `axis_left', `axis_top', and `axis_3d'.

Graphic option: axis_top

     Default value: `true'

     If `axis_top' is `true', the top axis is shown in 2d scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(axis_top = false,
                       explicit(x^3,x,-1,1))$

     See also `axis_bottom',  `axis_left', `axis_right', and `axis_3d'.

Graphic option: background_color

     Default value: `white'

     Sets the background color for terminals `gif', `png', `jpg', and
     `gif'. Default background color is white.

     See also `color'.

Graphic option: border

     Default value: `true'

     If `border' is `true', borders of polygons are painted according
     to `line_type' and `line_width'.

     This option affects the following graphic objects:
        * `gr2d': `polygon', `rectangle', and `ellipse'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(color       = brown,
                       line_width  = 8,
                       polygon([[3,2],[7,2],[5,5]]),
                       border      = false,
                       fill_color  = blue,
                       polygon([[5,2],[9,2],[7,5]]) )$

Graphic option: cbrange

     Default value: `auto'

     If `cbrange' is `auto', the range for the values which are colored
     when `enhanced3d' is not `false' is computed automatically. Values
     outside of the color range use color of the nearest extreme.

     When `enhanced3d' or `colorbox' is `false', option `cbrange' has
     no effect.

     If the user wants a specific interval for the colored values, it
     must be given as a Maxima list, as in `cbrange=[-2, 3]'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d (
                  enhanced3d     = true,
                  color          = green,
                  cbrange = [-3,10],
                  explicit(x^2+y^2, x,-2,2,y,-2,2)) $

     See also `enhanced3d', `colorbox' and `cbtics'.

Graphic option: cbtics

     Default value: `auto'

     This graphic option controls the way tic marks are drawn on the
     colorbox when option `enhanced3d' is not `false'.

     When `enhanced3d' or `colorbox' is `false', option `cbtics' has no
     effect.

     See `xtics' for a complete description.

     Example :

          (%i1) load(draw)$
          (%i2) draw3d (
                  enhanced3d = true,
                  color      = green,
                  cbtics  = {["High",10],["Medium",05],["Low",0]},
                  cbrange = [0, 10],
                  explicit(x^2+y^2, x,-2,2,y,-2,2)) $

     See also `enhanced3d', `colorbox' and `cbrange'.

Graphic option: color

     Default value: `blue'

     `color' specifies the color for plotting lines, points, borders of
     polygons and labels.

     Colors can be given as names or in hexadecimal rgb code.

     Available color names are: white            black            gray0            grey0
     gray10           grey10           gray20           grey20
     gray30           grey30           gray40           grey40
     gray50           grey50           gray60           grey60
     gray70           grey70           gray80           grey80
     gray90           grey90           gray100          grey100
     gray             grey             light_gray       light_grey
     dark_gray        dark_grey        red              light_red
     dark_red         yellow           light_yellow     dark_yellow
     green            light_green      dark_green       spring_green
     forest_green     sea_green        blue             light_blue
     dark_blue        midnight_blue    navy             medium_blue
     royalblue        skyblue          cyan             light_cyan
     dark_cyan        magenta          light_magenta    dark_magenta
     turquoise        light_turquoise  dark_turquoise   pink
     light_pink       dark_pink        coral            light_coral
     orange_red       salmon           light_salmon     dark_salmon
     aquamarine       khaki            dark_khaki       goldenrod
     light_goldenrod  dark_goldenrod   gold             beige
     brown            orange           dark_orange      violet
     dark_violet      plum             purple

     Cromatic componentes in hexadecimal code are introduced in the
     form `"#rrggbb"'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(explicit(x^2,x,_1,1), /* default is black */
                       color = red,
                       explicit(0.5 + x^2,x,-1,1),
                       color = blue,
                       explicit(1 + x^2,x,-1,1),
                       color = light_blue,
                       explicit(1.5 + x^2,x,-1,1),
                       color = "#23ab0f",
                       label(["This is a label",0,1.2])  )$

     See also `fill_color'.

Graphic option: colorbox

     Default value: `true'

     If `colorbox' is `true', a color scale without label is drawn
     together with `image' 2D objects, or coloured 3d objects. If
     `colorbox' is `false', no color scale is shown. If `colorbox' is a
     string, a color scale with label is drawn.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

     Color scale and images.

          (%i1) load(draw)$
          (%i2) im: apply('matrix,
                           makelist(makelist(random(200),i,1,30),i,1,30))$
          (%i3) draw2d(image(im,0,0,30,30))$
          (%i4) draw2d(colorbox = false, image(im,0,0,30,30))$

     Color scale and 3D coloured object.

          (%i1) load(draw)$
          (%i2) draw3d(
                  colorbox   = "Magnitude",
                  enhanced3d = true,
                  explicit(x^2+y^2,x,-1,1,y,-1,1))$

     See also `palette'.

Graphic option: columns

     Default value: 1

     `columns' is the number of columns in multiple plots.

     Since this is a global graphics option, its position in the scene
     description does not matter. It can be also used as an argument of
     function `draw'.

     Example:

          (%i1) load(draw)$
          (%i2) scene1: gr2d(title="Ellipse",
                             nticks=30,
                             parametric(2*cos(t),5*sin(t),t,0,2*%pi))$
          (%i3) scene2: gr2d(title="Triangle",
                             polygon([4,5,7],[6,4,2]))$
          (%i4) draw(scene1, scene2, columns = 2)$

Graphic option: contour

     Default value: `none'

     Option `contour' enables the user to select where to plot contour
     lines.  Possible values are:

        * `none': no contour lines are plotted.

        * `base': contour lines are projected on the xy plane.

        * `surface': contour lines are plotted on the surface.

        * `both': two contour lines are plotted: on the xy plane and on
          the surface.

        * `map': contour lines are projected on the xy plane, and the
          view point is set just in the vertical.

Graphic option: contour_levels

     Default value: 5

     This graphic option controls the way contours are drawn.
     `contour_levels' can be set to a positive integer number, a list
     of three numbers or an arbitrary set of numbers:

        * When option `contour_levels' is bounded to positive integer
          <n>, <n> contour lines will be drawn at equal intervals. By
          default, five equally spaced contours are plotted.

        * When option `contour_levels' is bounded to a list of length
          three of the form `[lowest,s,highest]', contour lines are
          plotted from `lowest' to `highest' in steps of `s'.

        * When option `contour_levels' is bounded to a set of numbers
          of the form `{n1, n2, ...}', contour lines are plotted at
          values `n1', `n2', ...

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Examples:

     Ten equally spaced contour lines. The actual number of levels can
     be adjusted to give simple labels.
          (%i1) load(draw)$
          (%i2) draw3d(color = green,
                       explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
                       contour_levels = 10,
                       contour        = both,
                       surface_hide   = true) $

     From -8 to 8 in steps of 4.
          (%i1) load(draw)$
          (%i2) draw3d(color = green,
                       explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
                       contour_levels = [-8,4,8],
                       contour        = both,
                       surface_hide   = true) $

     Isolines at levels -7, -6, 0.8 and 5.
          (%i1) load(draw)$
          (%i2) draw3d(color = green,
                       explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
                       contour_levels = {-7, -6, 0.8, 5},
                       contour        = both,
                       surface_hide   = true) $

     See also `contour'.

Graphic option: data_file_name

     Default value: `"data.gnuplot"'

     This is the name of the file with the numeric data needed by
     Gnuplot to build the requested plot.

     Since this is a global graphics option, its position in the scene
     description does not matter. It can be also used as an argument of
     function `draw'.

     See example in `gnuplot_file_name'.

Graphic option: delay

     Default value: 5

     This is the delay in 1/100 seconds of frames in animated gif files.

     Since this is a global graphics option, its position in the scene
     description does not matter. It can be also used as an argument of
     function `draw'.

     Example:

          (%i1) load(draw)$
          (%i2) draw(
                  delay     = 100,
                  file_name = "zzz",
                  terminal  = 'animated_gif,
                  gr2d(explicit(x^2,x,-1,1)),
                  gr2d(explicit(x^3,x,-1,1)),
                  gr2d(explicit(x^4,x,-1,1)));
          End of animation sequence
          (%o2)          [gr2d(explicit), gr2d(explicit), gr2d(explicit)]

     Option `delay' is only active in animated gif's; it is ignored in
     any other case.

     See also `terminal', `dimensions'.

Graphic option: dimensions

     Default value: `[600,500]'

     Dimensions of the output terminal. Its value is a list formed by
     the width and the height. The meaning of the two numbers depends on
     the terminal you are working with.

     With terminals `gif', `animated_gif', `png', `jpg', `svg',
     `screen', `wxt', and `aquaterm', the integers represent the number
     of points in each direction. If they are not intergers, they are
     rounded.

     With terminals `eps', `eps_color', `pdf', and `pdfcairo', both
     numbers represent hundredths of cm, which means that, by default,
     pictures in these formats are 6 cm in width and 5 cm in height.

     Since this is a global graphics option, its position in the scene
     description does not matter. It can be also used as an argument of
     function `draw'.

     Examples:

     Option `dimensions' applied to file output and to wxt canvas.

          (%i1) load(draw)$
          (%i2) draw2d(
                  dimensions = [300,300],
                  terminal   = 'png,
                  explicit(x^4,x,-1,1)) $
          (%i3) draw2d(
                  dimensions = [300,300],
                  terminal   = 'wxt,
                  explicit(x^4,x,-1,1)) $

     Option `dimensions' applied to eps output.  We want an eps file
     with A4 portrait dimensions.

          (%i1) load(draw)$
          (%i2) A4portrait: 100*[21, 29.7]$
          (%i3) draw3d(
                  dimensions = A4portrait,
                  terminal   = 'eps,
                  explicit(x^2-y^2,x,-2,2,y,-2,2)) $

Graphic option: draw_realpart

     Default value: `true'

     When `true', functions to be drawn are considered as complex
     functions whose real part value should be plotted; when `false',
     nothing will be plotted when the function does not give a real
     value.

     This option affects objects `explicit' and `parametric' in 2D and
     3D, and `parametric_surface'.

     Example:

     Option `draw_realpart' affects objects `explicit' and `parametric'.

          (%i1) load(draw)$
          (%i2) draw2d(
                  draw_realpart = false,
                  explicit(sqrt(x^2  - 4*x) - x, x, -1, 5),
                  color         = red,
                  draw_realpart = true,
                  parametric(x,sqrt(x^2  - 4*x) - x + 1, x, -1, 5) );

Graphic option: enhanced3d

     Default value: `none'

     If `enhanced3d' is `none', surfaces are not colored in 3D plots.
     In order to get a colored surface, a list must be assigned to
     option `enhanced3d', where the first element is an expression and
     the rest are the names of the variables or parameters used in that
     expression. A list such `[f(x,y,z), x, y, z]' means that point
     `[x,y,z]' of the surface is assigned number `f(x,y,z)', which will
     be colored according to the actual `palette'. For those 3D graphic
     objects defined in terms of parameters, it is possible to define
     the color number in terms of the parameters, as in `[f(u), u]', as
     in objects `parametric' and `tube', or `[f(u,v), u, v]', as in
     object `parametric_surface'.  While all 3D objects admit the model
     based on absolute coordinates, `[f(x,y,z), x, y, z]', only two of
     them, namely `explicit' and `elevation_grid', accept also models
     defined on the `[x,y]' coordinates, `[f(x,y), x, y]'. 3D graphic
     object `implicit' accepts only the `[f(x,y,z), x, y, z]' model.
     Object `points' accepts also the `[f(x,y,z), x, y, z]' model, but
     when points have a chronological nature, model `[f(k), k]' is also
     valid, being `k' an ordering parameter.

     When `enhanced3d' is assigned something different to `none',
     options `color' and `surface_hide' are ignored.

     The names of the variables defined in the lists may be different
     to those used in the definitions of the graphic objects.

     In order to maintain back compatibility, `enhanced3d = false' is
     equivalent to `enhanced3d = none', and `enhanced3d = true' is
     equivalent to `enhanced3d = [z, x, y, z]'.  If an expression is
     given to `enhanced3d', its variables must be the same used in the
     surface definition. This is not necessary when using lists.

     See option `palette' to learn how palettes are specified.

     Examples:

     `explicit' object with coloring defined by the `[f(x,y,z), x, y,
     z]' model.

          (%i1) load(draw)$
          (%i2) draw3d(
                   enhanced3d = [x-z/10,x,y,z],
                   palette    = gray,
                   explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$

     `explicit' object with coloring defined by the `[f(x,y), x, y]'
     model.  The names of the variables defined in the lists may be
     different to those used in the definitions of the graphic objects;
     in this case, `r' corresponds to `x', and `s' to `y'.

          (%i1) load(draw)$
          (%i2) draw3d(
                   enhanced3d = [sin(r*s),r,s],
                   explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$

     `parametric' object with coloring defined by the `[f(x,y,z), x, y,
     z]' model.

          (%i1) load(draw)$
          (%i2) draw3d(
                   nticks = 100,
                   line_width = 2,
                   enhanced3d = [if y>= 0 then 1 else 0, x, y, z],
                   parametric(sin(u)^2,cos(u),u,u,0,4*%pi)) $

     `parametric' object with coloring defined by the `[f(u), u]' model.
     In this case, `(u-1)^2' is a shortcut for `[(u-1)^2,u]'.

          (%i1) load(draw)$
          (%i2) draw3d(
                   nticks = 60,
                   line_width = 3,
                   enhanced3d = (u-1)^2,
                   parametric(cos(5*u)^2,sin(7*u),u-2,u,0,2))$

     `elevation_grid' object with coloring defined by the `[f(x,y), x,
     y]' model.

          (%i1) load(draw)$
          (%i2) m: apply(
                     matrix,
                     makelist(makelist(cos(i^2/80-k/30),k,1,30),i,1,20)) $
          (%i3) draw3d(
                   enhanced3d = [cos(x*y*10),x,y],
                   elevation_grid(m,-1,-1,2,2),
                   xlabel = "x",
                   ylabel = "y");

     `tube' object with coloring defined by the `[f(x,y,z), x, y, z]'
     model.

          (%i1) load(draw)$
          (%i2) draw3d(
                   enhanced3d = [cos(x-y),x,y,z],
                   palette = gray,
                   xu_grid = 50,
                   tube(cos(a), a, 0, 1, a, 0, 4*%pi) )$

     `tube' object with coloring defined by the `[f(u), u]' model.
     Here, `enhanced3d = -a' would be the shortcut for `enhanced3d =
     [-foo,foo]'.

          (%i1) load(draw)$
          (%i2) draw3d(
                   tube_extremes = [open, closed],
                   palette = [26,15,-2],
                   enhanced3d = [-foo, foo],
                   tube(a, a, a^2, 1, a, -2, 2) )$

     `implicit' and `points' objects with coloring defined by the
     `[f(x,y,z), x, y, z]' model.

          (%i1) load(draw)$
          (%i2) draw3d(
                   enhanced3d = [x-y,x,y,z],
                   implicit((x^2+y^2+z^2-1)*(x^2+(y-1.5)^2+z^2-0.5)=0.015,
                            x,-1,1,y,-1.2,2.3,z,-1,1)) $
          (%i3) m: makelist([random(1.0),random(1.0),random(1.0)],k,1,2000)$
          (%i4) draw3d(
                   point_type = filled_circle,
                   point_size = 2,
                   enhanced3d = [u+v-w,u,v,w],
                   points(m) ) $

     When points have a chronological nature, model `[f(k), k]' is also
     valid, being `k' an ordering parameter.

          (%i1) load(draw)$
          (%i2) m:makelist([random(1.0), random(1.0), random(1.0)],k,1,5)$
          (%i3) draw3d(
                   enhanced3d = [sin(j), j],
                   point_size = 3,
                   point_type = filled_circle,
                   points_joined = true,
                   points(m)) $

Graphic option: error_type

     Default value: `y'

     Depending on its value, which can be `x', `y', or `xy', graphic
     object `errors' will draw points with horizontal, vertical, or
     both, error bars. When `error_type=boxes', boxes will be drawn
     instead of crosses.

     See also `errors'.

Graphic option: file_name

     Default value: `"maxima_out"'

     This is the name of the file where terminals `png', `jpg', `gif',
     `eps', `eps_color', `pdf', `pdfcairo' and `svg' will save the
     graphic.

     Since this is a global graphics option, its position in the scene
     description does not matter. It can be also used as an argument of
     function `draw'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(file_name = "myfile",
                       explicit(x^2,x,-1,1),
                       terminal  = 'png)$

     See also `terminal', `dimensions'.

Graphic option: fill_color

     Default value: `"red"'

     `fill_color' specifies the color for filling polygons and 2d
     `explicit' functions.

     See `color' to learn how colors are specified.

Graphic option: fill_density

     Default value: 0

     `fill_density' is a number between 0 and 1 that specifies the
     intensity of the `fill_color' in `bars' objects.

     See `bars' for examples.

Graphic option: filled_func

     Default value: `false'

     Option `filled_func' controls how regions limited by functions
     should be filled. When `filled_func' is `true', the region bounded
     by the function defined with object `explicit' and the bottom of
     the graphic window is filled with `fill_color'. When `filled_func'
     contains a function expression, then the region bounded by this
     function and the function defined with object `explicit' will be
     filled. By default, explicit functions are not filled.

     This option affects only the 2d graphic object `explicit'.

     Example:

     Region bounded by an `explicit' object and the bottom of the
     graphic window.
          (%i1) load(draw)$
          (%i2) draw2d(fill_color  = red,
                       filled_func = true,
                       explicit(sin(x),x,0,10) )$

     Region bounded by an `explicit' object and the function defined by
     option `filled_func'. Note that the variable in `filled_func' must
     be the same as that used in `explicit'.
          (%i1) load(draw)$
          (%i2) draw2d(fill_color  = grey,
                       filled_func = sin(x),
                       explicit(-sin(x),x,0,%pi));

     See also `fill_color' and `explicit'.

Graphic option: font

     Default value: `""' (empty string)

     This option can be used to set the font face to be used by the
     terminal.  Only one font face and size can be used throughout the
     plot.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     See also `font_size'.

     Gnuplot doesn't handle fonts by itself, it leaves this task to the
     support libraries of the different terminals, each one with its own
     philosophy about it. A brief summary follows:

        * x11: Uses the normal x11 font server mechanism.

          Example:
               (%i1) load(draw)$
               (%i2) draw2d(font      = "Arial",
                            font_size = 20,
                            label(["Arial font, size 20",1,1]))$

        * windows: The windows terminal doesn't support changing of
          fonts from inside the plot.  Once the plot has been
          generated, the font can be changed right-clicking on the menu
          of the graph window.

        * png, jpeg, gif: The libgd library uses the font path stored
          in the environment variable `GDFONTPATH'; in this case, it is
          only necessary to set option `font' to the font's name. It is
          also possible to give the complete path to the font file.

          Examples:

          Option `font' can be given the complete path to the font file:
               (%i1) load(draw)$
               (%i2) path: "/usr/share/fonts/truetype/freefont/" $
               (%i3) file: "FreeSerifBoldItalic.ttf" $
               (%i4) draw2d(
                       font      = concat(path, file),
                       font_size = 20,
                       color     = red,
                       label(["FreeSerifBoldItalic font, size 20",1,1]),
                       terminal  = png)$

          If environment variable `GDFONTPATH' is set to the path where
          font files are allocated, it is possible to set graphic
          option `font' to the name of the font.
               (%i1) load(draw)$
               (%i2) draw2d(
                       font      = "FreeSerifBoldItalic",
                       font_size = 20,
                       color     = red,
                       label(["FreeSerifBoldItalic font, size 20",1,1]),
                       terminal  = png)$

        * Postscript: Standard Postscript fonts are:
          `"Times-Roman"', `"Times-Italic"', `"Times-Bold"',
          `"Times-BoldItalic"',
          `"Helvetica"', `"Helvetica-Oblique"', `"Helvetica-Bold"',
          `"Helvetic-BoldOblique"', `"Courier"', `"Courier-Oblique"',
          `"Courier-Bold"',
          and `"Courier-BoldOblique"'.

          Example:
               (%i1) load(draw)$
               (%i2) draw2d(
                       font      = "Courier-Oblique",
                       font_size = 15,
                       label(["Courier-Oblique font, size 15",1,1]),
                       terminal = eps)$

        * pdf: Uses same fonts as Postscript.

        * pdfcairo: Uses same fonts as wxt.

        * wxt: The pango library finds fonts via the `fontconfig'
          utility.

        * aqua: Default is `"Times-Roman"'.

     The gnuplot documentation is an important source of information
     about terminals and fonts.

Graphic option: font_size

     Default value: 10

     This option can be used to set the font size to be used by the
     terminal.  Only one font face and size can be used throughout the
     plot. `font_size' is active only when option `font' is not equal
     to the empty string.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     See also `font'.

Graphic option: gnuplot_file_name

     Default value: `"maxout.gnuplot"'

     This is the name of the file with the necessary commands to be
     processed by Gnuplot.

     Since this is a global graphics option, its position in the scene
     description does not matter. It can be also used as an argument of
     function `draw'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(
                 file_name = "my_file",
                 gnuplot_file_name = "my_commands_for_gnuplot",
                 data_file_name    = "my_data_for_gnuplot",
                 terminal          = png,
                 explicit(x^2,x,-1,1)) $

     See also `data_file_name'.

Graphic option: grid

     Default value: `false'

     If `grid' is `true', a grid will be drawn on the <xy> plane.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(grid = true,
                       explicit(exp(u),u,-2,2))$

Graphic option: head_angle

     Default value: 45

     `head_angle' indicates the angle, in degrees, between the arrow
     heads and the segment.

     This option is relevant only for `vector' objects.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(xrange      = [0,10],
                       yrange      = [0,9],
                       head_length = 0.7,
                       head_angle  = 10,
                       vector([1,1],[0,6]),
                       head_angle  = 20,
                       vector([2,1],[0,6]),
                       head_angle  = 30,
                       vector([3,1],[0,6]),
                       head_angle  = 40,
                       vector([4,1],[0,6]),
                       head_angle  = 60,
                       vector([5,1],[0,6]),
                       head_angle  = 90,
                       vector([6,1],[0,6]),
                       head_angle  = 120,
                       vector([7,1],[0,6]),
                       head_angle  = 160,
                       vector([8,1],[0,6]),
                       head_angle  = 180,
                       vector([9,1],[0,6]) )$

     See also `head_both', `head_length', and `head_type'.

Graphic option: head_both

     Default value: `false'

     If `head_both' is `true', vectors are plotted with two arrow heads.
     If `false', only one arrow is plotted.

     This option is relevant only for `vector' objects.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(xrange      = [0,8],
                       yrange      = [0,8],
                       head_length = 0.7,
                       vector([1,1],[6,0]),
                       head_both   = true,
                       vector([1,7],[6,0]) )$

     See also `head_length', `head_angle', and `head_type'.

Graphic option: head_length

     Default value: 2

     `head_length' indicates, in <x>-axis units, the length of arrow
     heads.

     This option is relevant only for `vector' objects.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(xrange      = [0,12],
                       yrange      = [0,8],
                       vector([0,1],[5,5]),
                       head_length = 1,
                       vector([2,1],[5,5]),
                       head_length = 0.5,
                       vector([4,1],[5,5]),
                       head_length = 0.25,
                       vector([6,1],[5,5]))$

     See also `head_both', `head_angle', and `head_type'.

Graphic option: head_type

     Default value: `filled'

     `head_type' is used to specify how arrow heads are plotted.
     Possible values are: `filled' (closed and filled arrow heads),
     `empty' (closed but not filled arrow heads), and `nofilled' (open
     arrow heads).

     This option is relevant only for `vector' objects.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(xrange      = [0,12],
                       yrange      = [0,10],
                       head_length = 1,
                       vector([0,1],[5,5]), /* default type */
                       head_type = 'empty,
                       vector([3,1],[5,5]),
                       head_type = 'nofilled,
                       vector([6,1],[5,5]))$

     See also `head_both', `head_angle', and `head_length'.

Graphic option: ip_grid

     Default value: `[50, 50]'

     `ip_grid' sets the grid for the first sampling in implicit plots.

     This option is relevant only for `implicit' objects.

Graphic option: ip_grid_in

     Default value: `[5, 5]'

     `ip_grid_in' sets the grid for the second sampling in implicit
     plots.

     This option is relevant only for `implicit' objects.

Graphic option: key

     Default value: `""' (empty string)

     `key' is the name of a function in the legend. If `key' is an
     empty string, no key is assigned to the function.

     This option affects the following graphic objects:
        * `gr2d': `points', `polygon', `rectangle', `ellipse',
          `vector', `explicit', `implicit', `parametric', and `polar'.

        * `gr3d': `points', `explicit', `parametric', and
          `parametric_surface'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(key   = "Sinus",
                       explicit(sin(x),x,0,10),
                       key   = "Cosinus",
                       color = red,
                       explicit(cos(x),x,0,10) )$

Graphic option: label_alignment

     Default value: `center'

     `label_alignment' is used to specify where to write labels with
     respect to the given coordinates. Possible values are: `center',
     `left', and `right'.

     This option is relevant only for `label' objects.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(xrange          = [0,10],
                       yrange          = [0,10],
                       points_joined   = true,
                       points([[5,0],[5,10]]),
                       color           = blue,
                       label(["Centered alignment (default)",5,2]),
                       label_alignment = 'left,
                       label(["Left alignment",5,5]),
                       label_alignment = 'right,
                       label(["Right alignment",5,8]))$

     See also `label_orientation', and `color'.

Graphic option: label_orientation

     Default value: `horizontal'

     `label_orientation' is used to specify orientation of labels.
     Possible values are: `horizontal', and `vertical'.

     This option is relevant only for `label' objects.

     Example:

     In this example, a dummy point is added to get an image.  Package
     `draw' needs always data to draw an scene.
          (%i1) load(draw)$
          (%i2) draw2d(xrange     = [0,10],
                       yrange     = [0,10],
                       point_size = 0,
                       points([[5,5]]),
                       color      = navy,
                       label(["Horizontal orientation (default)",5,2]),
                       label_orientation = 'vertical,
                       color             = "#654321",
                       label(["Vertical orientation",1,5]))$

     See also `label_alignment' and `color'.

Graphic option: line_type

     Default value: `solid'

     `line_type' indicates how lines are displayed; possible values are
     `solid' and `dots'.

     This option affects the following graphic objects:
        * `gr2d': `points', `polygon', `rectangle', `ellipse',
          `vector', `explicit', `implicit', `parametric' and `polar'.

        * `gr3d': `points', `explicit', `parametric' and
          `parametric_surface'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(line_type = dots,
                       explicit(1 + x^2,x,-1,1),
                       line_type = solid, /* default */
                       explicit(2 + x^2,x,-1,1))$

     See also `line_width'.

Graphic option: line_width

     Default value: 1

     `line_width' is the width of plotted lines.  Its value must be a
     positive number.

     This option affects the following graphic objects:
        * `gr2d': `points', `polygon', `rectangle', `ellipse',
          `vector', `explicit', `implicit', `parametric' and `polar'.

        * `gr3d': `points' and `parametric'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(explicit(x^2,x,-1,1), /* default width */
                       line_width = 5.5,
                       explicit(1 + x^2,x,-1,1),
                       line_width = 10,
                       explicit(2 + x^2,x,-1,1))$

     See also `line_type'.

Graphic option: logcb

     Default value: `false'

     If `logcb' is `true', the tics in the colorbox will be drawn in the
     logarithmic scale.

     When `enhanced3d' or `colorbox' is `false', option `logcb' has no
     effect.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d (
                  enhanced3d = true,
                  color      = green,
                  logcb = true,
                  logz  = true,
                  palette = [-15,24,-9],
                  explicit(exp(x^2-y^2), x,-2,2,y,-2,2)) $

     See also `enhanced3d', `colorbox' and `cbrange'.

Graphic option: logx

     Default value: `false'

     If `logx' is `true', the <x> axis will be drawn in the logarithmic
     scale.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(explicit(log(x),x,0.01,5),
                       logx = true)$

     See also `logy' and `logz'.

Graphic option: logy

     Default value: `false'

     If `logy' is `true', the <y> axis will be drawn in the logarithmic
     scale.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(logy = true,
                       explicit(exp(x),x,0,5))$

     See also `logx' and `logz'.

Graphic option: logz

     Default value: `false'

     If `logz' is `true', the <z> axis will be drawn in the logarithmic
     scale.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(logz = true,
                       explicit(exp(u^2+v^2),u,-2,2,v,-2,2))$

     See also `logx' and `logy'.

Graphic option: nticks

     Default value: 29

     In 2d, `nticks' gives the initial number of points used by the
     adaptive plotting routine for explicit objects. It is also the
     number of points that will be shown in parametric and polar curves.

     This option affects the following graphic objects:
        * `gr2d': `ellipse', `explicit', `parametric' and `polar'.

        * `gr3d': `parametric'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(transparent = true,
                       ellipse(0,0,4,2,0,180),
                       nticks = 5,
                       ellipse(0,0,4,2,180,180) )$

Graphic option: palette

     Default value: `color'

     `palette' indicates how to map gray levels onto color components.
     It works together with option `enhanced3d' in 3D graphics, who
     associates every point of a surfaces to a real number or gray
     level.  It also works with gray images. With `palette', levels are
     transformed into colors.

     There are two ways for defining these transformations.

     First, `palette' can be a vector of length three with components
     ranging from -36 to +36; each value is an index for a formula
     mapping the levels onto red, green and blue colors, respectively:
           0: 0               1: 0.5           2: 1
           3: x               4: x^2           5: x^3
           6: x^4             7: sqrt(x)       8: sqrt(sqrt(x))
           9: sin(90x)       10: cos(90x)     11: |x-0.5|
          12: (2x-1)^2       13: sin(180x)    14: |cos(180x)|
          15: sin(360x)      16: cos(360x)    17: |sin(360x)|
          18: |cos(360x)|    19: |sin(720x)|  20: |cos(720x)|
          21: 3x             22: 3x-1         23: 3x-2
          24: |3x-1|         25: |3x-2|       26: (3x-1)/2
          27: (3x-2)/2       28: |(3x-1)/2|   29: |(3x-2)/2|
          30: x/0.32-0.78125 31: 2*x-0.84     32: 4x;1;-2x+1.84;x/0.08-11.5
          33: |2*x - 0.5|    34: 2*x          35: 2*x - 0.5
          36: 2*x - 1
     negative numbers mean negative colour component.  `palette = gray'
     and `palette = color' are short cuts for `palette = [3,3,3]' and
     `palette = [7,5,15]', respectively.

     Second, `palette' can be a user defined lookup table. In this case,
     the format for building a lookup table of length `n' is
     `palette=[color_1, color_2, ..., color_n]', where `color_i' is a
     well formed color (see option `color'), such that `color_1' is
     assigned to the lowest gray level and `color_n' to the highest.
     The rest of colors are interpolated.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Examples:

     It works together with option `enhanced3d' in 3D graphics.

          (%i1) load(draw)$
          (%i2) draw3d(
                  enhanced3d = [z-x+2*y,x,y,z],
                  palette = [32, -8, 17],
                  explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$

     It also works with gray images.

          (%i1) load(draw)$
          (%i2) im: apply(
                     'matrix,
                      makelist(makelist(random(200),i,1,30),i,1,30))$
          (%i3) /* palette = color, default */
                draw2d(image(im,0,0,30,30))$
          (%i4) draw2d(palette = gray, image(im,0,0,30,30))$
          (%i5) draw2d(palette = [15,20,-4],
                       colorbox=false,
                       image(im,0,0,30,30))$

     `palette' can be a user defined lookup table.  In this example,
     low values of `x' are colored in red, and higher values in yellow.

          (%i1) load(draw)$
          (%i2) draw3d(
                   palette = [red, blue, yellow],
                   enhanced3d = x,
                   explicit(x^2+y^2,x,-1,1,y,-1,1)) $

     See also `colorbox' and `enhanced3d'.

Graphic option: point_size

     Default value: 1

     `point_size' sets the size for plotted points. It must be a non
     negative number.

     This option has no effect when graphic option `point_type' is set
     to `dot'.

     This option affects the following graphic objects:
        * `gr2d': `points'.

        * `gr3d': `points'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(points(makelist([random(20),random(50)],k,1,10)),
                  point_size = 5,
                  points(makelist(k,k,1,20),makelist(random(30),k,1,20)))$

Graphic option: point_type

     Default value: 1

     `point_type' indicates how isolated points are displayed; the
     value of this option can be any integer index greater or equal
     than -1, or the name of a point style: `$none' (-1), `dot' (0),
     `plus' (1), `multiply' (2), `asterisk' (3), `square' (4),
     `filled_square' (5), `circle' (6), `filled_circle' (7),
     `up_triangle' (8), `filled_up_triangle' (9), `down_triangle' (10),
     `filled_down_triangle' (11), `diamant' (12) and `filled_diamant'
     (13).

     This option affects the following graphic objects:
        * `gr2d': `points'.

        * `gr3d': `points'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(xrange = [0,10],
                       yrange = [0,10],
                       point_size = 3,
                       point_type = diamant,
                       points([[1,1],[5,1],[9,1]]),
                       point_type = filled_down_triangle,
                       points([[1,2],[5,2],[9,2]]),
                       point_type = asterisk,
                       points([[1,3],[5,3],[9,3]]),
                       point_type = filled_diamant,
                       points([[1,4],[5,4],[9,4]]),
                       point_type = 5,
                       points([[1,5],[5,5],[9,5]]),
                       point_type = 6,
                       points([[1,6],[5,6],[9,6]]),
                       point_type = filled_circle,
                       points([[1,7],[5,7],[9,7]]),
                       point_type = 8,
                       points([[1,8],[5,8],[9,8]]),
                       point_type = filled_diamant,
                       points([[1,9],[5,9],[9,9]]) )$

Graphic option: points_joined

     Default value: `false'

     When `points_joined' is `true', points are joined by lines; when
     `false', isolated points are drawn. A third possible value for
     this graphic option is `impulses'; in such case, vertical segments
     are drawn from points to the x-axis (2D) or to the xy-plane (3D).

     This option affects the following graphic objects:
        * `gr2d': `points'.

        * `gr3d': `points'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(xrange        = [0,10],
                       yrange        = [0,4],
                       point_size    = 3,
                       point_type    = up_triangle,
                       color         = blue,
                       points([[1,1],[5,1],[9,1]]),
                       points_joined = true,
                       point_type    = square,
                       line_type     = dots,
                       points([[1,2],[5,2],[9,2]]),
                       point_type    = circle,
                       color         = red,
                       line_width    = 7,
                       points([[1,3],[5,3],[9,3]]) )$

Graphic option: proportional_axes

     Default value: `none'

     When `proportional_axes' is equal to `xy' or `xyz', a 2D or 3D
     scene will be drawn with axes proportional to their relative
     lengths.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     This option works with Gnuplot version 4.2.6 or greater.

     Examples:

     Single 2D plot.

          (%i1) load(draw)$
          (%i2) draw2d(
                  ellipse(0,0,1,1,0,360),
                  transparent=true,
                  color = blue,
                  line_width = 4,
                  ellipse(0,0,2,1/2,0,360),
                  proportional_axes = xy) $

     Multiplot.

          (%i1) load(draw)$
          (%i2) draw(
                  terminal = wxt,
                  gr2d(proportional_axes = xy,
                       explicit(x^2,x,0,1)),
                  gr2d(explicit(x^2,x,0,1),
                       xrange = [0,1],
                       yrange = [0,2],
                       proportional_axes=xy),
                  gr2d(explicit(x^2,x,0,1))

Graphic option: surface_hide

     Default value: `false'

     If `surface_hide' is `true', hidden parts are not plotted in 3d
     surfaces.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw(columns=2,
                     gr3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3)),
                     gr3d(surface_hide = true,
                          explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3)) )$

Graphic option: terminal

     Default value: `screen'

     Selects the terminal to be used by Gnuplot; possible values are:
     `screen' (default), `png', `pngcairo', `jpg', `eps', `eps_color',
     `pdf', `pdfcairo', `gif', `animated_gif', `wxt', `svg', and
     `aquaterm'.

     Terminals `screen', `wxt' and `aquaterm' can be also defined as a
     list with two elements: the name of the terminal itself and a non
     negative integer number.  In this form, multiple windows can be
     opened at the same time, each with its corresponding number. This
     feature does not work in Windows platforms.

     Since this is a global graphics option, its position in the scene
     description does not matter. It can be also used as an argument of
     function `draw'.

     N.B. pdfcairo requires Gnuplot 4.3 or newer.  `pdf' requires
     Gnuplot to be compiled with the option `--enable-pdf' and libpdf
     must be installed. The pdf library is available from:
     `http://www.pdflib.com/en/download/pdflib-family/pdflib-lite/'

     Examples:

          (%i1) load(draw)$
          (%i2) /* screen terminal (default) */
                draw2d(explicit(x^2,x,-1,1))$
          (%i3) /* png file */
                draw2d(terminal  = 'png,
                       explicit(x^2,x,-1,1))$
          (%i4) /* jpg file */
                draw2d(terminal   = 'jpg,
                       dimensions = [300,300],
                       explicit(x^2,x,-1,1))$
          (%i5) /* eps file */
                draw2d(file_name = "myfile",
                       explicit(x^2,x,-1,1),
                       terminal  = 'eps)$
          (%i6) /* pdf file */
                draw2d(file_name = "mypdf",
                       dimensions = 100*[12.0,8.0],
                       explicit(x^2,x,-1,1),
                       terminal  = 'pdf)$
          (%i7) /* wxwidgets window */
                draw2d(explicit(x^2,x,-1,1),
                       terminal  = 'wxt)$

     Multiple windows.
          (%i1) load(draw)$
          (%i2) draw2d(explicit(x^5,x,-2,2), terminal=[screen, 3])$
          (%i3) draw2d(explicit(x^2,x,-2,2), terminal=[screen, 0])$

     An animated gif file.
          (%i1) load(draw)$
          (%i2) draw(
                  delay     = 100,
                  file_name = "zzz",
                  terminal  = 'animated_gif,
                  gr2d(explicit(x^2,x,-1,1)),
                  gr2d(explicit(x^3,x,-1,1)),
                  gr2d(explicit(x^4,x,-1,1)));
          End of animation sequence
          (%o2)          [gr2d(explicit), gr2d(explicit), gr2d(explicit)]

     Option `delay' is only active in animated gif's; it is ignored in
     any other case.

     See also `file_name', `dimensions' and `delay'.

Graphic option: title

     Default value: `""' (empty string)

     Option `title', a string, is the main title for the scene.  By
     default, no title is written.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(explicit(exp(u),u,-2,2),
                       title = "Exponential function")$

Graphic option: transform

     Default value: `none'

     If `transform' is `none', the space is not transformed and graphic
     objects are drawn as defined. When a space transformation is
     desired, a list must be assigned to option `transform'. In case of
     a 2D scene, the list takes the form `[f1(x,y), f2(x,y), x, y]'.
     In case of a 3D scene, the list is of the form `[f1(x,y,z),
     f2(x,y,z), f3(x,y,z), x, y, z]'.

     The names of the variables defined in the lists may be different
     to those used in the definitions of the graphic objects.

     Examples:

     Rotation in 2D.

          (%i1) load(draw)$
          (%i2) th : %pi / 4$
          (%i3) draw2d(
                  color = "#e245f0",
                  proportional_axes = 'xy,
                  line_width = 8,
                  triangle([3,2],[7,2],[5,5]),
                  border     = false,
                  fill_color = yellow,
                  transform  = [cos(th)*x - sin(th)*y,
                                sin(th)*x + cos(th)*y, x, y],
                  triangle([3,2],[7,2],[5,5]) )$

     Translation in 3D.

          (%i1) load(draw)$
          (%i2) draw3d(
                  color     = "#a02c00",
                  explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3),
                  transform = [x+10,y+10,z+10,x,y,z],
                  color     = blue,
                  explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3) )$

Graphic option: transparent

     Default value: `false'

     If `transparent' is `true', interior regions of polygons are
     filled according to `fill_color'.

     This option affects the following graphic objects:
        * `gr2d': `polygon', `rectangle', and `ellipse'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(polygon([[3,2],[7,2],[5,5]]),
                       transparent = true,
                       color       = blue,
                       polygon([[5,2],[9,2],[7,5]]) )$

Graphic option: tube_extremes

     Default value: `[open, open]'

     A list with two possible elements, `open' and `closed', indicating
     whether the extremes of a graphic object `tube' remain open or
     must be closed. By default, both extremes are left open.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(
                  tube_extremes = [open, closed],
                  tube(0, 0, a, 1,
                       a, 0, 8) )$

Graphic option: unit_vectors

     Default value: `false'

     If `unit_vectors' is `true', vectors are plotted with module 1.
     This is useful for plotting vector fields. If `unit_vectors' is
     `false', vectors are plotted with its original length.

     This option is relevant only for `vector' objects.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(xrange      = [-1,6],
                       yrange      = [-1,6],
                       head_length = 0.1,
                       vector([0,0],[5,2]),
                       unit_vectors = true,
                       color        = red,
                       vector([0,3],[5,2]))$

Graphic option: user_preamble

     Default value: `""' (empty string)

     Expert Gnuplot users can make use of this option to fine tune
     Gnuplot's behaviour by writing settings to be sent before the
     `plot' or `splot' command.

     The value of this option must be a string or a list of strings
     (one per line).

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

     The dumb terminal is not supported by package `draw', but it is
     possible to set it by making use of option `user_preamble',
          (%i1) load(draw)$
          (%i2) draw2d(explicit(exp(x)-1,x,-1,1),
                       parametric(cos(u),sin(u),u,0,2*%pi),
                       user_preamble="set terminal dumb")$

Graphic option: view

     Default value: `[60,30]'

     A pair of angles, measured in degrees, indicating the view
     direction in a 3D scene. The first angle is the vertical rotation
     around the <x> axis, in the range [0, 180]. The second one is the
     horizontal rotation around the <z> axis, in the range [0, 360].

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(view = [170, 360],
                       explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$

Graphic option: wired_surface

     Default value: `false'

     Indicates whether 3D surfaces in `enhanced3d' mode show the grid
     joinning the points or not.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(
                  enhanced3d    = [sin(x),x,y],
                  wired_surface = true,
                  explicit(x^2+y^2,x,-1,1,y,-1,1)) $

Graphic option: x_voxel

     Default value: 10

     `x_voxel' is the number of voxels in the x direction to be used by
     the marching cubes algorithm implemented by the 3d `implicit'
     object. It is also used by graphic object `region'.

Graphic option: xaxis

     Default value: `false'

     If `xaxis' is `true', the <x> axis is drawn.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(explicit(x^3,x,-1,1),
                       xaxis       = true,
                       xaxis_color = blue)$

     See also `xaxis_width', `xaxis_type' and `xaxis_color'.

Graphic option: xaxis_color

     Default value: `"black"'

     `xaxis_color' specifies the color for the <x> axis. See `color' to
     know how colors are defined.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(explicit(x^3,x,-1,1),
                       xaxis       = true,
                       xaxis_color = red)$

     See also `xaxis', `xaxis_width' and `xaxis_type'.

Graphic option: xaxis_secondary

     Default value: `false'

     If `xaxis_secondary' is `true', function values can be plotted with
     respect to the second <x> axis, which will be drawn on top of the
     scene.

     Note that this is a local graphics option which only affects to 2d
     plots.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(
                   key   = "Bottom x-axis",
                   explicit(x+1,x,1,2),
                   color = red,
                   key   = "Above x-axis",
                   xtics_secondary = true,
                   xaxis_secondary = true,
                   explicit(x^2,x,-1,1)) $

     See also `xrange_secondary', `xtics_secondary',
     `xtics_rotate_secondary', `xtics_axis_secondary' and
     `xaxis_secondary'.

Graphic option: xaxis_type

     Default value: `dots'

     `xaxis_type' indicates how the <x> axis is displayed; possible
     values are `solid' and `dots'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(explicit(x^3,x,-1,1),
                       xaxis       = true,
                       xaxis_type  = solid)$

     See also `xaxis', `xaxis_width' and `xaxis_color'.

Graphic option: xaxis_width

     Default value: 1

     `xaxis_width' is the width of the <x> axis.  Its value must be a
     positive number.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(explicit(x^3,x,-1,1),
                       xaxis       = true,
                       xaxis_width = 3)$

     See also `xaxis', `xaxis_type' and `xaxis_color'.

Graphic option: xlabel

     Default value: `""' (empty string)

     Option `xlabel', a string, is the label for the <x> axis.  By
     default, no label is written.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(xlabel = "Time",
                       explicit(exp(u),u,-2,2),
                       ylabel = "Population")$

     See also `ylabel', and `zlabel'.

Graphic option: xrange

     Default value: `auto'

     If `xrange' is `auto', the range for the <x> coordinate is
     computed automatically.

     If the user wants a specific interval for <x>, it must be given as
     a Maxima list, as in `xrange=[-2, 3]'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(xrange = [-3,5],
                       explicit(x^2,x,-1,1))$

     See also `yrange' and `zrange'.

Graphic option: xrange_secondary

     Default value: `auto'

     If `xrange_secondary' is `auto', the range for the second <x> axis
     is computed automatically.

     If the user wants a specific interval for the second <x> axis, it
     must be given as a Maxima list, as in `xrange_secondary=[-2, 3]'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     See also `xrange', `yrange', `zrange' and `yrange_secondary'.

Graphic option: xtics

     Default value: `auto'

     This graphic option controls the way tic marks are drawn on the
     <x> axis.

        * When option `xtics' is bounded to symbol <auto>, tic marks are
          drawn automatically.

        * When option `xtics' is bounded to symbol <none>, tic marks are
          not drawn.

        * When option `xtics' is bounded to a positive number, this is
          the distance between two consecutive tic marks.

        * When option `xtics' is bounded to a list of length three of
          the form `[start,incr,end]', tic marks are plotted from
          `start' to `end' at intervals of length `incr'.

        * When option `xtics' is bounded to a set of numbers of the
          form `{n1, n2, ...}', tic marks are plotted at values `n1',
          `n2', ...

        * When option `xtics' is bounded to a set of pairs of the form
          `{["label1", n1], ["label2", n2], ...}', tic marks
          corresponding to values `n1', `n2', ... are labeled with
          `"label1"', `"label2"', ..., respectively.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Examples:

     Disable tics.
          (%i1) load(draw)$
          (%i2) draw2d(xtics = 'none,
                       explicit(x^3,x,-1,1)  )$

     Tics every 1/4 units.
          (%i1) load(draw)$
          (%i2) draw2d(xtics = 1/4,
                       explicit(x^3,x,-1,1)  )$

     Tics from -3/4 to 3/4 in steps of 1/8.
          (%i1) load(draw)$
          (%i2) draw2d(xtics = [-3/4,1/8,3/4],
                       explicit(x^3,x,-1,1)  )$

     Tics at points -1/2, -1/4 and 3/4.
          (%i1) load(draw)$
          (%i2) draw2d(xtics = {-1/2,-1/4,3/4},
                       explicit(x^3,x,-1,1)  )$

     Labeled tics.
          (%i1) load(draw)$
          (%i2) draw2d(xtics = {["High",0.75],["Medium",0],["Low",-0.75]},
                       explicit(x^3,x,-1,1)  )$

     See also `ytics', and `ztics'.

Graphic option: xtics_axis

     Default value: `false'

     If `xtics_axis' is `true', tic marks and their labels are plotted
     just along the <x> axis, if it is `false' tics are plotted on the
     border.

     Since this is a global graphics option, its position in the scene
     description does not matter.

Graphic option: xtics_rotate

     Default value: `false'

     If `xtics_rotate' is `true', tic marks on the <x> axis are rotated
     90 degrees.

     Since this is a global graphics option, its position in the scene
     description does not matter.

Graphic option: xtics_rotate_secondary

     Default value: `false'

     If `xtics_rotate_secondary' is `true', tic marks on the secondary
     <x> axis are rotated 90 degrees.

     Since this is a global graphics option, its position in the scene
     description does not matter.

Graphic option: xtics_secondary

     Default value: `auto'

     This graphic option controls the way tic marks are drawn on the
     second <x> axis.

     See `xtics' for a complete description.

Graphic option: xtics_secondary_axis

     Default value: `false'

     If `xtics_secondary_axis' is `true', tic marks and their labels
     are plotted just along the secondary <x> axis, if it is `false'
     tics are plotted on the border.

     Since this is a global graphics option, its position in the scene
     description does not matter.

Graphic option: xu_grid

     Default value: 30

     `xu_grid' is the number of coordinates of the first variable (`x'
     in explicit and `u' in parametric 3d surfaces) to build the grid
     of sample points.

     This option affects the following graphic objects:
        * `gr3d': `explicit' and `parametric_surface'.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(xu_grid = 10,
                       yv_grid = 50,
                       explicit(x^2+y^2,x,-3,3,y,-3,3) )$

     See also `yv_grid'.

Graphic option: xy_file

     Default value: `""' (empty string)

     `xy_file' is the name of the file where the coordinates will be
     saved after clicking with the mouse button and hitting the 'x'
     key. By default, no coordinates are saved.

     Since this is a global graphics option, its position in the scene
     description does not matter.

Graphic option: xyplane

     Default value: `false'

     Allocates the xy-plane in 3D scenes. When `xyplane' is `false',
     the xy-plane is placed automatically; when it is a real number,
     the xy-plane intersects the z-axis at this level.  This option has
     no effect in 2D scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(xyplane = %e-2,
                       explicit(x^2+y^2,x,-1,1,y,-1,1))$

Graphic option: y_voxel

     Default value: 10

     `y_voxel' is the number of voxels in the y direction to be used by
     the marching cubes algorithm implemented by the 3d `implicit'
     object. It is also used by graphic object `region'.

Graphic option: yaxis

     Default value: `false'

     If `yaxis' is `true', the <y> axis is drawn.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(explicit(x^3,x,-1,1),
                       yaxis       = true,
                       yaxis_color = blue)$

     See also `yaxis_width', `yaxis_type' and `yaxis_color'.

Graphic option: yaxis_color

     Default value: `"black"'

     `yaxis_color' specifies the color for the <y> axis. See `color' to
     know how colors are defined.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(explicit(x^3,x,-1,1),
                       yaxis       = true,
                       yaxis_color = red)$

     See also `yaxis', `yaxis_width' and `yaxis_type'.

Graphic option: yaxis_secondary

     Default value: `false'

     If `yaxis_secondary' is `true', function values can be plotted with
     respect to the second <y> axis, which will be drawn on the right
     side of the scene.

     Note that this is a local graphics option which only affects to 2d
     plots.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(
                   explicit(sin(x),x,0,10),
                   yaxis_secondary = true,
                   ytics_secondary = true,
                   color = blue,
                   explicit(100*sin(x+0.1)+2,x,0,10));

     See also `yrange_secondary', `ytics_secondary',
     `ytics_rotate_secondary' and `ytics_axis_secondary'.

Graphic option: yaxis_type

     Default value: `dots'

     `yaxis_type' indicates how the <y> axis is displayed; possible
     values are `solid' and `dots'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(explicit(x^3,x,-1,1),
                       yaxis       = true,
                       yaxis_type  = solid)$

     See also `yaxis', `yaxis_width' and `yaxis_color'.

Graphic option: yaxis_width

     Default value: 1

     `yaxis_width' is the width of the <y> axis.  Its value must be a
     positive number.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(explicit(x^3,x,-1,1),
                       yaxis       = true,
                       yaxis_width = 3)$

     See also `yaxis', `yaxis_type' and `yaxis_color'.

Graphic option: ylabel

     Default value: `""' (empty string)

     Option `ylabel', a string, is the label for the <y> axis.  By
     default, no label is written.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(xlabel = "Time",
                       ylabel = "Population",
                       explicit(exp(u),u,-2,2) )$

     See also `xlabel', and `zlabel'.

Graphic option: yrange

     Default value: `auto'

     If `yrange' is `auto', the range for the <y> coordinate is
     computed automatically.

     If the user wants a specific interval for <y>, it must be given as
     a Maxima list, as in `yrange=[-2, 3]'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(yrange = [-2,3],
                       explicit(x^2,x,-1,1),
                       xrange = [-3,3])$

     See also `xrange', `yrange_secondary' and `zrange'.

Graphic option: yrange_secondary

     Default value: `auto'

     If `yrange_secondary' is `auto', the range for the second <y> axis
     is computed automatically.

     If the user wants a specific interval for the second <y> axis, it
     must be given as a Maxima list, as in `yrange_secondary=[-2, 3]'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(
                   explicit(sin(x),x,0,10),
                   yaxis_secondary = true,
                   ytics_secondary = true,
                   yrange = [-3, 3],
                   yrange_secondary = [-20, 20],
                   color = blue,
                   explicit(100*sin(x+0.1)+2,x,0,10)) $

     See also `xrange', `yrange' and `zrange'.

Graphic option: ytics

     Default value: `auto'

     This graphic option controls the way tic marks are drawn on the
     <y> axis.

     See `xtics' for a complete description.

Graphic option: ytics_axis

     Default value: `false'

     If `ytics_axis' is `true', tic marks and their labels are plotted
     just along the <y> axis, if it is `false' tics are plotted on the
     border.

     Since this is a global graphics option, its position in the scene
     description does not matter.

Graphic option: ytics_rotate

     Default value: `false'

     If `ytics_rotate' is `true', tic marks on the <y> axis are rotated
     90 degrees.

     Since this is a global graphics option, its position in the scene
     description does not matter.

Graphic option: ytics_rotate_secondary

     Default value: `false'

     If `ytics_rotate_secondary' is `true', tic marks on the secondary
     <y> axis are rotated 90 degrees.

     Since this is a global graphics option, its position in the scene
     description does not matter.

Graphic option: ytics_secondary

     Default value: `auto'

     This graphic option controls the way tic marks are drawn on the
     second <y> axis.

     See `xtics' for a complete description.

Graphic option: ytics_secondary_axis

     Default value: `false'

     If `ytics_secondary_axis' is `true', tic marks and their labels
     are plotted just along the secondary <y> axis, if it is `false'
     tics are plotted on the border.

     Since this is a global graphics option, its position in the scene
     description does not matter.

Graphic option: yv_grid

     Default value: 30

     `yv_grid' is the number of coordinates of the second variable (`y'
     in explicit and `v' in parametric 3d surfaces) to build the grid
     of sample points.

     This option affects the following graphic objects:
        * `gr3d': `explicit' and `parametric_surface'.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(xu_grid = 10,
                       yv_grid = 50,
                       explicit(x^2+y^2,x,-3,3,y,-3,3) )$

     See also `xu_grid'.

Graphic option: z_voxel

     Default value: 10

     `z_voxel' is the number of voxels in the z direction to be used by
     the marching cubes algorithm implemented by the 3d `implicit'
     object.

Graphic option: zaxis

     Default value: `false'

     If `zaxis' is `true', the <z> axis is drawn in 3D plots.  This
     option has no effect in 2D scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
                       zaxis       = true,
                       zaxis_type  = solid,
                       zaxis_color = blue)$

     See also `zaxis_width', `zaxis_type' and `zaxis_color'.

Graphic option: zaxis_color

     Default value: `"black"'

     `zaxis_color' specifies the color for the <z> axis. See `color' to
     know how colors are defined.  This option has no effect in 2D
     scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
                       zaxis       = true,
                       zaxis_type  = solid,
                       zaxis_color = red)$

     See also `zaxis', `zaxis_width' and `zaxis_type'.

Graphic option: zaxis_type

     Default value: `dots'

     `zaxis_type' indicates how the <z> axis is displayed; possible
     values are `solid' and `dots'.  This option has no effect in 2D
     scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
                       zaxis       = true,
                       zaxis_type  = solid)$

     See also `zaxis', `zaxis_width' and `zaxis_color'.

Graphic option: zaxis_width

     Default value: 1

     `zaxis_width' is the width of the <z> axis.  Its value must be a
     positive number. This option has no effect in 2D scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
                       zaxis       = true,
                       zaxis_type  = solid,
                       zaxis_width = 3)$

     See also `zaxis', `zaxis_type' and `zaxis_color'.

Graphic option: zlabel

     Default value: `""' (empty string)

     Option `zlabel', a string, is the label for the <z> axis.  By
     default, no label is written.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(zlabel = "Z variable",
                       ylabel = "Y variable",
                       explicit(sin(x^2+y^2),x,-2,2,y,-2,2),
                       xlabel = "X variable" )$

     See also `xlabel', and `ylabel'.

Graphic option: zrange

     Default value: `auto'

     If `zrange' is `auto', the range for the <z> coordinate is
     computed automatically.

     If the user wants a specific interval for <z>, it must be given as
     a Maxima list, as in `zrange=[-2, 3]'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(yrange = [-3,3],
                       zrange = [-2,5],
                       explicit(x^2+y^2,x,-1,1,y,-1,1),
                       xrange = [-3,3])$

     See also `xrange' and `yrange'.

Graphic option: ztics

     Default value: `auto'

     This graphic option controls the way tic marks are drawn on the
     <z> axis.

     See `xtics' for a complete description.

Graphic option: ztics_axis

     Default value: `false'

     If `ztics_axis' is `true', tic marks and their labels are plotted
     just along the <z> axis, if it is `false' tics are plotted on the
     border.

     Since this is a global graphics option, its position in the scene
     description does not matter.

Graphic option: ztics_rotate

     Default value: `false'

     If `ztics_rotate' is `true', tic marks on the <z> axis are rotated
     90 degrees.

     Since this is a global graphics option, its position in the scene
     description does not matter.

Graphic object: bars ([<x1>,<h1>,<w1>], [<x2>,<h2>,<w2>, ...])

     Draws vertical bars in 2D.

     2D

     `bars ([<x1>,<h1>,<w1>], [<x2>,<h2>,<w2>, ...])' draws bars
     centered at values <x1>, <x2>, ... with heights <h1>, <h2>, ...
     and widths <w1>, <w2>, ...

     This object is affected by the following graphic options: `key',
     `fill_color', `fill_density' and `line_width'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(
                 key          = "Group A",
                 fill_color   = blue,
                 fill_density = 0.2,
                 bars([0.8,5,0.4],[1.8,7,0.4],[2.8,-4,0.4]),
                 key          = "Group B",
                 fill_color   = red,
                 fill_density = 0.6,
                 line_width   = 4,
                 bars([1.2,4,0.4],[2.2,-2,0.4],[3.2,5,0.4]),
                 xaxis = true);

Graphic object: cylindrical (<radius>, <z>, <minz>, <maxz>, <azi>, <minazi>, <maxazi>)

     Draws 3D functions defined in cylindrical coordinates.

     3D

     `cylindrical(<radius>, <z>, <minz>, <maxz>, <azi>, <minazi>,
     <maxazi>)' plots the function `<radius>(<z>, <azi>)' defined in
     cylindrical coordinates, with variable <z> taking values from
     <minz> to <maxz> and azimuth <azi> taking values from <minazi> to
     <maxazi>.

     This object is affected by the following graphic options:
     `xu_grid', `yv_grid', `line_type', `key', `wired_surface',
     `enhanced3d' and `color'.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(cylindrical(1,z,-2,2,az,0,2*%pi))$

Graphic object: elevation_grid (<mat>,<x0>,<y0>,<width>,<height>)

     Draws matrix <mat> in 3D space. <z> values are taken from <mat>,
     the abscissas range from <x0> to <x0> + <width> and ordinates from
     <y0> to <y0> + <height>. Element a(1,1) is projected on point
     (x0,y0+height), a(1,n) on (x0+width,y0+height), a(m,1) on (x0,y0),
     and a(m,n) on (x0+width,y0).

     This object is affected by the following graphic options:
     `line_type', `line_width', `key', `wired_surface', `enhanced3d',
     and `color'.

     In older versions of Maxima, `elevation_grid' was called `mesh'.
     See also `mesh'.

     Example:

          (%i1) load(draw)$
          (%i2) m: apply(
                      matrix,
                      makelist(makelist(random(10.0),k,1,30),i,1,20)) $
          (%i3) draw3d(
                   color = blue,
                   elevation_grid(m,0,0,3,2),
                   xlabel = "x",
                   ylabel = "y",
                   surface_hide = true);

Graphic object: ellipse (<xc>, <yc>, <a>, <b>, <ang1>, <ang2>)

     Draws ellipses and circles in 2D.

     2D

     `ellipse (<xc>, <yc>, <a>, <b>, <ang1>, <ang2>)' plots an ellipse
     centered at `[<xc>, <yc>]' with horizontal and vertical semi axis
     <a> and <b>, respectively, starting at angle <ang1> with an
     amplitude equal to angle <ang2>.

     This object is affected by the following graphic options: `nticks',
     `transparent', `fill_color', `border', `line_width', `line_type',
     `key' and `color'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(transparent = false,
                       fill_color  = red,
                       color       = gray30,
                       transparent = false,
                       line_width  = 5,
                       ellipse(0,6,3,2,270,-270),
                       /* center (x,y), a, b, start & end in degrees */
                       transparent = true,
                       color       = blue,
                       line_width  = 3,
                       ellipse(2.5,6,2,3,30,-90),
                       xrange      = [-3,6],
                       yrange      = [2,9] )$

Graphic object: errors ([<x1>, <x2>, ...], [<y1>, <y2>, ...])

     Draws points with error bars, horizontally, vertically or both,
     depending on the value of option `error_type'.

     2D

     If `error_type = x', arguments to `errors' must be of the form
     `[x, y, xdelta]' or `[x, y, xlow, xhigh]'.  If `error_type = y',
     arguments must be of the form `[x, y, ydelta]' or `[x, y, ylow,
     yhigh]'.  If `error_type = xy' or `error_type = boxes', arguments
     to `errors' must be of the form `[x, y, xdelta, ydelta]' or `[x,
     y, xlow, xhigh, ylow, yhigh]'.

     See also `error_type'.

     This object is affected by the following graphic options:
     `error_type', `points_joined', `line_width', `key', `line_type',
     `color', `fill_density', `xaxis_secondary', and  `yaxis_secondary'.

     Option `fill_density' is only relevant when `error_type=boxes'.

     Examples:

     Horizontal error bars.

          (%i1) load(draw)$
          (%i2) draw2d(
                  error_type = y,
                  errors([[1,2,1], [3,5,3], [10,3,1], [17,6,2]]))$

     Vertical and horizontal error bars.

          (%i1) load(draw)$
          (%i2) draw2d(
                  error_type = xy,
                  points_joined = true,
                  color = blue,
                  errors([[1,2,1,2], [3,5,2,1], [10,3,1,1], [17,6,1/2,2]]));

Graphic object: explicit (<fcn>,<var>,<minval>,<maxval>)
Graphic object: explicit

          (<fcn>,<var1>,<minval1>,<maxval1>,<var2>,<minval2>,<maxval2>)
     Draws explicit functions in 2D and 3D.

     2D

     `explicit(<fcn>,<var>,<minval>,<maxval>)' plots explicit function
     <fcn>, with variable <var> taking values from <minval> to <maxval>.

     This object is affected by the following graphic options: `nticks',
     `adapt_depth', `draw_realpart', `line_width', `line_type', `key',
     `filled_func', `fill_color' and `color'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(line_width = 3,
                       color      = blue,
                       explicit(x^2,x,-3,3) )$
          (%i3) draw2d(fill_color  = brown,
                       filled_func = true,
                       explicit(x^2,x,-3,3) )$

     3D

     `explicit(<fcn>, <var1>, <minval1>, <maxval1>, <var2>, <minval2>,
     <maxval2>)' plots the explicit function <fcn>, with variable
     <var1> taking values from <minval1> to <maxval1> and variable
     <var2> taking values from <minval2> to <maxval2>.

     This object is affected by the following graphic options:
     `draw_realpart', `xu_grid', `yv_grid', `line_type', `line_width',
     `key', `wired_surface', `enhanced3d', and `color'.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(key   = "Gauss",
                       color = "#a02c00",
                       explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3),
                       yv_grid     = 10,
                       color = blue,
                       key   = "Plane",
                       explicit(x+y,x,-5,5,y,-5,5),
                       surface_hide = true)$

     See also `filled_func' for filled functions.

Graphic object: image (<im>,<x0>,<y0>,<width>,<height>)

     Renders images in 2D.

     2D

     `image (<im>,<x0>,<y0>,<width>,<height>)' plots image <im> in the
     rectangular region from vertex `(<x0>,<y0>)' to
     `(x0+<width>,y0+<height>)' on the real plane. Argument <im> must
     be a matrix of real numbers, a matrix of vectors of length three
     or a <picture> object.

     If <im> is a matrix of real numbers or a <levels picture> object,
     pixel values are interpreted according to graphic option `palette',
     which is a vector of length three with components ranging from -36
     to +36; each value is an index for a formula mapping the levels
     onto red, green and blue colors, respectively:
           0: 0               1: 0.5           2: 1
           3: x               4: x^2           5: x^3
           6: x^4             7: sqrt(x)       8: sqrt(sqrt(x))
           9: sin(90x)       10: cos(90x)     11: |x-0.5|
          12: (2x-1)^2       13: sin(180x)    14: |cos(180x)|
          15: sin(360x)      16: cos(360x)    17: |sin(360x)|
          18: |cos(360x)|    19: |sin(720x)|  20: |cos(720x)|
          21: 3x             22: 3x-1         23: 3x-2
          24: |3x-1|         25: |3x-2|       26: (3x-1)/2
          27: (3x-2)/2       28: |(3x-1)/2|   29: |(3x-2)/2|
          30: x/0.32-0.78125                  31: 2*x-0.84
          32: 4x;1;-2x+1.84;x/0.08-11.5
          33: |2*x - 0.5|    34: 2*x          35: 2*x - 0.5
          36: 2*x - 1
     negative numbers mean negative colour component.

     `palette = gray' and `palette = color' are short cuts for `palette
     = [3,3,3]' and `palette = [7,5,15]', respectively.

     If <im> is a matrix of vectors of length three or an <rgb picture>
     object, they are interpreted as red, green and blue color
     components.

     Examples:

     If <im> is a matrix of real numbers, pixel values are interpreted
     according to graphic option `palette'.
          (%i1) load(draw)$
          (%i2) im: apply(
                     'matrix,
                      makelist(makelist(random(200),i,1,30),i,1,30))$
          (%i3) /* palette = color, default */
                draw2d(image(im,0,0,30,30))$
          (%i4) draw2d(palette = gray, image(im,0,0,30,30))$
          (%i5) draw2d(palette = [15,20,-4],
                       colorbox=false,
                       image(im,0,0,30,30))$

     See also `colorbox'.

     If <im> is a matrix of vectors of length three, they are
     interpreted as red, green and blue color components.
          (%i1) load(draw)$
          (%i2) im: apply(
                      'matrix,
                       makelist(
                         makelist([random(300),
                                   random(300),
                                   random(300)],i,1,30),i,1,30))$
          (%i3) draw2d(image(im,0,0,30,30))$

     Package `draw' automatically loads package `picture'. In this
     example, a level picture object is built by hand and then rendered.
          (%i1) load(draw)$
          (%i2) im: make_level_picture([45,87,2,134,204,16],3,2);
          (%o2)       picture(level, 3, 2, {Array:  #(45 87 2 134 204 16)})
          (%i3) /* default color palette */
                draw2d(image(im,0,0,30,30))$
          (%i4) /* gray palette */
                draw2d(palette = gray,
                       image(im,0,0,30,30))$

     An xpm file is read and then rendered.
          (%i1) load(draw)$
          (%i2) im: read_xpm("myfile.xpm")$
          (%i3) draw2d(image(im,0,0,10,7))$

     See also `make_level_picture', `make_rgb_picture' and `read_xpm'.

     `http://www.telefonica.net/web2/biomates/maxima/gpdraw/image'
     contains more elaborated examples.

Graphic object: implicit (<fcn>,<x>,<xmin>,<xmax>,<y>,<ymin>,<ymax>)
Graphic object: implicit

          (<fcn>,<x>,<xmin>,<xmax>,<y>,<ymin>,<ymax>,<z>,<zmin>,<zmax>)
     Draws implicit functions in 2D and 3D.

     2D

     `implicit(<fcn>,<x>,<xmin>,<xmax>,<y>,<ymin>,<ymax>)' plots the
     implicit function defined by <fcn>, with variable <x> taking values
     from <xmin> to <xmax>, and variable <y> taking values from <ymin>
     to <ymax>.

     This object is affected by the following graphic options:
     `ip_grid', `ip_grid_in', `line_width', `line_type', `key' and
     `color'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(terminal  = eps,
                       grid      = true,
                       line_type = solid,
                       key       = "y^2=x^3-2*x+1",
                       implicit(y^2=x^3-2*x+1, x, -4,4, y, -4,4),
                       line_type = dots,
                       key       = "x^3+y^3 = 3*x*y^2-x-1",
                       implicit(x^3+y^3 = 3*x*y^2-x-1, x,-4,4, y,-4,4),
                       title     = "Two implicit functions" )$

     3D

     `implicit (<fcn>,<x>,<xmin>,<xmax>, <y>,<ymin>,<ymax>,
     <z>,<zmin>,<zmax>)' plots the implicit surface defined by <fcn>,
     with variable <x> taking values from <xmin> to <xmax>, variable
     <y> taking values from <ymin> to <ymax> and variable <z> taking
     values from <zmin> to <zmax>. This object implements the marching
     cubes algorithm.

     This object is affected by the following graphic options:
     `x_voxel', `y_voxel', `z_voxel', `line_width', `line_type', `key',
     `wired_surface', `enhanced3d',and `color'.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(
                  color=blue,
                  implicit((x^2+y^2+z^2-1)*(x^2+(y-1.5)^2+z^2-0.5)=0.015,
                           x,-1,1,y,-1.2,2.3,z,-1,1),
                  surface_hide=true);

Graphic object: label ([<string>,<x>,<y>],...)
Graphic object: label ([<string>,<x>,<y>,<z>],...)

     Writes labels in 2D and 3D.

     Colored labels work only with Gnuplot 4.3. This is a known bug in
     package `draw'.

     This object is affected by the following graphic options:
     `label_alignment', `label_orientation' and `color'.

     2D

     `label([<string>,<x>,<y>])' writes the <string> at point
     `[<x>,<y>]'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(yrange = [0.1,1.4],
                       color = red,
                       label(["Label in red",0,0.3]),
                       color = "#0000ff",
                       label(["Label in blue",0,0.6]),
                       color = light_blue,
                       label(["Label in light-blue",0,0.9],
                             ["Another light-blue",0,1.2])  )$

     3D

     `label([<string>,<x>,<y>,<z>])' writes the <string> at point
     `[<x>,<y>,<z>]'.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3),
                       color = red,
                       label(["UP 1",-2,0,3], ["UP 2",1.5,0,4]),
                       color = blue,
                       label(["DOWN 1",2,0,-3]) )$

Graphic object: mesh (<row_1>,<row_2>,...)

     Draws a quadrangular mesh in 3D.

     3D

     Argument <row_i> is a list of <n> 3D points of the form
     `[[x_i1,y_i1,z_i1], ...,[x_in,y_in,z_in]]', and all rows are of
     equal length. All these points define an arbitrary surface in 3D
     and in some sense it's a generalization of the `elevation_grid'
     object.

     This object is affected by the following graphic options:
     `line_type', `line_width', `color', `key', `wired_surface',
     `enhanced3d', and `transform'.

     Examples:

     A simple example.

          (%i1) load(draw)$
          (%i2) draw3d(
                   mesh([[1,1,3],   [7,3,1],[12,-2,4],[15,0,5]],
                        [[2,7,8],   [4,3,1],[10,5,8], [12,7,1]],
                        [[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $

     Plotting a triangle in 3D.

          (%i1) load(draw)$
          (%i2) draw3d(
                  line_width = 2,
                  mesh([[1,0,0],[0,1,0]],
                       [[0,0,1],[0,0,1]])) $

     Two quadrilaterals.

          (%i1) load(draw)$
          (%i2) draw3d(
                  surface_hide = true,
                  line_width   = 3,
                  color = red,
                  mesh([[0,0,0], [0,1,0]],
                       [[2,0,2], [2,2,2]]),
                  color = blue,
                  mesh([[0,0,2], [0,1,2]],
                       [[2,0,4], [2,2,4]])) $

Graphic object: parametric (<xfun>,<yfun>,<par>,<parmin>,<parmax>)
Graphic object: parametric

          (<xfun>,<yfun>,<zfun>,<par>,<parmin>,<parmax>)
     Draws parametric functions in 2D and 3D.

     This object is affected by the following graphic options: `nticks',
     `line_width', `line_type', `key', `color' and `enhanced3d'.

     2D

     The command `parametric(<xfun>, <yfun>, <par>, <parmin>,
     <parmax>)' plots the parametric function `[<xfun>, <yfun>]', with
     parameter <par> taking values from <parmin> to <parmax>.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(explicit(exp(x),x,-1,3),
                       color = red,
                       key   = "This is the parametric one!!",
                       parametric(2*cos(rrr),rrr^2,rrr,0,2*%pi))$

     3D

     `parametric(<xfun>, <yfun>, <zfun>, <par>, <parmin>, <parmax>)'
     plots the parametric curve `[<xfun>, <yfun>, <zfun>]', with
     parameter <par> taking values from <parmin> to <parmax>.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3),
                       color = royalblue,
                       parametric(cos(5*u)^2,sin(7*u),u-2,u,0,2),
                       color      = turquoise,
                       line_width = 2,
                       parametric(t^2,sin(t),2+t,t,0,2),
                       surface_hide = true,
                       title = "Surface & curves" )$

Graphic object: parametric_surface (<xfun>, <yfun>, <zfun>, <par1>, <par1min>, <par1max>, <par2>, <par2min>, <par2max>)

     Draws parametric surfaces in 3D.

     3D

     The command `parametric_surface(<xfun>, <yfun>, <zfun>, <par1>,
     <par1min>, <par1max>, <par2>, <par2min>, <par2max>)' plots the
     parametric surface `[<xfun>, <yfun>, <zfun>]', with parameter
     <par1> taking values from <par1min> to <par1max> and parameter
     <par2> taking values from <par2min> to <par2max>.

     This object is affected by the following graphic options:
     `draw_realpart', `xu_grid', `yv_grid', `line_type', `line_width',
     `key', `wired_surface', `enhanced3d',  and `color'.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(title          = "Sea shell",
                       xu_grid        = 100,
                       yv_grid        = 25,
                       view           = [100,20],
                       surface_hide   = true,
                       parametric_surface(0.5*u*cos(u)*(cos(v)+1),
                                     0.5*u*sin(u)*(cos(v)+1),
                                     u*sin(v) - ((u+3)/8*%pi)^2 - 20,
                                     u, 0, 13*%pi, v, -%pi, %pi) )$

Graphic object: points ([[<x1>,<y1>], [<x2>,<y2>],...])
Graphic object: points ([<x1>,<x2>,...], [<y1>,<y2>,...])
Graphic object: points ([<y1>,<y2>,...])
Graphic object: points ([[<x1>,<y1>,<z1>], [<x2>,<y2>,<z2>],...])
Graphic object: points ([<x1>,<x2>,...], [<y1>,<y2>,...], [<z1>,<z2>,...])
Graphic object: points (<matrix>)
Graphic object: points (<1d_y_array>)
Graphic object: points (<1d_x_array>, <1d_y_array>)
Graphic object: points (<1d_x_array>, <1d_y_array>, <1d_z_array>)
Graphic object: points (<2d_xy_array>)
Graphic object: points (<2d_xyz_array>)

     Draws points in 2D and 3D.

     This object is affected by the following graphic options:
     `point_size', `point_type', `points_joined', `line_width', `key',
     `line_type' and `color'. In 3D mode, it is also affected by
     `enhanced3d'.

     2D

     `points ([[<x1>,<y1>], [<x2>,<y2>],...])' or `points
     ([<x1>,<x2>,...], [<y1>,<y2>,...])' plots points `[x1,y1]',
     `[x2,y2]', etc. If abscissas are not given, they are set to
     consecutive positive integers, so that `points ([<y1>,<y2>,...])'
     draws points `[1,<y1>]', `[2,<y2>]', etc.  If <matrix> is a
     two-column or two-row matrix, `points (<matrix>)' draws the
     associated points. If <matrix> is a one-column or one-row matrix,
     abscissas are assigned automatically.

     If <1d_y_array> is a 1D lisp array of numbers, `points
     (<1d_y_array>)' plots them setting abscissas to consecutive
     positive integers. `points (<1d_x_array>, <1d_y_array>)' plots
     points with their coordinates taken from the two arrays passed as
     arguments. If <2d_xy_array> is a 2D array with two columns, or
     with two rows, `points (<2d_xy_array>)' plots the corresponding
     points on the plane.

     Examples:

     Two types of arguments for `points', a list of pairs and two lists
     of separate coordinates.
          (%i1) load(draw)$
          (%i2) draw2d(
                  key = "Small points",
                  points(makelist([random(20),random(50)],k,1,10)),
                  point_type    = circle,
                  point_size    = 3,
                  points_joined = true,
                  key           = "Great points",
                  points(makelist(k,k,1,20),makelist(random(30),k,1,20)),
                  point_type    = filled_down_triangle,
                  key           = "Automatic abscissas",
                  color         = red,
                  points([2,12,8]))$

     Drawing impulses.
          (%i1) load(draw)$
          (%i2) draw2d(
                  points_joined = impulses,
                  line_width    = 2,
                  color         = red,
                  points(makelist([random(20),random(50)],k,1,10)))$

     Array with ordinates.
          (%i1) load(draw)$
          (%i2) a: make_array (flonum, 100) $
          (%i3) for i:0 thru 99 do a[i]: random(1.0) $
          (%i4) draw2d(points(a)) $

     Two arrays with separate coordinates.
          (%i1) load(draw)$
          (%i2) x: make_array (flonum, 100) $
          (%i3) y: make_array (fixnum, 100) $
          (%i4) for i:0 thru 99 do (
                  x[i]: float(i/100),
                  y[i]: random(10) ) $
          (%i5) draw2d(points(x, y)) $

     A two-column 2D array.
          (%i1) load(draw)$
          (%i2) xy: make_array(flonum, 100, 2) $
          (%i3) for i:0 thru 99 do (
                  xy[i, 0]: float(i/100),
                  xy[i, 1]: random(10) ) $
          (%i4) draw2d(points(xy)) $

     Drawing an array filled with function `read_array'.
          (%i1) load(draw)$
          (%i2) a: make_array(flonum,100) $
          (%i3) read_array (file_search ("pidigits.data"), a) $
          (%i4) draw2d(points(a)) $

     3D

     `points([[<x1>, <y1>, <z1>], [<x2>, <y2>, <z2>], ...])' or
     `points([<x1>, <x2>, ...], [<y1>, <y2>, ...], [<z1>, <z2>,...])'
     plots points `[<x1>, <y1>, <z1>]', `[<x2>, <y2>, <z2>]', etc.  If
     <matrix> is a three-column or three-row matrix, `points
     (<matrix>)' draws the associated points.

     When arguments are lisp arrays, `points (<1d_x_array>,
     <1d_y_array>, <1d_z_array>)' takes coordinates from the three 1D
     arrays.  If <2d_xyz_array> is a 2D array with three columns, or
     with three rows, `points (<2d_xyz_array>)' plots the corresponding
     points.

     Examples:

     One tridimensional sample,
          (%i1) load(draw)$
          (%i2) load (numericalio)$
          (%i3) s2 : read_matrix (file_search ("wind.data"))$
          (%i4) draw3d(title = "Daily average wind speeds",
                       point_size = 2,
                       points(args(submatrix (s2, 4, 5))) )$

     Two tridimensional samples,
          (%i1) load(draw)$
          (%i2) load (numericalio)$
          (%i3) s2 : read_matrix (file_search ("wind.data"))$
          (%i4) draw3d(
                   title = "Daily average wind speeds. Two data sets",
                   point_size = 2,
                   key        = "Sample from stations 1, 2 and 3",
                   points(args(submatrix (s2, 4, 5))),
                   point_type = 4,
                   key        = "Sample from stations 1, 4 and 5",
                   points(args(submatrix (s2, 2, 3))) )$

     Unidimensional arrays,
          (%i1) load(draw)$
          (%i2) x: make_array (fixnum, 10) $
          (%i3) y: make_array (fixnum, 10) $
          (%i4) z: make_array (fixnum, 10) $
          (%i5) for i:0 thru 9 do (
                  x[i]: random(10),
                  y[i]: random(10),
                  z[i]: random(10) ) $
          (%i6) draw3d(points(x,y,z)) $

     Bidimensional colored array,
          (%i1) load(draw)$
          (%i2) xyz: make_array(fixnum, 10, 3) $
          (%i3) for i:0 thru 9 do (
                  xyz[i, 0]: random(10),
                  xyz[i, 1]: random(10),
                  xyz[i, 2]: random(10) ) $
          (%i4) draw3d(
                   enhanced3d = true,
                   points_joined = true,
                   points(xyz)) $

     Color numbers explicitly specified by the user.
          (%i1) load(draw)$
          (%i2) pts: makelist([t,t^2,cos(t)], t, 0, 15)$
          (%i3) col_num: makelist(k, k, 1, length(pts))$
          (%i4) draw3d(
                  enhanced3d = ['part(col_num,k),k],
                  point_size = 3,
                  point_type = filled_circle,
                  points(pts))$

Graphic object: polar (<radius>,<ang>,<minang>,<maxang>)

     Draws 2D functions defined in polar coordinates.

     2D

     `polar (<radius>,<ang>,<minang>,<maxang>)' plots function
     `<radius>(<ang>)' defined in polar coordinates, with variable
     <ang> taking values from <minang> to <maxang>.

     This object is affected by the following graphic options: `nticks',
     `line_width', `line_type', `key' and `color'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(user_preamble = "set grid polar",
                       nticks        = 200,
                       xrange        = [-5,5],
                       yrange        = [-5,5],
                       color         = blue,
                       line_width    = 3,
                       title         = "Hyperbolic Spiral",
                       polar(10/theta,theta,1,10*%pi) )$

Graphic object: polygon ([[<x1>, <y1>], [<x2>, <y2>], ...])
Graphic object: polygon ([<x1>, <x2>, ...], [<y1>, <y2>, ...])

     Draws polygons in 2D.

     2D

     The commands `polygon([[<x1>, <y1>], [<x2>, <y2>], ...])' or
     `polygon([<x1>, <x2>, ...], [<y1>, <y2>, ...])' plot on the plane
     a polygon with vertices `[<x1>, <y1>]', `[<x2>, <y2>]', etc.

     This object is affected by the following graphic options:
     `transparent', `fill_color', `border', `line_width', `key',
     `line_type' and `color'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(color      = "#e245f0",
                       line_width = 8,
                       polygon([[3,2],[7,2],[5,5]]),
                       border      = false,
                       fill_color  = yellow,
                       polygon([[5,2],[9,2],[7,5]]) )$

Graphic object: quadrilateral (<point_1>, <point_2>, <point_3>, <point_4>)

     Draws a quadrilateral.

     2D

     `quadrilateral([<x1>, <y1>], [<x2>, <y2>], [<x3>, <y3>], [<x4>,
     <y4>])' draws a quadrilateral with vertices `[<x1>, <y1>]',
     `[<x2>, <y2>]', `[<x3>, <y3>]', and `[<x4>, <y4>]'.

     This object is affected by the following graphic options:
     `transparent', `fill_color', `border', `line_width', `key',
     `xaxis_secondary', `yaxis_secondary', `line_type', `transform' and
     `color'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(
                  quadrilateral([1,1],[2,2],[3,-1],[2,-2]))$

     3D

     `quadrilateral([<x1>, <y1>, <z1>], [<x2>, <y2>, <z2>], [<x3>,
     <y3>, <z3>], [<x4>, <y4>, <z4>])' draws a quadrilateral with
     vertices `[<x1>, <y1>, <z1>]', `[<x2>, <y2>, <z2>]', `[<x3>, <y3>,
     <z3>]', and `[<x4>, <y4>, <z4>]'.

     This object is affected by the following graphic options:
     `line_type', `line_width', `color', `key', `enhanced3d', and
     `transform'.

Graphic object: rectangle ([<x1>,<y1>], [<x2>,<y2>])

     Draws rectangles in 2D.

     2D

     `rectangle ([<x1>,<y1>], [<x2>,<y2>])' draws a rectangle with
     opposite vertices `[<x1>,<y1>]' and `[<x2>,<y2>]'.

     This object is affected by the following graphic options:
     `transparent', `fill_color', `border', `line_width', `key',
     `line_type' and `color'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(fill_color  = red,
                       line_width  = 6,
                       line_type   = dots,
                       transparent = false,
                       fill_color  = blue,
                       rectangle([-2,-2],[8,-1]), /* opposite vertices */
                       transparent = true,
                       line_type   = solid,
                       line_width  = 1,
                       rectangle([9,4],[2,-1.5]),
                       xrange      = [-3,10],
                       yrange      = [-3,4.5] )$

Graphic object: region

          (<expr>,<var1>,<minval1>,<maxval1>,<var2>,<minval2>,<maxval2>)
     Plots a region on the plane defined by inequalities.

     2D <expr> is an expression formed by inequalities and boolean
     operators `and', `or', and `not'. The region is bounded by the
     rectangle defined by [<minval1>, <maxval1>] and [<minval2>,
     <maxval2>].

     This object is affected by the following graphic options:
     `fill_color', `key', `x_voxel', and `y_voxel'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(
                  x_voxel = 30,
                  y_voxel = 30,
                  region(x^2+y^2<1 and x^2+y^2 > 1/2,
                         x, -1.5, 1.5, y, -1.5, 1.5));

Graphic object: spherical (<radius>, <azi>, <minazi>, <maxazi>, <zen>, <minzen>, <maxzen>)

     Draws 3D functions defined in spherical coordinates.

     3D

     `spherical(<radius>, <azi>, <minazi>, <maxazi>, <zen>, <minzen>,
     <maxzen>)' plots the function `<radius>(<azi>, <zen>)' defined in
     spherical coordinates, with azimuth <azi> taking values from
     <minazi> to <maxazi> and zenith <zen> taking values from <minzen>
     to <maxzen>.

     This object is affected by the following graphic options:
     `xu_grid', `yv_grid', `line_type', `key', `wired_surface',
     `enhanced3d' and `color'.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(spherical(1,a,0,2*%pi,z,0,%pi))$

Graphic object: triangle (<point_1>, <point_2>, <point_3>)

     Draws a triangle.

     2D

     `triangle ([<x1>,<y1>], [<x2>,<y2>], [<x3>,<y3>])' draws a
     triangle with vertices `[<x1>,<y1>]', `[<x2>,<y2>]', and
     `[<x3>,<y3>]'.

     This object is affected by the following graphic options:
     `transparent', `fill_color', `border', `line_width', `key',
     `xaxis_secondary', `yaxis_secondary', `line_type', `transform',
     and `color'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(
                  triangle([1,1],[2,2],[3,-1]))$

     3D

     `triangle ([<x1>,<y1>,<z1>], [<x2>,<y2>,<z2>], [<x3>,<y3>,<z3>])'
     draws a triangle with vertices `[<x1>,<y1>,<z1>]',
     `[<x2>,<y2>,<z2>]', and `[<x3>,<y3>,<z3>]'.

     This object is affected by the following graphic options:
     `line_type', `line_width', `color', `key', `enhanced3d', and
     `transform'.

Graphic object: tube (<xfun>,<yfun>,<zfun>,<rfun>,<p>,<pmin>,<pmax>)

     Draws a tube in 3D with varying diameter.

     3D

     `[<xfun>,<yfun>,<zfun>]' is the parametric curve with parameter
     <p> taking values from <pmin> to <pmax>. Circles of radius <rfun>
     are placed with their centers on the parametric curve and
     perpendicular to it.

     This object is affected by the following graphic options:
     `xu_grid', `yv_grid', `line_type', `line_width', `key',
     `wired_surface', `enhanced3d', `color', and `tube_extremes'.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(
                  enhanced3d = true,
                  xu_grid = 50,
                  tube(cos(a), a, 0, cos(a/10)^2,
                       a, 0, 4*%pi) )$

Graphic object: vector ([<x>,<y>], [<dx>,<dy>])
Graphic object: vector ([<x>,<y>,<z>], [<dx>,<dy>,<dz>])

     Draws vectors in 2D and 3D.

     This object is affected by the following graphic options:
     `head_both', `head_length', `head_angle', `head_type',
     `line_width', `line_type', `key' and `color'.

     2D

     `vector([<x>,<y>], [<dx>,<dy>])' plots vector `[<dx>,<dy>]' with
     origin in `[<x>,<y>]'.

     Example:

          (%i1) load(draw)$
          (%i2) draw2d(xrange      = [0,12],
                       yrange      = [0,10],
                       head_length = 1,
                       vector([0,1],[5,5]), /* default type */
                       head_type = 'empty,
                       vector([3,1],[5,5]),
                       head_both = true,
                       head_type = 'nofilled,
                       line_type = dots,
                       vector([6,1],[5,5]))$

     3D

     `vector([<x>,<y>,<z>], [<dx>,<dy>,<dz>])' plots vector
     `[<dx>,<dy>,<dz>]' with origin in `[<x>,<y>,<z>]'.

     Example:

          (%i1) load(draw)$
          (%i2) draw3d(color = cyan,
                       vector([0,0,0],[1,1,1]/sqrt(3)),
                       vector([0,0,0],[1,-1,0]/sqrt(2)),
                       vector([0,0,0],[1,1,-2]/sqrt(6)) )$

48.3 Functions and Variables for pictures

Function: get_pixel (<pic>,<x>,<y>)

     Returns pixel from picture. Coordinates <x> and <y> range from 0 to
     `width-1' and `height-1', respectively.

Function: make_level_picture (<data>)
Function: make_level_picture (<data>,<width>,<height>)

     Returns a levels <picture> object. `make_level_picture (<data>)'
     builds the <picture> object from matrix <data>.
     `make_level_picture (<data>,<width>,<height>)' builds the object
     from a list of numbers; in this case, both the <width> and the
     <height> must be given.

     The returned <picture> object contains the following four parts:

       1. symbol `level'

       2. image width

       3. image height

       4. an integer array with pixel data ranging from 0 to 255.
          Argument <data> must contain only numbers ranged from 0 to
          255; negative numbers are substituted by 0, and those which
          are greater than 255 are set to 255.

     Example:

     Level picture from matrix.
          (%i1) load(draw)$
          (%i2) make_level_picture(matrix([3,2,5],[7,-9,3000]));
          (%o2)         picture(level, 3, 2, {Array:  #(3 2 5 7 0 255)})

     Level picture from numeric list.
          (%i1) load(draw)$
          (%i2) make_level_picture([-2,0,54,%pi],2,2);
          (%o2)            picture(level, 2, 2, {Array:  #(0 0 54 3)})

Function: make_rgb_picture (<redlevel>,<greenlevel>,<bluelevel>)

     Returns an rgb-coloured <picture> object. All three arguments must
     be levels picture; with red, green and blue levels.

     The returned <picture> object contains the following four parts:

       1. symbol `rgb'

       2. image width

       3. image height

       4. an integer array of length <3*width*height> with pixel data
          ranging from 0 to 255. Each pixel is represented by three
          consecutive numbers (red, green, blue).

     Example:

          (%i1) load(draw)$
          (%i2) red: make_level_picture(matrix([3,2],[7,260]));
          (%o2)           picture(level, 2, 2, {Array:  #(3 2 7 255)})
          (%i3) green: make_level_picture(matrix([54,23],[73,-9]));
          (%o3)           picture(level, 2, 2, {Array:  #(54 23 73 0)})
          (%i4) blue: make_level_picture(matrix([123,82],[45,32.5698]));
          (%o4)          picture(level, 2, 2, {Array:  #(123 82 45 33)})
          (%i5) make_rgb_picture(red,green,blue);
          (%o5) picture(rgb, 2, 2,
                        {Array:  #(3 54 123 2 23 82 7 73 45 255 0 33)})

Function: negative_picture (<pic>)

     Returns the negative of a (<level> or <rgb>) picture.

Function: picture_equalp (<x>,<y>)

     Returns `true' in case of equal pictures, and `false' otherwise.

Function: picturep (<x>)

     Returns `true' if the argument is a well formed image, and `false'
     otherwise.

Function: read_xpm (<xpm_file>)

     Reads a file in xpm and returns a picture object.

Function: rgb2level (<pic>)

     Transforms an <rgb> picture into a <level> one by averaging the
     red, green and blue channels.

Function: take_channel (<im>,<color>)

     If argument <color> is `red', `green' or `blue', function
     `take_channel' returns the corresponding color channel of picture
     <im>.  Example:

          (%i1) load(draw)$
          (%i2) red: make_level_picture(matrix([3,2],[7,260]));
          (%o2)           picture(level, 2, 2, {Array:  #(3 2 7 255)})
          (%i3) green: make_level_picture(matrix([54,23],[73,-9]));
          (%o3)           picture(level, 2, 2, {Array:  #(54 23 73 0)})
          (%i4) blue: make_level_picture(matrix([123,82],[45,32.5698]));
          (%o4)          picture(level, 2, 2, {Array:  #(123 82 45 33)})
          (%i5) make_rgb_picture(red,green,blue);
          (%o5) picture(rgb, 2, 2,
                        {Array:  #(3 54 123 2 23 82 7 73 45 255 0 33)})
          (%i6) take_channel(%,'green);  /* simple quote!!! */
          (%o6)           picture(level, 2, 2, {Array:  #(54 23 73 0)})

48.4 Functions and Variables for worldmap

This package automatically loads package `draw'.
Global variable: boundaries_array

     Default value: `false'

     `boundaries_array' is where the graphic object `geomap' looks for
     boundaries coordinates.

     Each component of `boundaries_array' is an array of floating point
     quantities, the coordinates of a polygonal segment or map boundary.

     See also `geomap'.

Function: numbered_boundaries (<nlist>)

     Draws a list of polygonal segments (boundaries), labeled by its
     numbers (`boundaries_array' coordinates). This is of great help
     when building new geographical entities.

     Example:

     Map of Europe labeling borders with their component number in
     `boundaries_array'.
          (%i1) load(worldmap)$
          (%i2) european_borders:
                     region_boundaries(-31.81,74.92,49.84,32.06)$
          (%i3) numbered_boundaries(european_borders)$

Function: make_poly_continent (<continent_name>)
Function: make_poly_continent (<country_list>)

     Makes the necessary polygons to draw a colored continent or a list
     of countries.

     Example:

          (%i1) load(worldmap)$
          (%i2) /* A continent */
                make_poly_continent(Africa)$
          (%i3) apply(draw2d, %)$
          (%i4) /* A list of countries */
                make_poly_continent([Germany,Denmark,Poland])$
          (%i5) apply(draw2d, %)$

Function: make_poly_country (<country_name>)

     Makes the necessary polygons to draw a colored country.  If
     islands exist, one country can be defined with more than just one
     polygon.

     Example:

          (%i1) load(worldmap)$
          (%i2) make_poly_country(India)$
          (%i3) apply(draw2d, %)$

Function: make_polygon (<nlist>)

     Returns a `polygon' object from boundary indices. Argument <nlist>
     is a list of components of `boundaries_array'.

     Example:

     Bhutan is defined by boundary numbers 171, 173 and 1143, so that
     `make_polygon([171,173,1143])' appends arrays of coordinates
     `boundaries_array[171]', `boundaries_array[173]' and
     `boundaries_array[1143]' and returns a `polygon' object suited to
     be plotted by `draw'. To avoid an error message, arrays must be
     compatible in the sense that any two consecutive arrays have two
     coordinates in the extremes in common. In this example, the two
     first components of `boundaries_array[171]' are equal to the last
     two coordinates of `boundaries_array[173]', and the two first of
     `boundaries_array[173]' are equal to the two first of
     `boundaries_array[1143]'; in conclussion, boundary numbers 171,
     173 and 1143 (in this order) are compatible and the colored
     polygon can be drawn.
          (%i1) load(worldmap)$
          (%i2) Bhutan;
          (%o2)                        [[171, 173, 1143]]
          (%i3) boundaries_array[171];
          (%o3) {Array:
                 #(88.750549 27.14727 88.806351 27.25305 88.901367 27.282221
                   88.917877 27.321039)}
          (%i4) boundaries_array[173];
          (%o4) {Array:
                 #(91.659554 27.76511 91.6008 27.66666 91.598022 27.62499
                   91.631348 27.536381 91.765533 27.45694 91.775253 27.4161
                   92.007751 27.471939 92.11441 27.28583 92.015259 27.168051
                   92.015533 27.08083 92.083313 27.02277 92.112183 26.920271
                   92.069977 26.86194 91.997192 26.85194 91.915253 26.893881
                   91.916924 26.85416 91.8358 26.863331 91.712479 26.799999
                   91.542191 26.80444 91.492188 26.87472 91.418854 26.873329
                   91.371353 26.800831 91.307457 26.778049 90.682457 26.77417
                   90.392197 26.903601 90.344131 26.894159 90.143044 26.75333
                   89.98996 26.73583 89.841919 26.70138 89.618301 26.72694
                   89.636093 26.771111 89.360786 26.859989 89.22081 26.81472
                   89.110237 26.829161 88.921631 26.98777 88.873016 26.95499
                   88.867737 27.080549 88.843307 27.108601 88.750549
                   27.14727)}
          (%i5) boundaries_array[1143];
          (%o5) {Array:
                 #(91.659554 27.76511 91.666924 27.88888 91.65831 27.94805
                   91.338028 28.05249 91.314972 28.096661 91.108856 27.971109
                   91.015808 27.97777 90.896927 28.05055 90.382462 28.07972
                   90.396088 28.23555 90.366074 28.257771 89.996353 28.32333
                   89.83165 28.24888 89.58609 28.139999 89.35997 27.87166
                   89.225517 27.795 89.125793 27.56749 88.971077 27.47361
                   88.917877 27.321039)}
          (%i6) Bhutan_polygon: make_polygon([171,173,1143])$
          (%i7) draw2d(Bhutan_polygon)$

Function: region_boundaries (<x1>,<y1>,<x2>,<y2>)

     Detects polygonal segments of global variable `boundaries_array'
     fully contained in the rectangle with vertices (<x1>,<y1>) -upper
     left- and (<x2>,<y2>) -bottom right-.

     Example:

     Returns segment numbers for plotting southern Italy.
          (%i1) load(worldmap)$
          (%i2) region_boundaries(10.4,41.5,20.7,35.4);
          (%o2)                [1846, 1863, 1864, 1881, 1888, 1894]
          (%i3) draw2d(geomap(%))$

Function: region_boundaries_plus (<x1>,<y1>,<x2>,<y2>)

     Detects polygonal segments of global variable `boundaries_array'
     containing at least one vertex in the rectangle defined by
     vertices (<x1>,<y1>) -upper left- and (<x2>,<y2>) -bottom right-.

     Example:

          (%i1) load(worldmap)$
          (%i2) region_boundaries_plus(10.4,41.5,20.7,35.4);
          (%o2) [1060, 1062, 1076, 1835, 1839, 1844, 1846, 1858,
                 1861, 1863, 1864, 1871, 1881, 1888, 1894, 1897]
          (%i3) draw2d(geomap(%))$

Graphic object: geomap (<numlist>)
Graphic object: geomap (<numlist>,<3Dprojection>)

     Draws cartographic maps in 2D and 3D.

     2D

     This function works together with global variable
     `boundaries_array'.

     Argument <numlist> is a list containing numbers or lists of
     numbers.  All these numbers must be integers greater or equal than
     zero, representing the components of global array
     `boundaries_array'.

     Each component of `boundaries_array' is an array of floating point
     quantities, the coordinates of a polygonal segment or map boundary.

     `geomap (<numlist>)' flattens its arguments and draws the
     associated boundaries in `boundaries_array'.

     This object is affected by the following graphic options:
     `line_width', `line_type' and `color'.

     Examples:

     A simple map defined by hand:
          (%i1) load(worldmap)$
          (%i2) /* Vertices of boundary #0: {(1,1),(2,5),(4,3)} */
             ( bnd0: make_array(flonum,6),
               bnd0[0]:1.0, bnd0[1]:1.0, bnd0[2]:2.0,
               bnd0[3]:5.0, bnd0[4]:4.0, bnd0[5]:3.0 )$
          (%i3) /* Vertices of boundary #1: {(4,3),(5,4),(6,4),(5,1)} */
             ( bnd1: make_array(flonum,8),
               bnd1[0]:4.0, bnd1[1]:3.0, bnd1[2]:5.0, bnd1[3]:4.0,
               bnd1[4]:6.0, bnd1[5]:4.0, bnd1[6]:5.0, bnd1[7]:1.0)$
          (%i4) /* Vertices of boundary #2: {(5,1), (3,0), (1,1)} */
             ( bnd2: make_array(flonum,6),
               bnd2[0]:5.0, bnd2[1]:1.0, bnd2[2]:3.0,
               bnd2[3]:0.0, bnd2[4]:1.0, bnd2[5]:1.0 )$
          (%i5) /* Vertices of boundary #3: {(1,1), (4,3)} */
             ( bnd3: make_array(flonum,4),
               bnd3[0]:1.0, bnd3[1]:1.0, bnd3[2]:4.0, bnd3[3]:3.0)$
          (%i6) /* Vertices of boundary #4: {(4,3), (5,1)} */
             ( bnd4: make_array(flonum,4),
               bnd4[0]:4.0, bnd4[1]:3.0, bnd4[2]:5.0, bnd4[3]:1.0)$
          (%i7) /* Pack all together in boundaries_array */
             ( boundaries_array: make_array(any,5),
               boundaries_array[0]: bnd0, boundaries_array[1]: bnd1,
               boundaries_array[2]: bnd2, boundaries_array[3]: bnd3,
               boundaries_array[4]: bnd4 )$
          (%i8) draw2d(geomap([0,1,2,3,4]))$

     The auxiliary package `worldmap' sets the global variable
     `boundaries_array' to real world boundaries in (longitude,
     latitude) coordinates. These data are in the public domain and
     come from `http://www-cger.nies.go.jp/grid-e/gridtxt/grid19.html'.
     Package `worldmap' defines also boundaries for countries,
     continents and coastlines as lists with the necessary components of
     `boundaries_array' (see file `share/draw/worldmap.mac' for more
     information). Package `worldmap' automatically loads package
     `worldmap'.
          (%i1) load(worldmap)$
          (%i2) c1: gr2d(geomap(Canada,United_States,
                                Mexico,Cuba))$
          (%i3) c2: gr2d(geomap(Africa))$
          (%i4) c3: gr2d(geomap(Oceania,China,Japan))$
          (%i5) c4: gr2d(geomap(France,Portugal,Spain,
                                Morocco,Western_Sahara))$
          (%i6) draw(columns  = 2,
                     c1,c2,c3,c4)$

     Package `worldmap' is also useful for plotting countries as
     polygons. In this case, graphic object `geomap' is no longer
     necessary and the `polygon' object is used instead. Since lists
     are now used and not arrays, maps rendering will be slower. See
     also `make_poly_country' and `make_poly_continent' to understand
     the following code.
          (%i1) load(worldmap)$
          (%i2) mymap: append(
             [color      = white],  /* borders are white */
             [fill_color = red],             make_poly_country(Bolivia),
             [fill_color = cyan],            make_poly_country(Paraguay),
             [fill_color = green],           make_poly_country(Colombia),
             [fill_color = blue],            make_poly_country(Chile),
             [fill_color = "#23ab0f"],       make_poly_country(Brazil),
             [fill_color = goldenrod],       make_poly_country(Argentina),
             [fill_color = "midnight-blue"], make_poly_country(Uruguay))$
          (%i3) apply(draw2d, mymap)$

     3D

     `geomap (<numlist>)' projects map boundaries on the sphere of
     radius 1 centered at (0,0,0). It is possible to change the sphere
     or the projection type by using `geomap
     (<numlist>,<3Dprojection>)'.

     Available 3D projections:

        * `[spherical_projection,<x>,<y>,<z>,<r>]': projects map
          boundaries on the sphere of radius <r> centered at
          (<x>,<y>,<z>).
               (%i1) load(worldmap)$
               (%i2) draw3d(geomap(Australia), /* default projection */
                            geomap(Australia,
                                   [spherical_projection,2,2,2,3]))$

        * `[cylindrical_projection,<x>,<y>,<z>,<r>,<rc>]': re-projects
          spherical map boundaries on the cylinder of radius <rc> and
          axis passing through the poles of the globe of radius <r>
          centered at (<x>,<y>,<z>).
               (%i1) load(worldmap)$
               (%i2) draw3d(geomap([America_coastlines,Eurasia_coastlines],
                                   [cylindrical_projection,2,2,2,3,4]))$

        * `[conic_projection,<x>,<y>,<z>,<r>,<alpha>]': re-projects
          spherical map boundaries on the cones of angle <alpha>, with
          axis passing through the poles of the globe of radius <r>
          centered at (<x>,<y>,<z>). Both the northern and southern
          cones are tangent to sphere.
               (%i1) load(worldmap)$
               (%i2) draw3d(geomap(World_coastlines,
                                   [conic_projection,0,0,0,1,90]))$

     See also
     `http://www.telefonica.net/web2/biomates/maxima/gpdraw/geomap' for
     more elaborated examples.

49 drawdf


Introduction to drawdf
Functions and Variables for drawdf

49.1 Introduction to drawdf

The function `drawdf' draws the direction field of a first-order
Ordinary Differential Equation (ODE) or a system of two autonomous
first-order ODE's.

   Since this is an additional package, in order to use it you must
first load it with `load(drawdf)'.  Drawdf is built upon the `draw'
package, which requires Gnuplot 4.2.

   To plot the direction field of a single ODE, the ODE must be written
in the form:
            dy
            -- = F(x,y)
            dx

   and the function <F> should be given as the argument for `drawdf'.
If the independent and dependent variables are not <x>, and <y>, as in
the equation above, then those two variables should be named explicitly
in a list given as an argument to the drawdf command (see the examples).

   To plot the direction field of a set of two autonomous ODE's, they
must be written in the form
            dx             dy
            -- = G(x,y)    -- = F(x,y)
            dt             dt

   and the argument for `drawdf' should be a list with the two
functions <G> and <F>, in that order; namely, the first expression in
the list will be taken to be the time derivative of the variable
represented on the horizontal axis, and the second expression will be
the time derivative of the variable represented on the vertical axis.
Those two variables do not have to be <x> and <y>, but if they are not,
then the second argument given to drawdf must be another list naming
the two variables, first the one on the horizontal axis and then the
one on the vertical axis.

   If only one ODE is given, `drawdf' will implicitly admit `x=t', and
`G(x,y)=1', transforming the non-autonomous equation into a system of
two autonomous equations.

49.2 Functions and Variables for drawdf

Function: drawdf (<dydx>, ...options and objects...)
Function: drawdf (<dvdu>, `['<u>,<v>`]', ...options and objects...)
Function: drawdf (<dvdu>, `['<u>,<umin>,<umax>`]', `['<v>,<vmin>,<vmax>`]', ...options and objects...)
Function: drawdf (`['<dxdt>,<dydt>`]', ...options and objects...)
Function: drawdf (`['<dudt>,<dvdt>`]', `['<u>,<v>`]', ...options

          and objects...)
Function: drawdf (`['<dudt>,<dvdt>`]', `['<u>,<umin>,<umax>`]', `['<v>,<vmin>,<vmax>`]', ...options and objects...)

     Function `drawdf' draws a 2D direction field with optional
     solution curves and other graphics using the `draw' package.

     The first argument specifies the derivative(s), and must be either
     an expression or a list of two expressions.  <dydx>, <dxdt> and
     <dydt> are expressions that depend on <x> and <y>.  <dvdu>, <dudt>
     and <dvdt> are expressions that depend on <u> and <v>.

     If the independent and dependent variables are not <x> and <y>,
     then their names must be specified immediately following the
     derivative(s), either as a list of two names `['<u>,<v>`]', or as
     two lists of the form `['<u>,<umin>,<umax>`]' and
     `['<v>,<vmin>,<vmax>`]'.

     The remaining arguments are graphic options, graphic objects, or
     lists containing graphic options and objects, nested to arbitrary
     depth.  The set of graphic options and objects supported by
     `drawdf' is a superset of those supported by `draw2d' and `gr2d'
     from the `draw' package.

     The arguments are interpreted sequentially: graphic options affect
     all following graphic objects.  Furthermore, graphic objects are
     drawn on the canvas in order specified, and may obscure graphics
     drawn earlier.  Some graphic options affect the global appearence
     of the scene.

     The additional graphic objects supported by `drawdf' include:
     `solns_at', `points_at', `saddles_at', `soln_at', `point_at', and
     `saddle_at'.

     The additional graphic options supported by `drawdf' include:
     `field_degree', `soln_arrows', `field_arrows', `field_grid',
     `field_color', `show_field', `tstep', `nsteps', `duration',
     `direction', `field_tstep', `field_nsteps', and `field_duration'.

     Commonly used graphic objects inherited from the `draw' package
     include: `explicit', `implicit', `parametric', `polygon',
     `points', `vector', `label', and all others supported by `draw2d'
     and `gr2d'.

     Commonly used graphic options inherited from the `draw' package
     include:
     `points_joined', `color', `point_type', `point_size', `line_width',
     `line_type', `key', `title', `xlabel', `ylabel', `user_preamble',
     `terminal', `dimensions', `file_name', and all others supported by
     `draw2d' and `gr2d'.

     See also `draw2d'.

     Users of wxMaxima or Imaxima may optionally use `wxdrawdf', which
     is identical to `drawdf' except that the graphics are drawn within
     the notebook using `wxdraw'.

     To make use of this function, write first `load(drawdf)'.

     Examples:

          (%i1) load(drawdf)$
          (%i2) drawdf(exp(-x)+y)$        /* default vars: x,y */
          (%i3) drawdf(exp(-t)+y, [t,y])$ /* default range: [-10,10] */
          (%i4) drawdf([y,-9*sin(x)-y/5], [x,1,5], [y,-2,2])$

     For backward compatibility, `drawdf' accepts most of the
     parameters supported by plotdf.

          (%i5) drawdf(2*cos(t)-1+y, [t,y], [t,-5,10], [y,-4,9],
                       [trajectory_at,0,0])$

     `soln_at' and `solns_at' draw solution curves passing through the
     specified points, using a slightly enhanced 4th-order Runge Kutta
     numerical integrator.

          (%i6) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
                       solns_at([0,0.1],[0,-0.1]),
                       color=blue, soln_at(0,0))$

     `field_degree=2' causes the field to be composed of quadratic
     splines, based on the first and second derivatives at each grid
     point.  `field_grid=['<COLS>,<ROWS>`]' specifies the number of
     columns and rows in the grid.

          (%i7) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
                       field_degree=2, field_grid=[20,15],
                       solns_at([0,0.1],[0,-0.1]),
                       color=blue, soln_at(0,0))$

     `soln_arrows=true' adds arrows to the solution curves, and (by
     default) removes them from the direction field.  It also changes
     the default colors to emphasize the solution curves.

          (%i8) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
                       soln_arrows=true,
                       solns_at([0,0.1],[0,-0.1],[0,0]))$

     `duration=40' specifies the time duration of numerical integration
     (default 10).  Integration will also stop automatically if the
     solution moves too far away from the plotted region, or if the
     derivative becomes complex or infinite.  Here we also specify
     `field_degree=2' to plot quadratic splines.  The equations below
     model a predator-prey system.

          (%i9) drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1],
                       field_degree=2, duration=40,
                       soln_arrows=true, point_at(1/2,1/2),
                       solns_at([0.1,0.2], [0.2,0.1], [1,0.8], [0.8,1],
                                [0.1,0.1], [0.6,0.05], [0.05,0.4],
                                [1,0.01], [0.01,0.75]))$

     `field_degree='solns' causes the field to be composed of many
     small solution curves computed by 4th-order Runge Kutta, with
     better results in this case.

          (%i10) drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1],
                        field_degree='solns, duration=40,
                        soln_arrows=true, point_at(1/2,1/2),
                        solns_at([0.1,0.2], [0.2,0.1], [1,0.8],
                                 [0.8,1], [0.1,0.1], [0.6,0.05],
                                 [0.05,0.4], [1,0.01], [0.01,0.75]))$

     `saddles_at' attempts to automatically linearize the equation at
     each saddle, and to plot a numerical solution corresponding to each
     eigenvector, including the separatrices.  `tstep=0.05' specifies
     the maximum time step for the numerical integrator (the default is
     0.1).  Note that smaller time steps will sometimes be used in
     order to keep the x and y steps small.  The equations below model
     a damped pendulum.

          (%i11) drawdf([y,-9*sin(x)-y/5], tstep=0.05,
                        soln_arrows=true, point_size=0.5,
                        points_at([0,0], [2*%pi,0], [-2*%pi,0]),
                        field_degree='solns,
                        saddles_at([%pi,0], [-%pi,0]))$

     `show_field=false' suppresses the field entirely.

          (%i12) drawdf([y,-9*sin(x)-y/5], tstep=0.05,
                        show_field=false, soln_arrows=true,
                        point_size=0.5,
                        points_at([0,0], [2*%pi,0], [-2*%pi,0]),
                        saddles_at([3*%pi,0], [-3*%pi,0],
                                   [%pi,0], [-%pi,0]))$

     `drawdf' passes all unrecognized parameters to `draw2d' or `gr2d',
     allowing you to combine the full power of the `draw' package with
     `drawdf'.

          (%i13) drawdf(x^2+y^2, [x,-2,2], [y,-2,2], field_color=gray,
                        key="soln 1", color=black, soln_at(0,0),
                        key="soln 2", color=red, soln_at(0,1),
                        key="isocline", color=green, line_width=2,
                        nticks=100, parametric(cos(t),sin(t),t,0,2*%pi))$

     `drawdf' accepts nested lists of graphic options and objects,
     allowing convenient use of makelist and other function calls to
     generate graphics.

          (%i14) colors : ['red,'blue,'purple,'orange,'green]$
          (%i15) drawdf([x-x*y/2, (x*y - 3*y)/4],
                        [x,2.5,3.5], [y,1.5,2.5],
                        field_color = gray,
                        makelist([ key   = concat("soln",k),
                                   color = colors[k],
                                   soln_at(3, 2 + k/20) ],
                                 k,1,5))$

50 dynamics


Introduction to dynamics
Functions and Variables for dynamics

50.1 Introduction to dynamics

The additional package `dynamics' includes several functions to create
various graphical representations of discrete dynamical systems and
fractals, and an implementation of the Runge-Kutta 4th-order numerical
method for solving systems of differential equations.

   To use the functions in this package you must first load it with
`load("dynamics")'.

   Changes introduced in Maxima 5.12

   Starting with Maxima 5.12, the dynamics package now uses the function
`plot2d' to do the graphs. The commands that produce graphics (with the
exception of `julia' and `mandelbrot') now accept any options of
`plot2d', including the option to change among the various graphical
interfaces, using different plot styles and colors, and representing
one or both axes in a logarithmic scale. The old options <domain>,
<pointsize>, <xcenter>, <xradius>, <ycenter>, <yradius>, <xaxislabel>
and <yaxislabel> are not accepted in this new version.

   All programs will now accept any variables names, and not just <x>
and <y> as in the older versions. Two required parameters have changes
in two of the programs: `evolution2d' now requires a list naming
explicitely the two independent variables, and the horizontal range for
`orbits' no longer requires a step size; the range should only specify
the variable name, and the minimum and maximum values; the number of
steps can now be changed with the option <nticks>.

50.2 Functions and Variables for dynamics

Function: chaosgame (`[['<x1>, <y1>`]'...`['<xm>, <ym>`]]', `['<x0>, <y0>`]', <b>, <n>, ..., options, ...);

     Implements the so-called chaos game: the initial point (<x0>,
     <y0>) is plotted and then one of the <m> points `['<x1>,
     <y1>`]'...`['<xm>, <ym>`]' will be selected at random. The next
     point plotted will be on the segment from the previous point
     plotted to the point chosen randomly, at a distance from the
     random point which will be <b> times that segment's length. The
     procedure is repeated <n> times.

Function: evolution (<F>, <y0>, <n>, ..., options, ...);

     Draws <n+1> points in a two-dimensional graph, where the horizontal
     coordinates of the points are the integers 0, 1, 2, ..., <n>, and
     the vertical coordinates are the corresponding values <y(n)> of the
     sequence defined by the recurrence relation
                  y(n+1) = F(y(n))

     With initial value <y(0)> equal to <y0>. <F> must be an expression
     that depends only on one variable (in the example, it depend on
     <y>, but any other variable can be used), <y0> must be a real
     number and <n> must be a positive integer.

Function: evolution2d (`['<F>, <G>`]', `['<u>, <v>`]', `['<u0>, <y0>`]', <n>, ..., options, ...);

     Shows, in a two-dimensional plot, the first <n+1> points in the
     sequence of points defined by the two-dimensional discrete
     dynamical system with recurrence relations
                  u(n+1) = F(u(n), v(n))    v(n+1) = G(u(n), v(n))

     With initial values <u0> and <v0>. <F> and <G> must be two
     expressions that depend only on two variables, <u> and <v>, which
     must be named explicitely in a list.

Function: ifs (`['<r1>, ..., <rm>`]', `['<A1>, ..., <Am>`]', `[['<x1>, <y1>`]', ..., `['<xm>, <ym>`]]', `['<x0>, <y0>`]',

          <n>, ..., options, ...);
     Implements the Iterated Function System method. This method is
     similar to the method described in the function `chaosgame', but
     instead of shrinking the segment from the current point to the
     randomly chosen point, the 2 components of that segment will be
     multiplied by the 2 by 2 matrix <Ai> that corresponds to the point
     chosen randomly.

     The random choice of one of the <m> attractive points can be made
     with a non-uniform probability distribution defined by the weights
     <r1>,...,<rm>. Those weights are given in cumulative form; for
     instance if there are 3 points with probabilities 0.2, 0.5 and
     0.3, the weights <r1>, <r2> and <r3> could be 2, 7 and 10.

Function: julia (<x>, <y>, ...<options>...)

     Creates a graphics file with the representation of the Julia set
     for the complex number (<x> + i <y>). The parameters <x> and <y>
     must be real. The file is created in the current directory or in
     the user's directory, using the XPM graphics format. The program
     may take several seconds to run and after it is finished, a
     message will be printed with the name of the file created.

     The points which do not belong to the Julia set are assigned
     different colors, according to the number of iterations it takes
     the sequence starting at that point to move out of the convergence
     circle of radius 2. The maximum number of iterations is set with
     the option <levels>; after that number of iterations, if the
     sequence is still inside the convergence circle, the point will be
     painted with the color defined by the option <color>.

     All the colors used for the points that do not belong to the Julia
     set will have the same <saturation> and <value>, but with different
     hue angles distributed uniformly between <hue> and (<hue> +
     <huerange>).

     <options> is an optional sequence of options. The list of accepted
     options is given in a section below.

Function: mandelbrot (<options>)

     Creates a graphics file with the representation of the Mandelbrot
     set. The file is created in the current directory or in the user's
     directory, using the XPM graphics format. The program may take
     several seconds to run and after it is finished, a message will be
     printed with the name of the file created.

     The points which do not belong to the Mandelbrot set are assigned
     different colors, according to the number of iterations it takes
     the sequence generated with that point to move out of the
     convergence circle o radius 2. The maximum number of iterations is
     set with the option <levels>; after that number of iterations, if
     the sequence is still inside the convergence circle, the point
     will be painted with the color defined by the option <color>.

     All the colors used for the points that do not belong to the
     Mandelbrot set will have the same <saturation> and <value>, but
     with different hue angles distributed uniformly between <hue> and
     (<hue> + <huerange>).

     <options> is an optional sequence of options. The list of accepted
     options is given in a section below.

Function: orbits (<F>, <y0>, <n1>, <n2>, [<x>, <x0>, <xf>, <xstep>], ...options...);

     Draws the orbits diagram for a family of one-dimensional discrete
     dynamical systems, with one parameter <x>; that kind of diagram is
     used to study the bifurcations of a one-dimensional discrete
     system.

     The function <F(y)> defines a sequence with a starting value of
     <y0>, as in the case of the function `evolution', but in this case
     that function will also depend on a parameter <x> that will take
     values in the interval from <x0> to <xf> with increments of
     <xstep>. Each value used for the parameter <x> is shown on the
     horizontal axis. The vertical axis will show the <n2> values of
     the sequence <y(n1+1)>,..., <y(n1+n2+1)> obtained after letting
     the sequence evolve <n1> iterations.

Function: rk (<ODE>, <var>, <initial>, <domain>)
Function: rk ([<ODE1>,...,<ODEm>], [<v1>,...,<vm>], [<init1>,...,<initm>], <domain>)

     The first form solves numerically one first-order ordinary
     differential equation, and the second form solves a system of m of
     those equations, using the 4th order Runge-Kutta method. <var>
     represents the dependent variable. <ODE> must be an expression
     that depends only on the independent and dependent variables and
     defines the derivative of the dependent variable with respect to
     the independent variable.

     The independent variable is specified with `domain', which must be
     a list of four elements as, for instance:
          [t, 0, 10, 0.1]
     the first element of the list identifies the independent variable,
     the second and third elements are the initial and final values for
     that variable, and the last element sets the increments that
     should be used within that interval.

     If <m> equations are going to be solved, there should be <m>
     dependent variables <v1>, <v2>, ..., <vm>. The initial values for
     those variables will be <init1>, <init2>, ..., <initm>.  There
     will still be just one independent variable defined by `domain',
     as in the previous case. <ODE1>, ..., <ODEm> are the expressions
     that define the derivatives of each dependent variable in terms of
     the independent variable. The only variables that may appear in
     those expressions are the independent variable and any of the
     dependent variables. It is important to give the derivatives
     <ODE1>, ..., <ODEm> in the list in exactly the same order used for
     the dependent variables; for instance, the third element in the
     list will be interpreted as the derivative of the third dependent
     variable.

     The program will try to integrate the equations from the initial
     value of the independent variable until its last value, using
     constant increments. If at some step one of the dependent
     variables takes an absolute value too large, the integration will
     be interrupted at that point. The result will be a list with as
     many elements as the number of iterations made. Each element in
     the results list is itself another list with <m>+1 elements: the
     value of the independent variable, followed by the values of the
     dependent variables corresponding to that point.

Function: staircase (<F>, <y0>, <n>, ...options...);

     Draws a staircase diagram for the sequence defined by the
     recurrence relation
                  y(n+1) = F(y(n))

     The interpretation and allowed values of the input parameters is
     the same as for the function `evolution'. A staircase diagram
     consists of a plot of the function <F(y)>, together with the line
     <G(y)> `=' <y>. A vertical segment is drawn from the point (<y0>,
     <y0>) on that line until the point where it intersects the
     function <F>. From that point a horizontal segment is drawn until
     it reaches the point (<y1>, <y1>) on the line, and the procedure
     is repeated <n> times until the point (<yn>, <yn>) is reached.

51 ezunits


Introduction to ezunits
Introduction to physical_constants
Functions and Variables for ezunits

51.1 Introduction to ezunits

`ezunits' is a package for working with dimensional quantities,
including some functions for dimensional analysis.  `ezunits' can carry
out arithmetic operations on dimensional quantities and unit
conversions.  The built-in units include Systeme Internationale (SI)
and US customary units, and other units can be declared.  See also
`physical_constants', a collection of physical constants.

   `load(ezunits)' loads this package.  `demo(ezunits)' displays
several examples.  The convenience function `known_units' returns a
list of the built-in and user-declared units, while
`display_known_unit_conversions' displays the set of known conversions
in an easy-to-read format.

   An expression a ` b represents a dimensional quantity, with `a'
indicating a nondimensional quantity and `b' indicating the dimensional
units.  A symbol can be used as a unit without declaring it as such;
unit symbols need not have any special properties.  The quantity and
unit of an expression a ` b can be extracted by the `qty' and `units'
functions, respectively.

   A symbol may be declared to be a dimensional quantity, with
specified quantity or specified units or both.

   An expression a ` b `` c converts from unit `b' to unit `c'.
`ezunits' has built-in conversions for SI base units, SI derived units,
and some non-SI units.  Unit conversions not already known to `ezunits'
can be declared.  The unit conversions known to `ezunits' are specified
by the global variable `known_unit_conversions', which comprises
built-in and user-defined conversions.  Conversions for products,
quotients, and powers of units are derived from the set of known unit
conversions.

   As Maxima generally prefers exact numbers (integers or rationals) to
inexact (float or bigfloat), so `ezunits' preserves exact numbers when
they appear in dimensional quantities.  All built-in unit conversions
are expressed in terms of exact numbers; inexact numbers in declared
conversions are coerced to exact.

   There is no preferred system for display of units; input units are
not converted to other units unless conversion is explicitly indicated.
`ezunits' recognizes the prefixes m-, k-, M, and G- (for milli-, kilo-,
mega-, and giga-) as applied to SI base units and SI derived units, but
such prefixes are applied only when indicated by an explicit conversion.

   Arithmetic operations on dimensional quantities are carried out by
conventional rules for such operations.

   * (x ` a) * (y ` b) is equal to (x * y) ` (a * b).

   * (x ` a) + (y ` a) is equal to (x + y) ` a.

   * (x ` a)^y is equal to x^y ` a^y when `y' is nondimensional.

   `ezunits' does not require that units in a sum have the same
dimensions; such terms are not added together, and no error is reported.

   `ezunits' includes functions for elementary dimensional analysis,
namely the fundamental dimensions and fundamental units of a
dimensional quantity, and computation of dimensionless quantities and
natural units.  The functions for dimensional analysis were adapted
from similar functions in another package, written by Barton Willis.

   For the purpose of dimensional analysis, a list of fundamental
dimensions and an associated list of fundamental units are maintained;
by default the fundamental dimensions are length, mass, time, charge,
temperature, and quantity, and the fundamental units are the associated
SI units, but other fundamental dimensions and units can be declared.

51.2 Introduction to physical_constants

`physical_constants' is a collection of physical constants, copied from
CODATA 2006 recommended values. [1] `load(physical_constants)' loads
this package, and loads `ezunits' also, if it is not already loaded.

   A physical constant is represented as a symbol which has a property
which is the constant value.  The constant value is a dimensional
quantity, as represented by `ezunits'.  The function `constvalue'
fetches the constant value; the constant value is not the ordinary
value of the symbol, so symbols of physical constants persist in
evaluated expressions until their values are fetched by `constvalue'.

   `physical_constants' includes some auxilliary information, namely, a
description string for each constant, an estimate of the error of its
numerical value, and a property for TeX display.  To identify physical
constants, each symbol has the `physical_constant' property;
`propvars(physical_constant)' therefore shows the list of all such
symbols.

   `physical_constants' comprises the following constants.

`%c'
     speed of light in vacuum

`%mu_0'
     magnetic constant

`%e_0'
     electric constant

`%Z_0'
     characteristic impedance of vacuum

`%G'
     Newtonian constant of gravitation

`%h'
     Planck constant

`%h_bar'
     Planck constant

`%m_P'
     Planck mass

`%T_P'
     Planck temperature

`%l_P'
     Planck length

`%t_P'
     Planck time

`%%e'
     elementary charge

`%Phi_0'
     magnetic flux quantum

`%G_0'
     conductance quantum

`%K_J'
     Josephson constant

`%R_K'
     von Klitzing constant

`%mu_B'
     Bohr magneton

`%mu_N'
     nuclear magneton

`%alpha'
     fine-structure constant

`%R_inf'
     Rydberg constant

`%a_0'
     Bohr radius

`%E_h'
     Hartree energy

`%ratio_h_me'
     quantum of circulation

`%m_e'
     electron mass

`%N_A'
     Avogadro constant

`%m_u'
     atomic mass constant

`%F'
     Faraday constant

`%R'
     molar gas constant

`%%k'
     Boltzmann constant

`%V_m'
     molar volume of ideal gas

`%n_0'
     Loschmidt constant

`%ratio_S0_R'
     Sackur-Tetrode constant (absolute entropy constant)

`%sigma'
     Stefan-Boltzmann constant

`%c_1'
     first radiation constant

`%c_1L'
     first radiation constant for spectral radiance

`%c_2'
     second radiation constant

`%b'
     Wien displacement law constant

`%b_prime'
     Wien displacement law constant

   References:

   [1] `http://physics.nist.gov/constants'

   Examples:

   The list of all symbols which have the `physical_constant' property.

     (%i1) load (physical_constants)$
     (%i2) propvars (physical_constant);
     (%o2) [%c, %mu_0, %e_0, %Z_0, %G, %h, %h_bar, %m_P, %T_P, %l_P,
     %t_P, %%e, %Phi_0, %G_0, %K_J, %R_K, %mu_B, %mu_N, %alpha,
     %R_inf, %a_0, %E_h, %ratio_h_me, %m_e, %N_A, %m_u, %F, %R, %%k,
     %V_m, %n_0, %ratio_S0_R, %sigma, %c_1, %c_1L, %c_2, %b, %b_prime]

   Properties of the physical constant `%c'.

     (%i1) load (physical_constants)$
     (%i2) constantp (%c);
     (%o2)                         true
     (%i3) get (%c, description);
     (%o3)               speed of light in vacuum
     (%i4) constvalue (%c);
                                           m
     (%o4)                     299792458 ` -
                                           s
     (%i5) get (%c, RSU);
     (%o5)                           0
     (%i6) tex (%c);
     $$c$$
     (%o6)                         false

   The energy equivalent of 1 pound-mass.  The symbol `%c' persists
until its value is fetched by `constvalue'.

     (%i1) load (physical_constants)$
     (%i2) m * %c^2;
                                     2
     (%o2)                         %c  m
     (%i3) %, m = 1 ` lbm;
                                   2
     (%o3)                       %c  ` lbm
     (%i4) constvalue (%);
                                                 2
                                            lbm m
     (%o4)              89875517873681764 ` ------
                                               2
                                              s
     (%i5) E : % `` J;
     Computing conversions to base units; may take a moment.
                          366838848464007200
     (%o5)                ------------------ ` J
                                  9
     (%i6) E `` GJ;
                           458548560580009
     (%o6)                 --------------- ` GJ
                              11250000
     (%i7) float (%);
     (%o7)              4.0759872051556356e+7 ` GJ

51.3 Functions and Variables for ezunits

Operator: `

     The dimensional quantity operator.  An expression a ` b represents
     a dimensional quantity, with `a' indicating a nondimensional
     quantity and `b' indicating the dimensional units.  A symbol can
     be used as a unit without declaring it as such; unit symbols need
     not have any special properties.  The quantity and unit of an
     expression a ` b can be extracted by the `qty' and `units'
     functions, respectively.

     Arithmetic operations on dimensional quantities are carried out by
     conventional rules for such operations.

        * (x ` a) * (y ` b) is equal to (x * y) ` (a * b).

        * (x ` a) + (y ` a) is equal to (x + y) ` a.

        * (x ` a)^y is equal to x^y ` a^y when `y' is nondimensional.

     `ezunits' does not require that units in a sum have the same
     dimensions; such terms are not added together, and no error is
     reported.

     `load(ezunits)' enables this operator.

     Examples:

     SI (Systeme Internationale) units.

          (%i1) load (ezunits)$
          (%i2) foo : 10 ` m;
          (%o2)                        10 ` m
          (%i3) qty (foo);
          (%o3)                          10
          (%i4) units (foo);
          (%o4)                           m
          (%i5) dimensions (foo);
          (%o5)                        length

     "Customary" units.

          (%i1) load (ezunits)$
          (%i2) bar : x ` acre;
          (%o2)                       x ` acre
          (%i3) dimensions (bar);
                                             2
          (%o3)                        length
          (%i4) fundamental_units (bar);
                                          2
          (%o4)                          m

     Units ad hoc.

          (%i1) load (ezunits)$
          (%i2) baz : 3 ` sheep + 8 ` goat + 1 ` horse;
          (%o2)           8 ` goat + 3 ` sheep + 1 ` horse
          (%i3) subst ([sheep = 3*goat, horse = 10*goat], baz);
          (%o3)                       27 ` goat
          (%i4) baz2 : 1000`gallon/fortnight;
                                          gallon
          (%o4)                   1000 ` ---------
                                         fortnight
          (%i5) subst (fortnight = 14*day, baz2);
                                    500   gallon
          (%o5)                     --- ` ------
                                     7     day

     Arithmetic operations on dimensional quantities.

          (%i1) load (ezunits)$
          (%i2) 100 ` kg + 200 ` kg;
          (%o2)                       300 ` kg
          (%i3) 100 ` m^3 - 100 ` m^3;
                                            3
          (%o3)                        0 ` m
          (%i4) (10 ` kg) * (17 ` m/s^2);
                                           kg m
          (%o4)                      170 ` ----
                                             2
                                            s
          (%i5) (x ` m) / (y ` s);
                                        x   m
          (%o5)                         - ` -
                                        y   s
          (%i6) (a ` m)^2;
                                        2    2
          (%o6)                        a  ` m

Operator: ``

     The unit conversion operator.  An expression a ` b `` c converts
     from unit `b' to unit `c'.  `ezunits' has built-in conversions for
     SI base units, SI derived units, and some non-SI units.  Unit
     conversions not already known to `ezunits' can be declared.  The
     unit conversions known to `ezunits' are specified by the global
     variable `known_unit_conversions', which comprises built-in and
     user-defined conversions.  Conversions for products, quotients,
     and powers of units are derived from the set of known unit
     conversions.

     There is no preferred system for display of units; input units are
     not converted to other units unless conversion is explicitly
     indicated.  `ezunits' does not attempt to simplify units by
     prefixes (milli-, centi-, deci-, etc) unless such conversion is
     explicitly indicated.

     `load(ezunits)' enables this operator.

     Examples:

     The set of known unit conversions.

          (%i1) load (ezunits)$
          (%i2) display2d : false$
          (%i3) known_unit_conversions;
          (%o3) {acre = 4840*yard^2,Btu = 1055*J,cfm = feet^3/minute,
                 cm = m/100,day = 86400*s,feet = 381*m/1250,ft = feet,
                 g = kg/1000,gallon = 757*l/200,GHz = 1000000000*Hz,
                 GOhm = 1000000000*Ohm,GPa = 1000000000*Pa,
                 GWb = 1000000000*Wb,Gg = 1000000*kg,Gm = 1000000000*m,
                 Gmol = 1000000*mol,Gs = 1000000000*s,ha = hectare,
                 hectare = 100*m^2,hour = 3600*s,Hz = 1/s,inch = feet/12,
                 km = 1000*m,kmol = 1000*mol,ks = 1000*s,l = liter,
                 lbf = pound_force,lbm = pound_mass,liter = m^3/1000,
                 metric_ton = Mg,mg = kg/1000000,MHz = 1000000*Hz,
                 microgram = kg/1000000000,micrometer = m/1000000,
                 micron = micrometer,microsecond = s/1000000,
                 mile = 5280*feet,minute = 60*s,mm = m/1000,
                 mmol = mol/1000,month = 2629800*s,MOhm = 1000000*Ohm,
                 MPa = 1000000*Pa,ms = s/1000,MWb = 1000000*Wb,
                 Mg = 1000*kg,Mm = 1000000*m,Mmol = 1000000000*mol,
                 Ms = 1000000*s,ns = s/1000000000,ounce = pound_mass/16,
                 oz = ounce,Ohm = s*J/C^2,
                 pound_force = 32*ft*pound_mass/s^2,
                 pound_mass = 200*kg/441,psi = pound_force/inch^2,
                 Pa = N/m^2,week = 604800*s,Wb = J/A,yard = 3*feet,
                 year = 31557600*s,C = s*A,F = C^2/J,GA = 1000000000*A,
                 GC = 1000000000*C,GF = 1000000000*F,GH = 1000000000*H,
                 GJ = 1000000000*J,GK = 1000000000*K,GN = 1000000000*N,
                 GS = 1000000000*S,GT = 1000000000*T,GV = 1000000000*V,
                 GW = 1000000000*W,H = J/A^2,J = m*N,kA = 1000*A,
                 kC = 1000*C,kF = 1000*F,kH = 1000*H,kHz = 1000*Hz,
                 kJ = 1000*J,kK = 1000*K,kN = 1000*N,kOhm = 1000*Ohm,
                 kPa = 1000*Pa,kS = 1000*S,kT = 1000*T,kV = 1000*V,
                 kW = 1000*W,kWb = 1000*Wb,mA = A/1000,mC = C/1000,
                 mF = F/1000,mH = H/1000,mHz = Hz/1000,mJ = J/1000,
                 mK = K/1000,mN = N/1000,mOhm = Ohm/1000,mPa = Pa/1000,
                 mS = S/1000,mT = T/1000,mV = V/1000,mW = W/1000,
                 mWb = Wb/1000,MA = 1000000*A,MC = 1000000*C,
                 MF = 1000000*F,MH = 1000000*H,MJ = 1000000*J,
                 MK = 1000000*K,MN = 1000000*N,MS = 1000000*S,
                 MT = 1000000*T,MV = 1000000*V,MW = 1000000*W,
                 N = kg*m/s^2,R = 5*K/9,S = 1/Ohm,T = J/(m^2*A),V = J/C,
                 W = J/s}

     Elementary unit conversions.

          (%i1) load (ezunits)$
          (%i2) 1 ` ft `` m;
          Computing conversions to base units; may take a moment.
                                      381
          (%o2)                       ---- ` m
                                      1250
          (%i3) %, numer;
          (%o3)                      0.3048 ` m
          (%i4) 1 ` kg `` lbm;
                                      441
          (%o4)                       --- ` lbm
                                      200
          (%i5) %, numer;
          (%o5)                      2.205 ` lbm
          (%i6) 1 ` W `` Btu/hour;
                                     720   Btu
          (%o6)                      --- ` ----
                                     211   hour
          (%i7) %, numer;
                                                  Btu
          (%o7)               3.412322274881517 ` ----
                                                  hour
          (%i8) 100 ` degC `` degF;
          (%o8)                      212 ` degF
          (%i9) -40 ` degF `` degC;
          (%o9)                     (- 40) ` degC
          (%i10) 1 ` acre*ft `` m^3;
                                  60228605349    3
          (%o10)                  ----------- ` m
                                   48828125
          (%i11) %, numer;
                                                    3
          (%o11)                1233.48183754752 ` m

     Coercing quantities in feet and meters to one or the other.

          (%i1) load (ezunits)$
          (%i2) 100 ` m + 100 ` ft;
          (%o2)                  100 ` m + 100 ` ft
          (%i3) (100 ` m + 100 ` ft) `` ft;
          Computing conversions to base units; may take a moment.
                                     163100
          (%o3)                      ------ ` ft
                                      381
          (%i4) %, numer;
          (%o4)                428.0839895013123 ` ft
          (%i5) (100 ` m + 100 ` ft) `` m;
                                      3262
          (%o5)                       ---- ` m
                                       25
          (%i6) %, numer;
          (%o6)                      130.48 ` m

     Dimensional analysis to find fundamental dimensions and
     fundamental units.

          (%i1) load (ezunits)$
          (%i2) foo : 1 ` acre * ft;
          (%o2)                      1 ` acre ft
          (%i3) dimensions (foo);
                                             3
          (%o3)                        length
          (%i4) fundamental_units (foo);
                                          3
          (%o4)                          m
          (%i5) foo `` m^3;
          Computing conversions to base units; may take a moment.
                                  60228605349    3
          (%o5)                   ----------- ` m
                                   48828125
          (%i6) %, numer;
                                                    3
          (%o6)                 1233.48183754752 ` m

     Declared unit conversions.

          (%i1) load (ezunits)$
          (%i2) declare_unit_conversion (MMBtu = 10^6*Btu, kW = 1000*W);
          (%o2)                         done
          (%i3) declare_unit_conversion (kWh = kW*hour, MWh = 1000*kWh,
                                         bell = 1800*s);
          (%o3)                         done
          (%i4) 1 ` kW*s `` MWh;
          Computing conversions to base units; may take a moment.
                                       1
          (%o4)                     ------- ` MWh
                                    3600000
          (%i5) 1 ` kW/m^2 `` MMBtu/bell/ft^2;
                                 1306449      MMBtu
          (%o5)                 ---------- ` --------
                                8242187500          2
                                             bell ft

Function: constvalue (<x>)
Function: declare_constvalue (<a>, <x>)
Function: remove_constvalue (<a>)

     Returns the declared constant value of a symbol, or value of an
     expression with declared constant values substituted for symbols.

     Constant values are declared by `declare_constvalue'.  Note that
     constant values as recognized by `constvalue' are separate from
     values declared by `numerval' and recognized by `constantp'.

     The `physical_units' package declares constant values for a number
     of physical constants.

     `remove_constvalue' reverts the effect of `declare_constvalue'.

     `load(ezunits)' loads these functions.

     Examples:

     Constant value of a physical constant.

          (%i1) load (physical_constants)$
          (%i2) constvalue (%G);
                                               3
                                              m
          (%o2)                    6.67428 ` -----
                                                 2
                                             kg s
          (%i3) get ('%G, 'description);
          (%o3)           Newtonian constant of gravitation

     Declaring a new constant.

          (%i1) load (ezunits)$
          (%i2) declare_constvalue (FOO, 100 ` lbm / acre);
                                           lbm
          (%o2)                      100 ` ----
                                           acre
          (%i3) FOO * (50 ` acre);
          (%o3)                     50 FOO ` acre
          (%i4) constvalue (%);
          (%o4)                      5000 ` lbm

Function: units (<x>)
Function: declare_units (<a>, <u>)

     Returns the units of a dimensional quantity <x>, or returns 1 if
     <x> is nondimensional.

     <x> may be a literal dimensional expression a ` b, a symbol with
     declared units via `declare_units', or an expression containing
     either or both of those.

     `declare_units' declares that `units(<a>)' should return <u>,
     where <u> is an expression.

     `load(ezunits)' loads these functions.

     Examples:

     `units' applied to literal dimensional expressions.

          (%i1) load (ezunits)$
          (%i2) foo : 100 ` kg;
          (%o2)                       100 ` kg
          (%i3) bar : x ` m/s;
                                            m
          (%o3)                         x ` -
                                            s
          (%i4) units (foo);
          (%o4)                          kg
          (%i5) units (bar);
                                          m
          (%o5)                           -
                                          s
          (%i6) units (foo * bar);
                                        kg m
          (%o6)                         ----
                                         s
          (%i7) units (foo / bar);
                                        kg s
          (%o7)                         ----
                                         m
          (%i8) units (foo^2);
                                           2
          (%o8)                          kg

     `units' applied to symbols with declared units.

          (%i1) load (ezunits)$
          (%i2) units (aa);
          (%o2)                           1
          (%i3) declare_units (aa, J);
          (%o3)                           J
          (%i4) units (aa);
          (%o4)                           J
          (%i5) units (aa^2);
                                          2
          (%o5)                          J
          (%i6) foo : 100 ` kg;
          (%o6)                       100 ` kg
          (%i7) units (aa * foo);
          (%o7)                         kg J

Function: qty (<x>)
Function: declare_qty (<a>, <x>)

     `qty' returns the nondimensional part of a dimensional quantity
     <x>, or returns <x> if <x> is nondimensional.  <x> may be a
     literal dimensional expression a ` b, a symbol with declared
     quantity, or an expression containing either or both of those.

     `declare_qty' declares that `qty(<a>)' should return <x>, where
     <x> is a nondimensional quantity.

     `load(ezunits)' loads these functions.

     Examples:

     `qty' applied to literal dimensional expressions.

          (%i1) load (ezunits)$
          (%i2) foo : 100 ` kg;
          (%o2)                       100 ` kg
          (%i3) qty (foo);
          (%o3)                          100
          (%i4) bar : v ` m/s;
                                            m
          (%o4)                         v ` -
                                            s
          (%i5) foo * bar;
                                            kg m
          (%o5)                     100 v ` ----
                                             s
          (%i6) qty (foo * bar);
          (%o6)                         100 v

     `qty' applied to symbols with declared quantity.

          (%i1) load (ezunits)$
          (%i2) declare_qty (aa, xx);
          (%o2)                          xx
          (%i3) qty (aa);
          (%o3)                          xx
          (%i4) qty (aa^2);
                                           2
          (%o4)                          xx
          (%i5) foo : 100 ` kg;
          (%o5)                       100 ` kg
          (%i6) qty (aa * foo);
          (%o6)                        100 xx

Function: unitp (<x>)

     Returns `true' if <x> is a literal dimensional expression, a
     symbol declared dimensional, or an expression in which the main
     operator is declared dimensional.  `unitp' returns `false'
     otherwise.

     `load(ezunits)' loads this function.

     Examples:

     `unitp' applied to a literal dimensional expression.

          (%i1) load (ezunits)$
          (%i2) unitp (100 ` kg);
          (%o2)                         true

     `unitp' applied to a symbol declared dimensional.

          (%i1) load (ezunits)$
          (%i2) unitp (foo);
          (%o2)                         false
          (%i3) declare (foo, dimensional);
          (%o3)                         done
          (%i4) unitp (foo);
          (%o4)                         true

     `unitp' applied to an expression in which the main operator is
     declared dimensional.

          (%i1) load (ezunits)$
          (%i2) unitp (bar (x, y, z));
          (%o2)                         false
          (%i3) declare (bar, dimensional);
          (%o3)                         done
          (%i4) unitp (bar (x, y, z));
          (%o4)                         true

Function: declare_unit_conversion (<u> = <v>, ...)

     Appends equations <u> = <v>, ... to the list of unit conversions
     known to the unit conversion operator ``.  <u> and <v> are both
     multiplicative terms, in which any variables are units, or both
     literal dimensional expressions.

     At present, it is necessary to express conversions such that the
     left-hand side of each equation is a simple unit (not a
     multiplicative expression) or a literal dimensional expression
     with the quantity equal to 1 and the unit being a simple unit.
     This limitation might be relaxed in future versions.

     `known_unit_conversions' is the list of known unit conversions.

     `load(ezunits)' loads this function.

     Examples:

     Unit conversions expressed by equations of multiplicative terms.

          (%i1) load (ezunits)$
          (%i2) declare_unit_conversion (nautical_mile = 1852 * m,
                                         fortnight = 14 * day);
          (%o2)                         done
          (%i3) 100 ` nautical_mile / fortnight `` m/s;
          Computing conversions to base units; may take a moment.
                                      463    m
          (%o3)                       ---- ` -
                                      3024   s

     Unit conversions expressed by equations of literal dimensional
     expressions.

          (%i1) load (ezunits)$
          (%i2) declare_unit_conversion (1 ` fluid_ounce = 2 ` tablespoon);
          (%o2)                         done
          (%i3) declare_unit_conversion (1 ` tablespoon = 3 ` teaspoon);
          (%o3)                         done
          (%i4) 15 ` fluid_ounce `` teaspoon;
          Computing conversions to base units; may take a moment.
          (%o4)                     90 ` teaspoon

Function: declare_dimensions (<a_1>, <d_1>, ..., <a_n>, <d_n>)
Function: remove_dimensions (<a_1>, ..., <a_n>)

     `declare_dimensions' declares <a_1>, ..., <a_n> to have dimensions
     <d_1>, ..., <d_n>, respectively.

     Each <a_k> is a symbol or a list of symbols.  If it is a list,
     then every symbol in <a_k> is declared to have dimension <d_k>.

     `remove_dimensions' reverts the effect of `declare_dimensions'.

     `load(ezunits)' loads these functions.

     Examples:

          (%i1) load (ezunits) $
          (%i2) declare_dimensions ([x, y, z], length, [t, u], time);
          (%o2)                         done
          (%i3) dimensions (y^2/u);
                                             2
                                       length
          (%o3)                        -------
                                        time
          (%i4) fundamental_units (y^2/u);
          0 errors, 0 warnings
                                          2
                                         m
          (%o4)                          --
                                         s

Function: declare_fundamental_dimensions (<d_1>, <d_2>, <d_3>, ...)
Function: remove_fundamental_dimensions (<d_1>, <d_2>, <d_3>, ...)
Global variable: fundamental_dimensions

     `declare_fundamental_dimensions' declares fundamental dimensions.
     Symbols <d_1>, <d_2>, <d_3>, ... are appended to the list of
     fundamental dimensions, if they are not already on the list.

     `remove_fundamental_dimensions' reverts the effect of
     `declare_fundamental_dimensions'.

     `fundamental_dimensions' is the list of fundamental dimensions.
     By default, the list comprises several physical dimensions.

     `load(ezunits)' loads these functions.

     Examples:

          (%i1) load (ezunits) $
          (%i2) fundamental_dimensions;
          (%o2) [length, mass, time, current, temperature, quantity]
          (%i3) declare_fundamental_dimensions (money, cattle, happiness);
          (%o3)                         done
          (%i4) fundamental_dimensions;
          (%o4) [length, mass, time, current, temperature, quantity,
                                                  money, cattle, happiness]
          (%i5) remove_fundamental_dimensions (cattle, happiness);
          (%o5)                         done
          (%i6) fundamental_dimensions;
          (%o6) [length, mass, time, current, temperature, quantity, money]

Function: declare_fundamental_units (<u_1>, <d_1>, ..., <u_n>, <d_n>)
Function: remove_fundamental_units (<u_1>, ..., <u_n>)

     `declare_fundamental_units' declares <u_1>, ..., <u_n> to have
     dimensions <d_1>, ..., <d_n>, respectively.  All arguments must be
     symbols.

     After calling `declare_fundamental_units', `dimensions(<u_k>)'
     returns <d_k> for each argument <u_1>, ..., <u_n>, and
     `fundamental_units(<d_k>)' returns <u_k> for each argument <d_1>,
     ..., <d_n>.

     `remove_fundamental_units' reverts the effect of
     `declare_fundamental_units'.

     `load(ezunits)' loads these functions.

     Examples:

          (%i1) load (ezunits) $
          (%i2) declare_fundamental_dimensions (money, cattle, happiness);
          (%o2)                         done
          (%i3) declare_fundamental_units (dollar, money, goat, cattle,
                                           smile, happiness);
          (%o3)                 [dollar, goat, smile]
          (%i4) dimensions (100 ` dollar/goat/km^2);
                                       money
          (%o4)                    --------------
                                                2
                                   cattle length
          (%i5) dimensions (x ` smile/kg);
                                      happiness
          (%o5)                       ---------
                                        mass
          (%i6) fundamental_units (money*cattle/happiness);
          0 errors, 0 warnings
                                     dollar goat
          (%o6)                      -----------
                                        smile

Function: dimensions (<x>)
Function: dimensions_as_list (<x>)

     `dimensions' returns the dimensions of the dimensional quantity <x>
     as an expression comprising products and powers of base dimensions.

     `dimensions_as_list' returns the dimensions of the dimensional
     quantity <x> as a list, in which each element is an integer which
     indicates the power of the corresponding base dimension in the
     dimensions of <x>.

     `load(ezunits)' loads these functions.

     Examples:

          (%i1) load (ezunits)$
          (%i2) dimensions (1000 ` kg*m^2/s^3);
                                          2
                                    length  mass
          (%o2)                     ------------
                                           3
                                       time
          (%i3) declare_units (foo, acre*ft/hour);
                                       acre ft
          (%o3)                        -------
                                        hour
          (%i4) dimensions (foo);
                                             3
                                       length
          (%o4)                        -------
                                        time

          (%i1) load (ezunits)$
          (%i2) fundamental_dimensions;
          (%o2)  [length, mass, time, charge, temperature, quantity]
          (%i3) dimensions_as_list (1000 ` kg*m^2/s^3);
          (%o3)                 [2, 1, - 3, 0, 0, 0]
          (%i4) declare_units (foo, acre*ft/hour);
                                       acre ft
          (%o4)                        -------
                                        hour
          (%i5) dimensions_as_list (foo);
          (%o5)                 [3, 0, - 1, 0, 0, 0]

Function: fundamental_units (<x>)
Function: fundamental_units ()

     `fundamental_units(<x>)' returns the units associated with the
     fundamental dimensions of <x>.  as determined by `dimensions(<x>)'.

     <x> may be a literal dimensional expression a ` b, a symbol with
     declared units via `declare_units', or an expression containing
     either or both of those.

     `fundamental_units()' returns the list of all known fundamental
     units, as declared by `declare_fundamental_units'.

     `load(ezunits)' loads this function.

     Examples:

          (%i1) load (ezunits)$
          (%i2) fundamental_units ();
          (%o2)                 [m, kg, s, A, K, mol]
          (%i3) fundamental_units (100 ` mile/hour);
                                          m
          (%o3)                           -
                                          s
          (%i4) declare_units (aa, g/foot^2);
                                          g
          (%o4)                         -----
                                            2
                                        foot
          (%i5) fundamental_units (aa);
                                         kg
          (%o5)                          --
                                          2
                                         m

Function: dimensionless (<L>)

     Returns a basis for the dimensionless quantities which can be
     formed from a list <L> of dimensional quantities.

     `load(ezunits)' loads this function.

     Examples:

          (%i1) load (ezunits) $
          (%i2) dimensionless ([x ` m, y ` m/s, z ` s]);
          0 errors, 0 warnings
          0 errors, 0 warnings
                                         y z
          (%o2)                         [---]
                                          x

     Dimensionless quantities derived from fundamental physical
     quantities.  Note that the first element on the list is
     proportional to the fine-structure constant.

          (%i1) load (ezunits) $
          (%i2) load (physical_constants) $
          (%i3) dimensionless([%h_bar, %m_e, %m_P, %%e, %c, %e_0]);
          0 errors, 0 warnings
          0 errors, 0 warnings
                                        2
                                     %%e        %m_e
          (%o3)                [--------------, ----]
                                %c %e_0 %h_bar  %m_P

Function: natural_unit (<expr>, [<v_1>, ..., <v_n>])

     Finds exponents <e_1>, ..., <e_n> such that `dimension(<expr>) =
     dimension(<v_1>^<e_1> ... <v_n>^<e_n>)'.

     `load(ezunits)' loads this function.

     Examples:

52 f90


Functions and Variables for f90

52.1 Functions and Variables for f90

Function: f90 (<expr_1>, ..., <expr_n>)

     Prints one or more expressions <expr_1>, ..., <expr_n> as a
     Fortran 90 program.  Output is printed to the standard output.

     `f90' prints output in the so-called "free form" input format for
     Fortran 90: there is no special attention to column positions.
     Long lines are split at a fixed width with the ampersand `&'
     continuation character.

     `load(f90)' loads this function.  See also the function `fortran'.

     Examples:

          (%i1) load (f90)$
          (%i2) foo : expand ((xxx + yyy + 7)^4);
                   4            3         3        2    2             2
          (%o2) yyy  + 4 xxx yyy  + 28 yyy  + 6 xxx  yyy  + 84 xxx yyy
                    2        3             2
           + 294 yyy  + 4 xxx  yyy + 84 xxx  yyy + 588 xxx yyy + 1372 yyy
                4         3          2
           + xxx  + 28 xxx  + 294 xxx  + 1372 xxx + 2401
          (%i3) f90 ('foo = foo);
          foo = yyy**4+4*xxx*yyy**3+28*yyy**3+6*xxx**2*yyy**2+84*xxx*yyy**2&
          +294*yyy**2+4*xxx**3*yyy+84*xxx**2*yyy+588*xxx*yyy+1372*yyy+xxx**&
          4+28*xxx**3+294*xxx**2+1372*xxx+2401
          (%o3)                         false

     Multiple expressions.  Capture standard output into a file via the
     `with_stdout'  function.

          (%i1) load (f90)$
          (%i2) foo : sin (3*x + 1) - cos (7*x - 2);
          (%o2)              sin(3 x + 1) - cos(7 x - 2)
          (%i3) with_stdout ("foo.f90",
                             f90 (x=0.25, y=0.625, 'foo=foo, 'stop, 'end));
          (%o3)                         false
          (%i4) printfile ("foo.f90");
          x = 0.25
          y = 0.625
          foo = sin(3*x+1)-cos(7*x-2)
          stop
          end
          (%o4)                        foo.f90

53 finance


Introduction to finance
Functions and Variables for finance

53.1 Introduction to finance

This is the Finance Package (Ver 0.1).

   In all the functions, <rate> is the compound interest rate, <num> is
the number of periods and must be postivive and <flow> refers to cash
flow so if you have an Output the flow is negative and positive for
Inputs.

   Note that before using the functions defined in this package, you
have to load it writing `load(finance)$'.

   Author: Nicolas Guarin Zapata.

53.2 Functions and Variables for finance

Function: days360 (<year1>,<month1>,<day1>,<year2>,<month2>,<day2>)

     Calculates the distance between 2 dates, assuming 360 days years,
     30 days months.

     Example:

          (%i1) load(finance)$
          (%i2) days360(2008,12,16,2007,3,25);
          (%o2)                      - 621

Function: fv (<rate>,<PV>,<num>)

     We can calculate the future value of a Present one given a certain
     interest rate.  <rate> is the interest rate, <PV> is the present
     value and <num> is the number of periods.

     Example:

          (%i1) load(finance)$
          (%i2) fv(0.12,1000,3);
          (%o2)                     1404.928

Function: pv (<rate>,<FV>,<num>)

     We can calculate the present value of a Future one given a certain
     interest rate.  <rate> is the interest rate, <FV> is the future
     value and <num> is the number of periods.

     Example:

          (%i1) load(finance)$
          (%i2) pv(0.12,1000,3);
          (%o2)                711.7802478134108

Function: graph_flow (<val>)

     Plots the money flow in a time line, the positive values are in
     blue and upside; the negative ones are in red and downside.  The
     direction of the flow is given by the sign of the value.  <val> is
     a list of flow values.

     Example:

          (%i1) load(finance)$
          (%i2) graph_flow([-5000,-3000,800,1300,1500,2000])$

Function: annuity_pv (<rate>,<PV>,<num>)

     We can calculate the annuity knowing the present value (like an
     ammount), it is a constant and periodic payment. <rate> is the
     interest rate, <PV> is the present value and <num> is the number
     of periods.

     Example:

          (%i1) load(finance)$
          (%i2) annuity_pv(0.12,5000,10);
          (%o2)                884.9208207992202

Function: annuity_fv (<rate>,<FV>,<num>)

     We can calculate the annuity knowing the desired value (future
     value), it is a constant and periodic payment. <rate> is the
     interest rate, <FV> is the future value and <num> is the number of
     periods.

     Example:

          (%i1) load(finance)$
          (%i2) annuity_fv(0.12,65000,10);
          (%o2)                3703.970670389863

Function: geo_annuity_pv (<rate>,<growing_rate>,<PV>,<num>)

     We can calculate the annuity knowing the present value (like an
     ammount), in a growing periodic payment. <rate> is the interest
     rate, <growing_rate> is the growing rate, <PV> is the present
     value and <num> is the number of periods.

     Example:

          (%i1) load(finance)$
          (%i2) geo_annuity_pv(0.14,0.05,5000,10);
          (%o2)                802.6888176505123

Function: geo_annuity_fv (<rate>,<growing_rate>,<FV>,<num>)

     We can calculate the annuity knowing the desired value (future
     value), in a growing periodic payment. <rate> is the interest
     rate, <growing_rate> is the growing rate, <FV> is the future value
     and <num> is the number of periods.

     Example:

          (%i1) load(finance)$
          (%i2) geo_annuity_fv(0.14,0.05,5000,10);
          (%o2)                216.5203395312695

Function: amortization (<rate>,<ammount>,<num>)

     Amortization table determinated by a specific rate.  <rate> is the
     interest rate, <ammount> is the ammount value, and <num> is the
     number of periods.

     Example:

          (%i1) load(finance)$
          (%i2) amortization(0.05,56000,12)$
                "n"    "Balance"     "Interest"   "Amortization"  "Payment"
               0.000     56000.000         0.000         0.000         0.000
               1.000     52481.777      2800.000      3518.223      6318.223
               2.000     48787.643      2624.089      3694.134      6318.223
               3.000     44908.802      2439.382      3878.841      6318.223
               4.000     40836.019      2245.440      4072.783      6318.223
               5.000     36559.597      2041.801      4276.422      6318.223
               6.000     32069.354      1827.980      4490.243      6318.223
               7.000     27354.599      1603.468      4714.755      6318.223
               8.000     22404.106      1367.730      4950.493      6318.223
               9.000     17206.088      1120.205      5198.018      6318.223
              10.000     11748.170       860.304      5457.919      6318.223
              11.000      6017.355       587.408      5730.814      6318.223
              12.000         0.000       300.868      6017.355      6318.223

Function: arit_amortization (<rate>,<increment>,<ammount>,<num>)

     The amortization table determinated by a specific rate and with
     growing payment can be claculated by `arit_amortization'.  Notice
     that the payment is not constant, it presents an arithmetic
     growing, increment is then the difference between two consecutive
     rows in the "Payment" column.  <rate> is the interest rate,
     <increment> is the increment, <ammount> is the ammount value, and
     <num> is the number of periods.

     Example:

          (%i1) load(finance)$
          (%i2) arit_amortization(0.05,1000,56000,12)$
                "n"    "Balance"     "Interest"   "Amortization"  "Payment"
               0.000     56000.000         0.000         0.000         0.000
               1.000     57403.679      2800.000     -1403.679      1396.321
               2.000     57877.541      2870.184      -473.863      2396.321
               3.000     57375.097      2893.877       502.444      3396.321
               4.000     55847.530      2868.755      1527.567      4396.321
               5.000     53243.586      2792.377      2603.945      5396.321
               6.000     49509.443      2662.179      3734.142      6396.321
               7.000     44588.594      2475.472      4920.849      7396.321
               8.000     38421.703      2229.430      6166.892      8396.321
               9.000     30946.466      1921.085      7475.236      9396.321
              10.000     22097.468      1547.323      8848.998     10396.321
              11.000     11806.020      1104.873     10291.448     11396.321
              12.000        -0.000       590.301     11806.020     12396.321

Function: geo_amortization (<rate>,<growing_rate>,<ammount>,<num>)

     The amortization table determinated by rate, ammount, and number
     of periods can be found by `geo_amortization'.  Notice that the
     payment is not constant, it presents a geometric growing,
     <growing_rate> is then the quotient between two consecutive rows
     in the "Payment" column.  <rate> is the interest rate, <ammount>
     is the ammount value, and <num> is the number of periods.

     Example:

          (%i1) load(finance)$
          (%i2) geo_amortization(0.05,0.03,56000,12)$
                "n"    "Balance"     "Interest"   "Amortization"  "Payment"
               0.000     56000.000         0.000         0.000         0.000
               1.000     53365.296      2800.000      2634.704      5434.704
               2.000     50435.816      2668.265      2929.480      5597.745
               3.000     47191.930      2521.791      3243.886      5765.677
               4.000     43612.879      2359.596      3579.051      5938.648
               5.000     39676.716      2180.644      3936.163      6116.807
               6.000     35360.240      1983.836      4316.475      6300.311
               7.000     30638.932      1768.012      4721.309      6489.321
               8.000     25486.878      1531.947      5152.054      6684.000
               9.000     19876.702      1274.344      5610.176      6884.520
              10.000     13779.481       993.835      6097.221      7091.056
              11.000      7164.668       688.974      6614.813      7303.787
              12.000         0.000       358.233      7164.668      7522.901

Function: saving (<rate>,<ammount>,<num>)

     The table that represents the values in a constant and periodic
     saving can be found by `saving'.  <ammount> represents the desired
     quantity and num the number of periods to save.

     Example:

          (%i1) load(finance)$
          (%i2) saving(0.15,12000,15)$
                "n"    "Balance"     "Interest"   "Payment"
               0.000         0.000         0.000         0.000
               1.000       252.205         0.000       252.205
               2.000       542.240        37.831       252.205
               3.000       875.781        81.336       252.205
               4.000      1259.352       131.367       252.205
               5.000      1700.460       188.903       252.205
               6.000      2207.733       255.069       252.205
               7.000      2791.098       331.160       252.205
               8.000      3461.967       418.665       252.205
               9.000      4233.467       519.295       252.205
              10.000      5120.692       635.020       252.205
              11.000      6141.000       768.104       252.205
              12.000      7314.355       921.150       252.205
              13.000      8663.713      1097.153       252.205
              14.000     10215.474      1299.557       252.205
              15.000     12000.000      1532.321       252.205

Function: npv (<rate>,<val>)

     Calculates de present value of a value series to evaluate the
     viability in a project.  <flowValues> es una lista con los valores
     para cada periodo.

     Example:

          (%i1) load(finance)$
          (%i2) npv(0.25,[100,500,323,124,300]);
          (%o2)                714.4703999999999

Function: irr (<val>,<IO>)

     IRR (Internal Rate of Return) is the value of rate which makes Net
     Present Value zero.  <flowValues> los valores para cada periodo
     (para periodos mayores a 0) y <I0> el valor para el periodo cero.

     Example:

          (%i1) load(finance)$
          (%i2) res:irr([-5000,0,800,1300,1500,2000],0)$
          (%i3) rhs(res[1][1]);
          (%o3)                .03009250374237132

Function: benefit_cost (<rate>,<input>,<output>)

     Calculates the ratio Benefit/Cost. Benefit is the Net Present
     Value (NPV) of the inputs, and Cost is the Net Present Value (NPV)
     of the outputs.  Notice that if there is not an input or output
     value in a specific period, the input/output would be a zero for
     that period.  <rate> is the interest rate, <input> is a list of
     input values, and <output> is a list of output values.

     Example:

          (%i1) load(finance)$
          (%i2) benefit_cost(0.24,[0,300,500,150],[100,320,0,180]);
          (%o2)               1.427249324905784

54 fractals


Introduction to fractals
Definitions for IFS fractals
Definitions for complex fractals
Definitions for Koch snowflakes
Definitions for Peano maps

54.1 Introduction to fractals

This package defines some well known fractals:

   - with random IFS (Iterated Function System): the Sierpinsky
triangle, a Tree and a Fern

   - Complex Fractals: the  Mandelbrot and Julia Sets

   - the Koch snowflake sets

   - Peano maps: the Sierpinski and Hilbert maps

   Author: Jose' Rami'rez Labrador.

   For questions, suggestions and  bugs, please feel free to contact me
at

   pepe DOT ramirez AAATTT uca DOT es

54.2 Definitions for IFS fractals

Some fractals can be generated by iterative applications of contractive
affine transformations in a random way; see

   Hoggar S. G., "Mathematics for computer graphics", Cambridge
University Press 1994.

   We define a list with several contractive affine transformations,
and  we randomly select the transformation in a recursive way.  The
probability of the choice of a transformation must be related with the
contraction ratio.

   You can change the transformations and find another fractal
Function: sierpinskiale (<n>)

     Sierpinski Triangle: 3 contractive maps; .5 contraction constant
     and translations; all maps have the same contraction ratio.
     Argument <n> must be great enougth, 10000 or greater.

     Example:

          (%i1) load(fractals)$
          (%i2) n: 10000$
          (%i3) plot2d([discrete,sierpinskiale(n)], [style,dots])$

Function: treefale (<n>)

     3 contractive maps all with the same contraction ratio.  Argument
     <n> must be great enougth, 10000 or greater.

     Example:

          (%i1) load(fractals)$
          (%i2) n: 10000$
          (%i3) plot2d([discrete,treefale(n)], [style,dots])$

Function: fernfale (<n>)

     4 contractive maps, the probability to choice a transformation
     must be related with the contraction ratio. Argument <n> must be
     great enougth, 10000 or greater.

     Example:

          (%i1) load(fractals)$
          (%i2) n: 10000$
          (%i3) plot2d([discrete,fernfale(n)], [style,dots])$

54.3 Definitions for complex fractals

Function: mandelbrot_set (<x>, <y>)

     Mandelbrot set.

     Example:

     This program is time consuming because it must make a lot of
     operations; the computing time is also related with the number of
     grid points.

          (%i1) load(fractals)$
          (%i2) plot3d (mandelbrot_set, [x, -2.5, 1], [y, -1.5, 1.5],
                          [gnuplot_preamble, "set view map"],
                          [gnuplot_pm3d, true],
                          [grid, 150, 150])$

Function: julia_set (<x>, <y>)

     Julia sets.

     This program is time consuming because it must make a lot of
     operations; the computing time is also related with the number of
     grid points.

     Example:

          (%i1) load(fractals)$
          (%i2) plot3d (julia_set, [x, -2, 1], [y, -1.5, 1.5],
                          [gnuplot_preamble, "set view map"],
                          [gnuplot_pm3d, true],
                          [grid, 150, 150])$

     See also `julia_parameter'.

Optional variable: julia_parameter

     Default value: `%i'

     Complex parameter for Julia fractals.  Its default value is `%i';
     we  suggest the  values `-.745+%i*.113002', `-.39054-%i*.58679',
     `-.15652+%i*1.03225', `-.194+%i*.6557' and `.011031-%i*.67037'.

Function: julia_sin (<x>, <y>)

     While function `julia_set' implements the transformation
     `julia_parameter+z^2', function `julia_sin' implements
     `julia_parameter*sin(z)'. See source code for more details.

     This program runs slowly  because it calculates a lot of sines.

     Example:

     This program is time consuming because it must make a lot of
     operations; the computing time is also related with the number of
     grid points.

          (%i1) load(fractals)$
          (%i2) julia_parameter:1+.1*%i$
          (%i3) plot3d (julia_sin, [x, -2, 2], [y, -3, 3],
                          [gnuplot_preamble, "set view map"],
                          [gnuplot_pm3d, true],
                          [grid, 150, 150])$

     See also `julia_parameter'.

54.4 Definitions for Koch snowflakes

Function: snowmap (<ent>, <nn>)

     Koch snowflake sets. Function `snowmap' plots the snow Koch map
     over the vertex of an initial closed polygonal, in the complex
     plane. Here the orientation of the polygon is important. Argument
     <nn> is the number of recursive applications of Koch
     transformation; <nn> must be small (5 or 6).

     Examples:

          (%i1) load(fractals)$
          (%i2) plot2d([discrete,
                        snowmap([1,exp(%i*%pi*2/3),exp(-%i*%pi*2/3),1],4)])$
          (%i3) plot2d([discrete,
                        snowmap([1,exp(-%i*%pi*2/3),exp(%i*%pi*2/3),1],4)])$
          (%i4) plot2d([discrete, snowmap([0,1,1+%i,%i,0],4)])$
          (%i5) plot2d([discrete, snowmap([0,%i,1+%i,1,0],4)])$

54.5 Definitions for Peano maps

Continuous curves that cover an area. Warning: the number of points
exponentially grows with <n>.
Function: hilbertmap (<nn>)

     Hilbert map. Argument <nn> must be small (5, for example).  Maxima
     can crash if <nn> is 7 or greater.

     Example:

          (%i1) load(fractals)$
          (%i2) plot2d([discrete,hilbertmap(6)])$

Function: sierpinskimap (<nn>)

     Sierpinski map. Argument <nn> must be small (5, for example).
     Maxima can crash if <nn> is 7 or greater.

     Example:

          (%i1) load(fractals)$
          (%i2) plot2d([discrete,sierpinskimap(6)])$

55 ggf


Functions and Variables for ggf

55.1 Functions and Variables for ggf

Option variable: GGFINFINITY

     Default value: 3

     This is an option variable for function `ggf'.

     When computing the continued fraction of the generating function,
     a partial quotient having a degree (strictly) greater than
     <GGFINFINITY> will be discarded and the current convergent will be
     considered as the exact value of the generating function; most
     often the degree of all partial quotients will be 0 or 1; if you
     use a greater value, then you should give enough terms in order to
     make the computation accurate enough.

     See also `ggf'.

Option variable: GGFCFMAX

     Default value: 3

     This is an option variable for function `ggf'.

     When computing the continued fraction of the generating function,
     if no good result has been found (see the <GGFINFINITY> flag)
     after having computed <GGFCFMAX> partial quotients, the generating
     function will be considered as not being a fraction of two
     polynomials and the function will exit. Put freely a greater value
     for more complicated generating functions.

     See also `ggf'.

Function: ggf (<l>)

     Compute the generating function (if it is a fraction of two
     polynomials) of a sequence, its first terms being given. <l> is a
     list of numbers.

     The solution is returned as a fraction of two polynomials.  If no
     solution has been found, it returns with `done'.

     This function is controlled by global variables <GGFINFINITY> and
     <GGFCFMAX>. See also <GGFINFINITY> and <GGFCFMAX>.

     To use this function write first `load("ggf")'.

56 graphs


Introduction to graphs
Functions and Variables for graphs

56.1 Introduction to graphs

The `graphs' package provides graph and digraph data structure for
Maxima. Graphs and digraphs are simple (have no multiple edges nor
loops), although digraphs can have a directed edge from <u> to <v> and
a directed edge from <v> to <u>.

   Internally graphs are represented by adjacency lists and implemented
as a lisp structures. Vertices are identified by their ids (an id is an
integer). Edges/arcs are represented by lists of length 2. Labels can be
assigned to vertices of graphs/digraphs and weights can be assigned to
edges/arcs of graphs/digraphs.

   There is a `draw_graph' function for drawing graphs. Graphs are
drawn using a force based vertex positioning algorithm. `draw_graph'
can also use graphviz programs available from
`http://www.graphviz.org'. `draw_graph' is based on the maxima `draw'
package.

   To use the `graphs' package, first load it with `load(graphs)'.

56.2 Functions and Variables for graphs

Function: create_graph (<v_list>, <e_list>)
Function: create_graph (<n>, <e_list>)
Function: create_graph (<v_list>, <e_list>, <directed>)

     Creates a new graph on the set of vertices <v_list> and with edges
     <e_list>.

     <v_list> is a list of vertices (`[v1, v2,..., vn]') or a list of
     vertices together with vertex labels (`[[v1,l1], [v2,l2],...,
     [vn,ln]]').

     <n> is the number of vertices. Vertices will be identified by
     integers from 0 to n-1.

     <e_list> is a list of edges (`[e1, e2,..., em]') or a list of
     edges together with edge-weights (`[[e1, w1], ..., [em, wm]]').

     If <directed> is not `false', a directed graph will be returned.

     Example 1: create a cycle on 3 vertices:
          (%i1) load (graphs)$
          (%i2) g : create_graph([1,2,3], [[1,2], [2,3], [1,3]])$
          (%i3) print_graph(g)$
          Graph on 3 vertices with 3 edges.
          Adjacencies:
            3 :  1  2
            2 :  3  1
            1 :  3  2

     Example 2: create a cycle on 3 vertices with edge weights:
          (%i1) load (graphs)$
          (%i2) g : create_graph([1,2,3], [[[1,2], 1.0], [[2,3], 2.0],
                                    [[1,3], 3.0]])$
          (%i3) print_graph(g)$
          Graph on 3 vertices with 3 edges.
          Adjacencies:
            3 :  1  2
            2 :  3  1
            1 :  3  2

     Example 3: create a directed graph:
          (%i1) load (graphs)$
          (%i2) d : create_graph(
                  [1,2,3,4],
                  [
                   [1,3], [1,4],
                   [2,3], [2,4]
                  ],
                  'directed = true)$
          (%i3) print_graph(d)$
          Digraph on 4 vertices with 4 arcs.
          Adjacencies:
            4 :
            3 :
            2 :  4  3
            1 :  4  3

Function: copy_graph (<g>)

     Returns a copy of the graph <g>.

Function: circulant_graph (<n>, <d>)

     Returns the circulant graph with parameters <n> and <d>.

     Example:
          (%i1) load (graphs)$
          (%i2) g : circulant_graph(10, [1,3])$
          (%i3) print_graph(g)$
          Graph on 10 vertices with 20 edges.
          Adjacencies:
            9 :  2  6  0  8
            8 :  1  5  9  7
            7 :  0  4  8  6
            6 :  9  3  7  5
            5 :  8  2  6  4
            4 :  7  1  5  3
            3 :  6  0  4  2
            2 :  9  5  3  1
            1 :  8  4  2  0
            0 :  7  3  9  1

Function: clebsch_graph ()

     Returns the Clebsch graph.

Function: complement_graph (<g>)

     Returns the complement of the graph <g>.

Function: complete_bipartite_graph (<n>, <m>)

     Returns the complete bipartite graph on <n+m> vertices.

Function: complete_graph (<n>)

     Returns the complete graph on <n> vertices.

Function: cycle_digraph (<n>)

     Returns the directed cycle on <n> vertices.

Function: cycle_graph (<n>)

     Returns the cycle on <n> vertices.

Function: cuboctahedron_graph (<n>)

     Returns the cuboctahedron graph.

Function: cube_graph (<n>)

     Returns the <n>-dimensional cube.

Function: dodecahedron_graph ()

     Returns the dodecahedron graph.

Function: empty_graph (<n>)

     Returns the empty graph on <n> vertices.

Function: flower_snark (<n>)

     Returns the flower graph on <4n> vertices.

     Example:
          (%i1) load (graphs)$
          (%i2) f5 : flower_snark(5)$
          (%i3) chromatic_index(f5);
          (%o3)                           4

Function: from_adjacency_matrix (<A>)

     Returns the graph represented by its adjacency matrix <A>.

Function: frucht_graph ()

     Returns the Frucht graph.

Function: graph_product (<g1>, <g1>)

     Returns the direct product of graphs <g1> and <g2>.

     Example:
          (%i1) load (graphs)$
          (%i2) grid : graph_product(path_graph(3), path_graph(4))$
          (%i3) draw_graph(grid)$

Function: graph_union (<g1>, <g1>)

     Returns the union (sum) of graphs <g1> and <g2>.

Function: grid_graph (<n>, <m>)

     Returns the <n x m> grid.

Function: great_rhombicosidodecahedron_graph ()

     Returns the great rhombicosidodecahedron graph.

Function: great_rhombicuboctahedron_graph ()

     Returns the great rhombicuboctahedron graph.

Function: grotzch_graph ()

     Returns the Grotzch graph.

Function: heawood_graph ()

     Returns the Heawood graph.

Function: icosahedron_graph ()

     Returns the icosahedron graph.

Function: icosidodecahedron_graph ()

     Returns the icosidodecahedron graph.

Function: induced_subgraph (<V>, <g>)

     Returns the graph induced on the subset <V> of vertices of the
     graph <g>.

     Example:
          (%i1) load (graphs)$
          (%i2) p : petersen_graph()$
          (%i3) V : [0,1,2,3,4]$
          (%i4) g : induced_subgraph(V, p)$
          (%i5) print_graph(g)$
          Graph on 5 vertices with 5 edges.
          Adjacencies:
            4 :  3  0
            3 :  2  4
            2 :  1  3
            1 :  0  2
            0 :  1  4

Function: line_graph (<g>)

     Returns the line graph of the graph <g>.

Function: make_graph (<vrt>, <f>)
Function: make_graph (<vrt>, <f>, <oriented>)

     Creates a graph using a predicate function <f>.

     <vrt> is a list/set of vertices or an integer. If <vrt> is an
     integer, then vertices of the graph will be integers from 1 to
     <vrt>.

     <f> is a predicate function. Two vertices <a> and <b> will be
     connected if `f(a,b)=true'.

     If <directed> is not <false>, then the graph will be directed.

     Example 1:
          (%i1) load(graphs)$
          (%i2) g : make_graph(powerset({1,2,3,4,5}, 2), disjointp)$
          (%i3) is_isomorphic(g, petersen_graph());
          (%o3)                         true
          (%i4) get_vertex_label(1, g);
          (%o4)                        {1, 2}

     Example 2:
          (%i1) load(graphs)$
          (%i2) f(i, j) := is (mod(j, i)=0)$
          (%i3) g : make_graph(20, f, directed=true)$
          (%i4) out_neighbors(4, g);
          (%o4)                    [8, 12, 16, 20]
          (%i5) in_neighbors(18, g);
          (%o5)                    [1, 2, 3, 6, 9]

Function: mycielski_graph (<g>)

     Returns the mycielskian graph of the graph <g>.

Function: new_graph ()

     Returns the graph with no vertices and no edges.

Function: path_digraph (<n>)

     Returns the directed path on <n> vertices.

Function: path_graph (<n>)

     Returns the path on <n> vertices.

Function: petersen_graph ()
Function: petersen_graph (<n>, <d>)

     Returns the petersen graph <P_{n,d}>. The default values for <n>
     and <d> are `n=5' and `d=2'.

Function: random_bipartite_graph (<a>, <b>, <p>)

     Returns a random bipartite graph on `a+b' vertices. Each edge is
     present with probability <p>.

Function: random_digraph (<n>, <p>)

     Returns a random directed graph on <n> vertices. Each arc is
     present with probability <p>.

Function: random_regular_graph (<n>)
Function: random_regular_graph (<n>, <d>)

     Returns a random <d>-regular graph on <n> vertices. The default
     value for <d> is `d=3'.

Function: random_graph (<n>, <p>)

     Returns a random graph on <n> vertices. Each edge is present with
     probability <p>.

Function: random_graph1 (<n>, <m>)

     Returns a random graph on <n> vertices and random <m> edges.

Function: random_network (<n>, <p>, <w>)

     Returns a random network on <n> vertices. Each arc is present with
     probability <p> and has a weight in the range `[0,w]'. The
     function returns a list `[network, source, sink]'.

     Example:
          (%i1) load (graphs)$
          (%i2) [net, s, t] : random_network(50, 0.2, 10.0);
          (%o2)                   [DIGRAPH, 50, 51]
          (%i3) max_flow(net, s, t)$
          (%i4) first(%);
          (%o4)                   27.65981397932507

Function: random_tournament (<n>)

     Returns a random tournament on <n> vertices.

Function: random_tree (<n>)

     Returns a random tree on <n> vertices.

Function: small_rhombicosidodecahedron_graph ()

     Returns the small rhombicosidodecahedron graph.

Function: small_rhombicuboctahedron_graph ()

     Returns the small rhombicuboctahedron graph.

Function: snub_cube_graph ()

     Returns the snub cube graph.

Function: snub_dodecahedron_graph ()

     Returns the snub dodecahedron graph.

Function: truncated_cube_graph ()

     Returns the truncated cube graph.

Function: truncated_dodecahedron_graph ()

     Returns the truncated dodecahedron graph.

Function: truncated_icosahedron_graph ()

     Returns the truncated icosahedron graph.

Function: truncated_tetrahedron_graph ()

     Returns the truncated tetrahedron graph.

Function: tutte_graph ()

     Returns the Tutte graph.

Function: underlying_graph (<g>)

     Returns the underlying graph of the directed graph <g>.

Function: wheel_graph (<n>)

     Returns the wheel graph on <n+1> vertices.

Function: adjacency_matrix (<gr>)

     Returns the adjacency matrix of the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) c5 : cycle_graph(4)$
          (%i3) adjacency_matrix(c5);
                                   [ 0  1  0  1 ]
                                   [            ]
                                   [ 1  0  1  0 ]
          (%o3)                    [            ]
                                   [ 0  1  0  1 ]
                                   [            ]
                                   [ 1  0  1  0 ]

Function: average_degree (<gr>)

     Returns the average degree of vertices in the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) average_degree(grotzch_graph());
                                         40
          (%o2)                          --
                                         11

Function: biconnected_components (<gr>)

     Returns the (vertex sets of) 2-connected components of the graph
     <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g : create_graph(
                      [1,2,3,4,5,6,7],
                      [
                       [1,2],[2,3],[2,4],[3,4],
                       [4,5],[5,6],[4,6],[6,7]
                      ])$
          (%i3) biconnected_components(g);
          (%o3)        [[6, 7], [4, 5, 6], [1, 2], [2, 3, 4]]

Function: bipartition (<gr>)

     Returns a bipartition of the vertices of the graph <gr> or an empty
     list if <gr> is not bipartite.

     Example:

          (%i1) load (graphs)$
          (%i2) h : heawood_graph()$
          (%i3) [A,B]:bipartition(h);
          (%o3)  [[8, 12, 6, 10, 0, 2, 4], [13, 5, 11, 7, 9, 1, 3]]
          (%i4) draw_graph(h, show_vertices=A, program=circular)$

Function: chromatic_index (<gr>)

     Returns the chromatic index of the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) p : petersen_graph()$
          (%i3) chromatic_index(p);
          (%o3)                           4

Function: chromatic_number (<gr>)

     Returns the chromatic number of the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) chromatic_number(cycle_graph(5));
          (%o2)                           3
          (%i3) chromatic_number(cycle_graph(6));
          (%o3)                           2

Function: clear_edge_weight (<e>, <gr>)

     Removes the weight of the edge  <e> in the graph <gr>.

     Example:

          (%i1) load (graphs)$
          (%i2) g : create_graph(3, [[[0,1], 1.5], [[1,2], 1.3]])$
          (%i3) get_edge_weight([0,1], g);
          (%o3)                          1.5
          (%i4) clear_edge_weight([0,1], g)$
          (%i5) get_edge_weight([0,1], g);
          (%o5)                           1

Function: clear_vertex_label (<v>, <gr>)

     Removes the label of the vertex <v> in the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g : create_graph([[0,"Zero"], [1, "One"]], [[0,1]])$
          (%i3) get_vertex_label(0, g);
          (%o3)                         Zero
          (%i4) clear_vertex_label(0, g);
          (%o4)                         done
          (%i5) get_vertex_label(0, g);
          (%o5)                         false

Function: connected_components (<gr>)

     Returns the (vertex sets of) connected components of the graph
     <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g: graph_union(cycle_graph(5), path_graph(4))$
          (%i3) connected_components(g);
          (%o3)            [[1, 2, 3, 4, 0], [8, 7, 6, 5]]

Function: diameter (<gr>)

     Returns the diameter of the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) diameter(dodecahedron_graph());
          (%o2)                           5

Function: edge_coloring (<gr>)

     Returns an optimal coloring of the edges of the graph <gr>.

     The function returns the chromatic index and a list representing
     the coloring of the edges of <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) p : petersen_graph()$
          (%i3) [ch_index, col] : edge_coloring(p);
          (%o3) [4, [[[0, 5], 3], [[5, 7], 1], [[0, 1], 1], [[1, 6], 2],
          [[6, 8], 1], [[1, 2], 3], [[2, 7], 4], [[7, 9], 2], [[2, 3], 2],
          [[3, 8], 3], [[5, 8], 2], [[3, 4], 1], [[4, 9], 4], [[6, 9], 3],
          [[0, 4], 2]]]
          (%i4) assoc([0,1], col);
          (%o4)                           1
          (%i5) assoc([0,5], col);
          (%o5)                           3

Function: degree_sequence (<gr>)

     Returns the list of vertex degrees of the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) degree_sequence(random_graph(10, 0.4));
          (%o2)            [2, 2, 2, 2, 2, 2, 3, 3, 3, 3]

Function: edge_connectivity (<gr>)

     Returns the edge-connectivity of the graph <gr>.

     See also `min_edge_cut'.

Function: edges (<gr>)

     Returns the list of edges (arcs) in a (directed) graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) edges(complete_graph(4));
          (%o2)   [[2, 3], [1, 3], [1, 2], [0, 3], [0, 2], [0, 1]]

Function: get_edge_weight (<e>, <gr>)
Function: get_edge_weight (<e>, <gr>, <ifnot>)

     Returns the weight of the edge <e> in the graph <gr>.

     If there is no weight assigned to the edge, the function returns
     1. If the edge is not present in the graph, the function signals
     an error or returns the optional argument <ifnot>.

     Example:
          (%i1) load (graphs)$
          (%i2) c5 : cycle_graph(5)$
          (%i3) get_edge_weight([1,2], c5);
          (%o3)                           1
          (%i4) set_edge_weight([1,2], 2.0, c5);
          (%o4)                         done
          (%i5) get_edge_weight([1,2], c5);
          (%o5)                          2.0

Function: get_vertex_label (<v>, <gr>)

     Returns the label of the vertex <v> in the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g : create_graph([[0,"Zero"], [1, "One"]], [[0,1]])$
          (%i3) get_vertex_label(0, g);
          (%o3)                         Zero

Function: graph_charpoly (<gr>, <x>)

     Returns the characteristic polynomial (in variable <x>) of the
     graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) p : petersen_graph()$
          (%i3) graph_charpoly(p, x), factor;
                                             5        4
          (%o3)               (x - 3) (x - 1)  (x + 2)

Function: graph_center (<gr>)

     Returns the center of the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g : grid_graph(5,5)$
          (%i3) graph_center(g);
          (%o3)                         [12]

Function: graph_eigenvalues (<gr>)

     Returns the eigenvalues of the graph <gr>. The function returns
     eigenvalues in the same format as maxima `eigenvalue' function.

     Example:
          (%i1) load (graphs)$
          (%i2) p : petersen_graph()$
          (%i3) graph_eigenvalues(p);
          (%o3)               [[3, - 2, 1], [1, 4, 5]]

Function: graph_periphery (<gr>)

     Returns the periphery of the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g : grid_graph(5,5)$
          (%i3) graph_periphery(g);
          (%o3)                    [24, 20, 4, 0]

Function: graph_size (<gr>)

     Returns the number of edges in the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) p : petersen_graph()$
          (%i3) graph_size(p);
          (%o3)                          15

Function: graph_order (<gr>)

     Returns the number of vertices in the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) p : petersen_graph()$
          (%i3) graph_order(p);
          (%o3)                          10

Function: girth (<gr>)

     Returns the length of the shortest cycle in <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g : heawood_graph()$
          (%i3) girth(g);
          (%o3)                           6

Function: hamilton_cycle (<gr>)

     Returns the Hamilton cycle of the graph <gr> or an empty list if
     <gr> is not hamiltonian.

     Example:
          (%i1) load (graphs)$
          (%i2) c : cube_graph(3)$
          (%i3) hc : hamilton_cycle(c);
          (%o3)              [7, 3, 2, 6, 4, 0, 1, 5, 7]
          (%i4) draw_graph(c, show_edges=vertices_to_cycle(hc))$

Function: hamilton_path (<gr>)

     Returns the Hamilton path of the graph <gr> or an empty list if
     <gr> does not have a Hamilton path.

     Example:
          (%i1) load (graphs)$
          (%i2) p : petersen_graph()$
          (%i3) hp : hamilton_path(p);
          (%o3)            [0, 5, 7, 2, 1, 6, 8, 3, 4, 9]
          (%i4) draw_graph(p, show_edges=vertices_to_path(hp))$

Function: isomorphism (<gr1>, <gr2>)

     Returns a an isomorphism between graphs/digraphs <gr1> and <gr2>.
     If <gr1> and <gr2> are not isomorphic, it returns an empty list.

     Example:
          (%i1) load (graphs)$
          (%i2) clk5:complement_graph(line_graph(complete_graph(5)))$
          (%i3) isomorphism(clk5, petersen_graph());
          (%o3) [9 -> 0, 2 -> 1, 6 -> 2, 5 -> 3, 0 -> 4, 1 -> 5, 3 -> 6,
                                                    4 -> 7, 7 -> 8, 8 -> 9]

Function: in_neighbors (<v>, <gr>)

     Returns the list of in-neighbors of the vertex <v> in the directed
     graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) p : path_digraph(3)$
          (%i3) in_neighbors(2, p);
          (%o3)                          [1]
          (%i4) out_neighbors(2, p);
          (%o4)                          []

Function: is_biconnected (<gr>)

     Returns `true' if <gr> is 2-connected and `false' otherwise.

     Example:
          (%i1) load (graphs)$
          (%i2) is_biconnected(cycle_graph(5));
          (%o2)                         true
          (%i3) is_biconnected(path_graph(5));
          (%o3)                         false

Function: is_bipartite (<gr>)

     Returns `true' if <gr> is bipartite (2-colorable) and `false'
     otherwise.

     Example:
          (%i1) load (graphs)$
          (%i2) is_bipartite(petersen_graph());
          (%o2)                         false
          (%i3) is_bipartite(heawood_graph());
          (%o3)                         true

Function: is_connected (<gr>)

     Returns `true' if the graph <gr> is connected and `false'
     otherwise.

     Example:
          (%i1) load (graphs)$
          (%i2) is_connected(graph_union(cycle_graph(4), path_graph(3)));
          (%o2)                         false

Function: is_digraph (<gr>)

     Returns `true' if <gr> is a directed graph and `false' otherwise.

     Example:
          (%i1) load (graphs)$
          (%i2) is_digraph(path_graph(5));
          (%o2)                         false
          (%i3) is_digraph(path_digraph(5));
          (%o3)                         true

Function: is_edge_in_graph (<e>, <gr>)

     Returns `true' if <e> is an edge (arc) in the (directed) graph <g>
     and `false' otherwise.

     Example:
          (%i1) load (graphs)$
          (%i2) c4 : cycle_graph(4)$
          (%i3) is_edge_in_graph([2,3], c4);
          (%o3)                         true
          (%i4) is_edge_in_graph([3,2], c4);
          (%o4)                         true
          (%i5) is_edge_in_graph([2,4], c4);
          (%o5)                         false
          (%i6) is_edge_in_graph([3,2], cycle_digraph(4));
          (%o6)                         false

Function: is_graph (<gr>)

     Returns `true' if <gr> is a graph and `false' otherwise.

     Example:
          (%i1) load (graphs)$
          (%i2) is_graph(path_graph(5));
          (%o2)                         true
          (%i3) is_graph(path_digraph(5));
          (%o3)                         false

Function: is_graph_or_digraph (<gr>)

     Returns `true' if <gr> is a graph or a directed graph and `false'
     otherwise.

     Example:
          (%i1) load (graphs)$
          (%i2) is_graph_or_digraph(path_graph(5));
          (%o2)                         true
          (%i3) is_graph_or_digraph(path_digraph(5));
          (%o3)                         true

Function: is_isomorphic (<gr1>, <gr2>)

     Returns `true' if graphs/digraphs <gr1> and <gr2> are isomorphic
     and `false' otherwise.

     See also `isomorphism'.

     Example:
          (%i1) load (graphs)$
          (%i2) clk5:complement_graph(line_graph(complete_graph(5)))$
          (%i3) is_isomorphic(clk5, petersen_graph());
          (%o3)                         true

Function: is_planar (<gr>)

     Returns `true' if <gr> is a planar graph and `false' otherwise.

     The algorithm used is the Demoucron's algorithm, which is a
     quadratic time algorithm.

     Example:
          (%i1) load (graphs)$
          (%i2) is_planar(dodecahedron_graph());
          (%o2)                         true
          (%i3) is_planar(petersen_graph());
          (%o3)                         false
          (%i4) is_planar(petersen_graph(10,2));
          (%o4)                         true

Function: is_sconnected (<gr>)

     Returns `true' if the directed graph <gr> is strongly connected and
     `false' otherwise.

     Example:
          (%i1) load (graphs)$
          (%i2) is_sconnected(cycle_digraph(5));
          (%o2)                         true
          (%i3) is_sconnected(path_digraph(5));
          (%o3)                         false

Function: is_vertex_in_graph (<v>, <gr>)

     Returns `true' if <v> is a vertex in the graph <g> and `false'
     otherwise.

     Example:
          (%i1) load (graphs)$
          (%i2) c4 : cycle_graph(4)$
          (%i3) is_vertex_in_graph(0, c4);
          (%o3)                         true
          (%i4) is_vertex_in_graph(6, c4);
          (%o4)                         false

Function: is_tree (<gr>)

     Returns `true' if <gr> is a tree and `false'  otherwise.

     Example:
          (%i1) load (graphs)$
          (%i2) is_tree(random_tree(4));
          (%o2)                         true
          (%i3) is_tree(graph_union(random_tree(4), random_tree(5)));
          (%o3)                         false

Function: laplacian_matrix (<gr>)

     Returns the laplacian matrix of the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) laplacian_matrix(cycle_graph(5));
                             [  2   - 1   0    0   - 1 ]
                             [                         ]
                             [ - 1   2   - 1   0    0  ]
                             [                         ]
          (%o2)              [  0   - 1   2   - 1   0  ]
                             [                         ]
                             [  0    0   - 1   2   - 1 ]
                             [                         ]
                             [ - 1   0    0   - 1   2  ]

Function: max_clique (<gr>)

     Returns a maximum clique of the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g : random_graph(100, 0.5)$
          (%i3) max_clique(g);
          (%o3)          [6, 12, 31, 36, 52, 59, 62, 63, 80]

Function: max_degree (<gr>)

     Returns the maximal degree of vertices of the graph <gr> and a
     vertex of maximal degree.

     Example:
          (%i1) load (graphs)$
          (%i2) g : random_graph(100, 0.02)$
          (%i3) max_degree(g);
          (%o3)                        [6, 79]
          (%i4) vertex_degree(95, g);
          (%o4)                           2

Function: max_flow (<net>, <s>, <t>)

     Returns a maximum flow through the network <net> with the source
     <s> and the sink <t>.

     The function returns the value of the maximal flow and a list
     representing the weights of the arcs in the optimal flow.

     Example:
          (%i1) load (graphs)$
          (%i2) net : create_graph(
            [1,2,3,4,5,6],
            [[[1,2], 1.0],
             [[1,3], 0.3],
             [[2,4], 0.2],
             [[2,5], 0.3],
             [[3,4], 0.1],
             [[3,5], 0.1],
             [[4,6], 1.0],
             [[5,6], 1.0]],
            directed=true)$
          (%i3) [flow_value, flow] : max_flow(net, 1, 6);
          (%o3) [0.7, [[[1, 2], 0.5], [[1, 3], 0.2], [[2, 4], 0.2],
          [[2, 5], 0.3], [[3, 4], 0.1], [[3, 5], 0.1], [[4, 6], 0.3],
          [[5, 6], 0.4]]]
          (%i4) fl : 0$
          (%i5) for u in out_neighbors(1, net)
               do fl : fl + assoc([1, u], flow)$
          (%i6) fl;
          (%o6)                          0.7

Function: max_independent_set (<gr>)

     Returns a maximum independent set of the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) d : dodecahedron_graph()$
          (%i3) mi : max_independent_set(d);
          (%o3)             [0, 3, 5, 9, 10, 11, 18, 19]
          (%i4) draw_graph(d, show_vertices=mi)$

Function: max_matching (<gr>)

     Returns a maximum matching of the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) d : dodecahedron_graph()$
          (%i3) m : max_matching(d);
          (%o3) [[5, 7], [8, 9], [6, 10], [14, 19], [13, 18], [12, 17],
                                         [11, 16], [0, 15], [3, 4], [1, 2]]
          (%i4) draw_graph(d, show_edges=m)$

Function: min_degree (<gr>)

     Returns the minimum degree of vertices of the graph <gr> and a
     vertex of minimum degree.

     Example:
          (%i1) load (graphs)$
          (%i2) g : random_graph(100, 0.1)$
          (%i3) min_degree(g);
          (%o3)                        [3, 49]
          (%i4) vertex_degree(21, g);
          (%o4)                           9

Function: min_edge_cut (<gr>)

     Returns the minimum edge cut in the graph <gr>.

     See also `edge_connectivity'.

Function: min_vertex_cover (<gr>)

     Returns the minimum vertex cover of the graph <gr>.

Function: min_vertex_cut (<gr>)

     Returns the minimum vertex cut in the graph <gr>.

     See also `vertex_connectivity'.

Function: minimum_spanning_tree (<gr>)

     Returns the minimum spanning tree of the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g : graph_product(path_graph(10), path_graph(10))$
          (%i3) t : minimum_spanning_tree(g)$
          (%i4) draw_graph(g, show_edges=edges(t))$

Function: neighbors (<v>, <gr>)

     Returns the list of neighbors of the vertex <v> in the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) p : petersen_graph()$
          (%i3) neighbors(3, p);
          (%o3)                       [4, 8, 2]

Function: odd_girth (<gr>)

     Returns the length of the shortest odd cycle in the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g : graph_product(cycle_graph(4), cycle_graph(7))$
          (%i3) girth(g);
          (%o3)                           4
          (%i4) odd_girth(g);
          (%o4)                           7

Function: out_neighbors (<v>, <gr>)

     Returns the list of out-neighbors of the vertex <v> in the directed
     graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) p : path_digraph(3)$
          (%i3) in_neighbors(2, p);
          (%o3)                          [1]
          (%i4) out_neighbors(2, p);
          (%o4)                          []

Function: planar_embedding (<gr>)

     Returns the list of facial walks in a planar embedding of <gr> and
     `false' if <gr> is not a planar graph.

     The graph <gr> must be biconnected.

     The algorithm used is the Demoucron's algorithm, which is a
     quadratic time algorithm.

     Example:
          (%i1) load (graphs)$
          (%i2) planar_embedding(grid_graph(3,3));
          (%o2) [[3, 6, 7, 8, 5, 2, 1, 0], [4, 3, 0, 1], [3, 4, 7, 6],
                                                [8, 7, 4, 5], [1, 2, 5, 4]]

Function: print_graph (<gr>)

     Prints some information about the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) c5 : cycle_graph(5)$
          (%i3) print_graph(c5)$
          Graph on 5 vertices with 5 edges.
          Adjacencies:
            4 :  0  3
            3 :  4  2
            2 :  3  1
            1 :  2  0
            0 :  4  1
          (%i4) dc5 : cycle_digraph(5)$
          (%i5) print_graph(dc5)$
          Digraph on 5 vertices with 5 arcs.
          Adjacencies:
            4 :  0
            3 :  4
            2 :  3
            1 :  2
            0 :  1
          (%i6) out_neighbors(0, dc5);
          (%o6)                          [1]

Function: radius (<gr>)

     Returns the radius of the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) radius(dodecahedron_graph());
          (%o2)                           5

Function: set_edge_weight (<e>, <w>, <gr>)

     Assigns the weight <w> to the edge <e> in the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g : create_graph([1, 2], [[[1,2], 1.2]])$
          (%i3) get_edge_weight([1,2], g);
          (%o3)                          1.2
          (%i4) set_edge_weight([1,2], 2.1, g);
          (%o4)                         done
          (%i5) get_edge_weight([1,2], g);
          (%o5)                          2.1

Function: set_vertex_label (<v>, <l>, <gr>)

     Assigns the label <l> to the vertex <v> in the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g : create_graph([[1, "One"], [2, "Two"]], [[1,2]])$
          (%i3) get_vertex_label(1, g);
          (%o3)                          One
          (%i4) set_vertex_label(1, "oNE", g);
          (%o4)                         done
          (%i5) get_vertex_label(1, g);
          (%o5)                          oNE

Function: shortest_path (<u>, <v>, <gr>)

     Returns the shortest path from <u> to <v> in the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) d : dodecahedron_graph()$
          (%i3) path : shortest_path(0, 7, d);
          (%o3)                   [0, 1, 19, 13, 7]
          (%i4) draw_graph(d, show_edges=vertices_to_path(path))$

Function: shortest_weighted_path (<u>, <v>, <gr>)

     Returns the length of the shortest weighted path and the shortest
     weighted path from <u> to <v> in the graph <gr>.

     The length of a weighted path is the sum of edge weights of edges
     in the path. If an edge has no weight, then it has a default
     weight 1.

     Example:

          (%i1) load (graphs)$
          (%i2) g: petersen_graph(20, 2)$
          (%i3) for e in edges(g) do set_edge_weight(e, random(1.0), g)$
          (%i4) shortest_weighted_path(0, 10, g);
          (%o4) [2.575143920268482, [0, 20, 38, 36, 34, 32, 30, 10]]

Function: strong_components (<gr>)

     Returns the strong components of a directed graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) t : random_tournament(4)$
          (%i3) strong_components(t);
          (%o3)                 [[1], [0], [2], [3]]
          (%i4) vertex_out_degree(3, t);
          (%o4)                           3

Function: topological_sort (<dag>)

     Returns a topological sorting of the vertices of a directed graph
     <dag> or an empty list if <dag> is not a directed acyclic graph.

     Example:
          (%i1) load (graphs)$
          (%i2) g:create_graph(
                   [1,2,3,4,5],
                   [
                    [1,2], [2,5], [5,3],
                    [5,4], [3,4], [1,3]
                   ],
                   directed=true)$
          (%i3) topological_sort(g);
          (%o3)                    [1, 2, 5, 3, 4]

Function: vertex_connectivity (<g>)

     Returns the vertex connectivity of the graph <g>.

     See also `min_vertex_cut'.

Function: vertex_degree (<v>, <gr>)

     Returns the degree of the vertex <v> in the graph <gr>.

Function: vertex_distance (<u>, <v>, <gr>)

     Returns the length of the shortest path between <u> and <v> in the
     (directed) graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) d : dodecahedron_graph()$
          (%i3) vertex_distance(0, 7, d);
          (%o3)                           4
          (%i4) shortest_path(0, 7, d);
          (%o4)                   [0, 1, 19, 13, 7]

Function: vertex_eccentricity (<v>, <gr>)

     Returns the eccentricity of the vertex <v> in the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g:cycle_graph(7)$
          (%i3) vertex_eccentricity(0, g);
          (%o3)                           3

Function: vertex_in_degree (<v>, <gr>)

     Returns the in-degree of the vertex <v> in the directed graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) p5 : path_digraph(5)$
          (%i3) print_graph(p5)$
          Digraph on 5 vertices with 4 arcs.
          Adjacencies:
            4 :
            3 :  4
            2 :  3
            1 :  2
            0 :  1
          (%i4) vertex_in_degree(4, p5);
          (%o4)                           1
          (%i5) in_neighbors(4, p5);
          (%o5)                          [3]

Function: vertex_out_degree (<v>, <gr>)

     Returns the out-degree of the vertex <v> in the directed graph
     <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) t : random_tournament(10)$
          (%i3) vertex_out_degree(0, t);
          (%o3)                           2
          (%i4) out_neighbors(0, t);
          (%o4)                        [7, 1]

Function: vertices (<gr>)

     Returns the list of vertices in the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) vertices(complete_graph(4));
          (%o2)                     [3, 2, 1, 0]

Function: vertex_coloring (<gr>)

     Returns an optimal coloring of the vertices of the graph <gr>.

     The function returns the chromatic number and a list representing
     the coloring of the vertices of <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) p:petersen_graph()$
          (%i3) vertex_coloring(p);
          (%o3) [3, [[0, 2], [1, 3], [2, 2], [3, 3], [4, 1], [5, 3],
                                           [6, 1], [7, 1], [8, 2], [9, 2]]]

Function: wiener_index (<gr>)

     Returns the Wiener index of the graph <gr>.

     Example:
          (%i2) wiener_index(dodecahedron_graph());
          (%o2)                          500

Function: add_edge (<e>, <gr>)

     Adds the edge <e> to the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) p : path_graph(4)$
          (%i3) neighbors(0, p);
          (%o3)                          [1]
          (%i4) add_edge([0,3], p);
          (%o4)                         done
          (%i5) neighbors(0, p);
          (%o5)                        [3, 1]

Function: add_edges (<e_list>, <gr>)

     Adds all edges in the list <e_list> to the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g : empty_graph(3)$
          (%i3) add_edges([[0,1],[1,2]], g)$
          (%i4) print_graph(g)$
          Graph on 3 vertices with 2 edges.
          Adjacencies:
            2 :  1
            1 :  2  0
            0 :  1

Function: add_vertex (<v>, <gr>)

     Adds the vertex <v> to the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g : path_graph(2)$
          (%i3) add_vertex(2, g)$
          (%i4) print_graph(g)$
          Graph on 3 vertices with 1 edges.
          Adjacencies:
            2 :
            1 :  0
            0 :  1

Function: add_vertices (<v_list>, <gr>)

     Adds all vertices in the list <v_list> to the graph <gr>.

Function: connect_vertices (<v_list>, <u_list>, <gr>)

     Connects all vertices from the list <v_list> with the vertices in
     the list <u_list> in the graph <gr>.

     <v_list> and <u_list> can be single vertices or lists of vertices.

     Example:
          (%i1) load (graphs)$
          (%i2) g : empty_graph(4)$
          (%i3) connect_vertices(0, [1,2,3], g)$
          (%i4) print_graph(g)$
          Graph on 4 vertices with 3 edges.
          Adjacencies:
            3 :  0
            2 :  0
            1 :  0
            0 :  3  2  1

Function: contract_edge (<e>, <gr>)

     Contracts the edge <e> in the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) g: create_graph(
                8, [[0,3],[1,3],[2,3],[3,4],[4,5],[4,6],[4,7]])$
          (%i3) print_graph(g)$
          Graph on 8 vertices with 7 edges.
          Adjacencies:
            7 :  4
            6 :  4
            5 :  4
            4 :  7  6  5  3
            3 :  4  2  1  0
            2 :  3
            1 :  3
            0 :  3
          (%i4) contract_edge([3,4], g)$
          (%i5) print_graph(g)$
          Graph on 7 vertices with 6 edges.
          Adjacencies:
            7 :  3
            6 :  3
            5 :  3
            3 :  5  6  7  2  1  0
            2 :  3
            1 :  3
            0 :  3

Function: remove_edge (<e>, <gr>)

     Removes the edge <e> from the graph <gr>.

     Example:
          (%i1) load (graphs)$
          (%i2) c3 : cycle_graph(3)$
          (%i3) remove_edge([0,1], c3)$
          (%i4) print_graph(c3)$
          Graph on 3 vertices with 2 edges.
          Adjacencies:
            2 :  0  1
            1 :  2
            0 :  2

Function: remove_vertex (<v>, <gr>)

     Removes the vertex <v> from the graph <gr>.

Function: dimacs_export (<gr>, <fl>)
Function: dimacs_export (<gr>, <fl>, <comment1>, ..., <commentn>)

     Exports the graph into the file <fl> in the DIMACS format. Optional
     comments will be added to the top of the file.

Function: dimacs_import (<fl>)

     Returns the graph from file <fl> in the DIMACS format.

Function: graph6_decode (<str>)

     Returns the graph encoded in the graph6 format in the string <str>.

Function: graph6_encode (<gr>)

     Returns a string which encodes the graph <gr> in the graph6 format.

Function: graph6_export (<gr_list>, <fl>)

     Exports graphs in the list <gr_list> to the file <fl> in the
     graph6 format.

Function: graph6_import (<fl>)

     Returns a list of graphs from the file <fl> in the graph6 format.

Function: sparse6_decode (<str>)

     Returns the graph encoded in the sparse6 format in the string
     <str>.

Function: sparse6_encode (<gr>)

     Returns a string which encodes the graph <gr> in the sparse6
     format.

Function: sparse6_export (<gr_list>, <fl>)

     Exports graphs in the list <gr_list> to the file <fl> in the
     sparse6 format.

Function: sparse6_import (<fl>)

     Returns a list of graphs from the file <fl> in the sparse6 format.

Function: draw_graph (<graph>)
Function: draw_graph (<graph>, <option1>, ..., <optionk>)

     Draws the graph using the `draw' package.

     The algorithm used to position vertices is specified by the
     optional argument <program>. The default value is
     `program=spring_embedding'. <draw_graph> can also use the graphviz
     programs for positioning vertices, but graphviz must be installed
     separately.

     Example 1:

          (%i1) load (graphs)$
          (%i2) g:grid_graph(10,10)$
          (%i3) m:max_matching(g)$
          (%i4) draw_graph(g,
             spring_embedding_depth=100,
             show_edges=m, edge_type=dots,
             vertex_size=0)$

     Example 2:

          (%i1) load (graphs)$
          (%i2) g:create_graph(16,
              [
               [0,1],[1,3],[2,3],[0,2],[3,4],[2,4],
               [5,6],[6,4],[4,7],[6,7],[7,8],[7,10],[7,11],
               [8,10],[11,10],[8,9],[11,12],[9,15],[12,13],
               [10,14],[15,14],[13,14]
              ])$
          (%i3) t:minimum_spanning_tree(g)$
          (%i4) draw_graph(
              g,
              show_edges=edges(t),
              show_edge_width=4,
              show_edge_color=green,
              vertex_type=filled_square,
              vertex_size=2
              )$

     Example 3:

          (%i1) load (graphs)$
          (%i2) g:create_graph(16,
              [
               [0,1],[1,3],[2,3],[0,2],[3,4],[2,4],
               [5,6],[6,4],[4,7],[6,7],[7,8],[7,10],[7,11],
               [8,10],[11,10],[8,9],[11,12],[9,15],[12,13],
               [10,14],[15,14],[13,14]
              ])$
          (%i3) mi : max_independent_set(g)$
          (%i4) draw_graph(
              g,
              show_vertices=mi,
              show_vertex_type=filled_up_triangle,
              show_vertex_size=2,
              edge_color=cyan,
              edge_width=3,
              show_id=true,
              text_color=brown
              )$

     Example 4:

          (%i1) load (graphs)$
          (%i2) net : create_graph(
              [0,1,2,3,4,5],
              [
               [[0,1], 3], [[0,2], 2],
               [[1,3], 1], [[1,4], 3],
               [[2,3], 2], [[2,4], 2],
               [[4,5], 2], [[3,5], 2]
              ],
              directed=true
              )$
          (%i3) draw_graph(
              net,
              show_weight=true,
              vertex_size=0,
              show_vertices=[0,5],
              show_vertex_type=filled_square,
              head_length=0.2,
              head_angle=10,
              edge_color="dark-green",
              text_color=blue
              )$

     Example 5:

          (%i1) load(graphs)$
          (%i2) g: petersen_graph(20, 2);
          (%o2)                         GRAPH
          (%i3) draw_graph(g, redraw=true, program=planar_embedding);
          (%o3)                         done

     Example 6:

          (%i1) load(graphs)$
          (%i2) t: tutte_graph();
          (%o2)                         GRAPH
          (%i3) draw_graph(t, redraw=true,
                              fixed_vertices=[1,2,3,4,5,6,7,8,9]);
          (%o3)                         done

Option variable: draw_graph_program

     Default value: <spring_embedding>

     The default value for the program used to position vertices in
     `draw_graph' program.

draw_graph option: show_id

     Default value: <false>

     If <true> then ids of the vertices are displayed.

draw_graph option: show_label

     Default value: <false>

     If <true> then labels of the vertices are displayed.

draw_graph option: label_alignment

     Default value: <center>

     Determines how to align the labels/ids of the vertices. Can be
     `left', `center' or `right'.

draw_graph option: show_weight

     Default value: <false>

     If <true> then weights of the edges are displayed.

draw_graph option: vertex_type

     Default value: <circle>

     Defines how vertices are displayed. See the <point_type> option for
     the `draw' package for possible values.

draw_graph option: vertex_size

     The size of vertices.

draw_graph option: vertex_color

     The color used for displaying vertices.

draw_graph option: show_vertices

     Default value: []

     Display selected vertices in the using a different color.

draw_graph option: show_vertex_type

     Defines how vertices specified in <show_vertices> are displayed.
     See the <point_type> option for the `draw' package for possible
     values.

draw_graph option: show_vertex_size

     The size of vertices in <show_vertices>.

draw_graph option: show_vertex_color

     The color used for displaying vertices in the <show_vertices> list.

draw_graph option: vertex_partition

     Default value: []

     A partition `[[v1,v2,...],...,[vk,...,vn]]' of the vertices of the
     graph. The vertices of each list in the partition will be drawn in
     a different color.

draw_graph option: vertex_coloring

     Specifies coloring of the vertices. The coloring <col> must be
     specified in the format as returned by <vertex_coloring>.

draw_graph option: edge_color

     The color used for displaying edges.

draw_graph option: edge_width

     The width of edges.

draw_graph option: edge_type

     Defines how edges are displayed. See the <line_type> option for the
     `draw' package.

draw_graph option: show_edges

     Display edges specified in the list <e_list> using a different
     color.

draw_graph option: show_edge_color

     The color used for displaying edges in the <show_edges> list.

draw_graph option: show_edge_width

     The width of edges in <show_edges>.

draw_graph option: show_edge_type

     Defines how edges in <show_edges> are displayed. See the
     <line_type> option for the `draw' package.

draw_graph option: edge_partition

     A partition `[[e1,e2,...],...,[ek,...,em]]' of edges of the graph.
     The edges of each list in the partition will be drawn using a
     different color.

draw_graph option: edge_coloring

     The coloring of edges. The coloring must be specified in the
     format as returned by the function <edge_coloring>.

draw_graph option: redraw

     Default value: <false>

     If `true', vertex positions are recomputed even if the positions
     have been saved from a previous drawing of the graph.

draw_graph option: head_angle

     Default value: 15

     The angle for the arrows displayed on arcs (in directed graphs).

draw_graph option: head_length

     Default value: 0.1

     The length for the arrows displayed on arcs (in directed graphs).

draw_graph option: spring_embedding_depth

     Default value: 50

     The number of iterations in the spring embedding graph drawing
     algorithm.

draw_graph option: terminal

     The terminal used for drawing (see the <terminal> option in the
     `draw' package).

draw_graph option: file_name

     The filename of the drawing if terminal is not screen.

draw_graph option: program

     Defines the program used for positioning vertices of the graph.
     Can be one of the graphviz programs (dot, neato, twopi, circ, fdp),
     <circular>, <spring_embedding> or <planar_embedding>.
     <planar_embedding> is only available for 2-connected planar
     graphs. When `program=spring_embedding', a set of vertices with
     fixed position can be specified with the <fixed_vertices> option.

draw_graph option: fixed_vertices

     Specifies a list of vertices which will have positions fixed along
     a regular polygon.  Can be used when `program=spring_embedding'.

Function: vertices_to_path (<v_list>)

     Converts a list <v_list> of vertices to a list of edges of the path
     defined by <v_list>.

Function: vertices_to_cycle (<v_list>)

     Converts a list <v_list> of vertices to a list of edges of the
     cycle defined by <v_list>.

57 grobner


* Introduction to grobner ::
* Functions and Variables for grobner ::

57.1 Introduction to grobner

`grobner' is a package for working with Groebner bases in Maxima.

A tutorial on _Groebner Bases_ can be found at

`http://www.geocities.com/CapeCanaveral/Hall/3131/'

To use the following functions you must load the `grobner.lisp' package.

     load(grobner);

A demo can be started by
     demo("grobner.demo");

or
     batch("grobner.demo")

Some of the calculation in the demo will take a lot of time therefore
the output `grobner-demo.output' of the demo can be found in the same
directory as the demo file.

57.2 Functions and Variables for grobner

Option variable: poly_monomial_order

     Default value: `lex'

     This global switch controls which monomial order is used in
     polynomial and Groebner Bases calculations. If not set, `lex' will
     be used.

Option variable: poly_coefficient_ring

     Default value: `expression_ring'

     This switch indicates the coefficient ring of the polynomials that
     will be used in grobner calculations. If not set, _maxima's_
     general expression ring will be used. This variable may be set to
     `ring_of_integers' if desired.

Option variable: poly_primary_elimination_order

     Default value: `false'

     Name of the default order for eliminated variables in
     elimination-based functions. If not set, `lex' will be used.

Option variable: poly_secondary_elimination_order

     Default value: `false'

     Name of the default order for kept variables in elimination-based
     functions. If not set, `lex' will be used.

Option variable: poly_elimination_order

     Default value: `false'

     Name of the default elimination order used in elimination
     calculations. If set, it overrides the settings in variables
     `poly_primary_elimination_order' and
     `poly_secondary_elimination_order'.  The user must ensure that
     this is a true elimination order valid for the number of
     eliminated variables.

Option variable: poly_return_term_list

     Default value: `false'

     If set to `true', all functions in this package will return each
     polynomial as a list of terms in the current monomial order rather
     than a _maxima_ general expression.

Option variable: poly_grobner_debug

     Default value: `false'

     If set to `true', produce debugging and tracing output.

Option variable: poly_grobner_algorithm

     Default value: `buchberger'

     Possible values:
        * `buchberger'

        * `parallel_buchberger'

        * `gebauer_moeller'

     The name of the algorithm used to find the Groebner Bases.

Option variable: poly_top_reduction_only

     Default value: `false'

     If not `false', use top reduction only whenever possible. Top
     reduction means that division algorithm stops after the first
     reduction.

Function: poly_add (<poly1>, <poly2>, <varlist>)

     Adds two polynomials <poly1> and <poly2>.

          (%i1) poly_add(z+x^2*y,x-z,[x,y,z]);
                                              2
          (%o1)                              x  y + x

Function: poly_subtract (<poly1>, <poly2>, <varlist>)

     Subtracts a polynomial <poly2> from <poly1>.

          (%i1) poly_subtract(z+x^2*y,x-z,[x,y,z]);
                                                2
          (%o1)                          2 z + x  y - x

Function: poly_multiply (<poly1>, <poly2>, <varlist>)

     Returns the product of polynomials <poly1> and <poly2>.

          (%i2) poly_multiply(z+x^2*y,x-z,[x,y,z])-(z+x^2*y)*(x-z),expand;
          (%o1)                                  0

Function: poly_s_polynomial (<poly1>, <poly2>, <varlist>)

     Returns the _syzygy polynomial_ (_S-polynomial_) of two
     polynomials <poly1> and <poly2>.

Function: poly_primitive_part (<poly1>, <varlist>)

     Returns the polynomial <poly> divided by the GCD of its
     coefficients.

          (%i1) poly_primitive_part(35*y+21*x,[x,y]);
          (%o1)                              5 y + 3 x

Function: poly_normalize (<poly>, <varlist>)

     Returns the polynomial <poly> divided by the leading coefficient.
     It assumes that the division is possible, which may not always be
     the case in rings which are not fields.

Function: poly_expand (<poly>, <varlist>)

     This function parses polynomials to internal form and back. It is
     equivalent to `expand(<poly>)' if <poly> parses correctly to a
     polynomial. If the representation is not compatible with a
     polynomial in variables <varlist>, the result is an error.  It can
     be used to test whether an expression correctly parses to the
     internal representation. The following examples illustrate that
     indexed and transcendental function variables are allowed.

          (%i1) poly_expand((x-y)*(y+x),[x,y]);
                                               2    2
          (%o1)                               x  - y
          (%i2) poly_expand((y+x)^2,[x,y]);
                                          2            2
          (%o2)                          y  + 2 x y + x
          (%i3) poly_expand((y+x)^5,[x,y]);
                            5      4         2  3       3  2      4      5
          (%o3)            y  + 5 x y  + 10 x  y  + 10 x  y  + 5 x  y + x
          (%i4) poly_expand(-1-x*exp(y)+x^2/sqrt(y),[x]);
                                                    2
                                            y      x
          (%o4)                       - x %e  + ------- - 1
                                                 sqrt(y)

          (%i5) poly_expand(-1-sin(x)^2+sin(x),[sin(x)]);
                                          2
          (%o5)                      - sin (x) + sin(x) - 1

Function: poly_expt (<poly>, <number>, <varlist>)

     exponentitates <poly> by a positive integer <number>. If <number>
     is not a positive integer number an error will be raised.

          (%i1) poly_expt(x-y,3,[x,y])-(x-y)^3,expand;
          (%o1)                                  0

Function: poly_content (<poly>. <varlist>)

     `poly_content' extracts the GCD of its coefficients

          (%i1) poly_content(35*y+21*x,[x,y]);
          (%o1)                                  7

Function: poly_pseudo_divide (<poly>, <polylist>, <varlist>)

     Pseudo-divide a polynomial <poly> by the list of n polynomials
     <polylist>. Return multiple values. The first value is a list of
     quotients a. The second value is the remainder r. The third
     argument is a scalar coefficient c, such that c*poly can be
     divided by <polylist> within the ring of coefficients, which is
     not necessarily a field. Finally, the fourth value is an integer
     count of the number of reductions performed. The resulting objects
     satisfy the equation:

     c*poly=sum(a[i]*polylist[i],i=1...n)+r.

Function: poly_exact_divide (<poly1>, <poly2>, <varlist>)

     Divide a polynomial <poly1> by another polynomial <poly2>. Assumes
     that exact division with no remainder is possible. Returns the
     quotient.

Function: poly_normal_form (<poly>, <polylist>, <varlist>)

     `poly_normal_form' finds the normal form of a polynomial <poly>
     with respect to a set of polynomials <polylist>.

Function: poly_buchberger_criterion (<polylist>, <varlist>)

     Returns `true' if <polylist> is a Groebner basis with respect to
     the current term order, by using the Buchberger criterion: for
     every two polynomials h1 and h2 in <polylist> the S-polynomial
     S(h1,h2) reduces to 0 modulo <polylist>.

Function: poly_buchberger (<polylist_fl> <varlist>)

     `poly_buchberger' performs the Buchberger algorithm on a list of
     polynomials and returns the resulting Groebner basis.

Function: poly_reduction (<polylist>, <varlist>)

     `poly_reduction' reduces a list of polynomials <polylist>, so that
     each polynomial is fully reduced with respect to the other
     polynomials.

Function: poly_minimization (<polylist>, <varlist>)

     Returns a sublist of the polynomial list <polylist> spanning the
     same monomial ideal as <polylist> but minimal, i.e. no leading
     monomial of a polynomial in the sublist divides the leading
     monomial of another polynomial.

Function: poly_normalize_list (<polylist>, <varlist>)

     `poly_normalize_list' applies `poly_normalize' to each polynomial
     in the list.  That means it divides every polynomial in a list
     <polylist> by its leading coefficient.

Function: poly_grobner (<polylist>, <varlist>)

     Returns a Groebner basis of the ideal span by the polynomials
     <polylist>. Affected by the global flags.

Function: poly_reduced_grobner (<polylist>, <varlist>)

     Returns a reduced Groebner basis of the ideal span by the
     polynomials <polylist>. Affected by the global flags.

Function: poly_depends_p (<poly>, <var>, <varlist>)

     `poly_depends' tests whether a polynomial depends on a variable
     <var>.

Function: poly_elimination_ideal (<polylist>, <number>, <varlist>)

     `poly_elimination_ideal' returns the grobner basis of the
     number-th elimination ideal of an ideal specified as a list of
     generating polynomials (not necessarily Groebner basis).

Function: poly_colon_ideal (<polylist1>, <polylist2>, <varlist>)

     Returns the reduced Groebner basis of the colon ideal

     I(polylist1):I(polylist2)

     where polylist1 and polylist2 are two lists of polynomials.

Function: poly_ideal_intersection (<polylist1>, <polylist2>, <varlist>)

     `poly_ideal_intersection' returns the intersection of two ideals.

Function: poly_lcm (<poly1>, <poly2>, <varlist>)

     Returns the lowest common multiple of <poly1> and <poly2>.

Function: poly_gcd (<poly1>, <poly2>, <varlist>)

     Returns the greatest common divisor of <poly1> and <poly2>.

     See also `ezgcd',  `gcd',  `gcdex',  and `gcdivide'.

     Example:

          (%i1) p1:6*x^3+19*x^2+19*x+6;
                                  3       2
          (%o1)                6 x  + 19 x  + 19 x + 6
          (%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                            5       4       3       2
          (%o2)          6 x  + 13 x  + 12 x  + 13 x  + 6 x
          (%i3) poly_gcd(p1, p2, [x]);
                                      2
          (%o3)                    6 x  + 13 x + 6

Function: poly_grobner_equal (<polylist1>, <polylist2>, <varlist>)

     `poly_grobner_equal' tests whether two Groebner Bases generate the
     same ideal.  Returns `true' if two lists of polynomials
     <polylist1> and <polylist2>, assumed to be Groebner Bases,
     generate the same ideal, and `false' otherwise.  This is
     equivalent to checking that every polynomial of the first basis
     reduces to 0 modulo the second basis and vice versa. Note that in
     the example below the first list is not a Groebner basis, and thus
     the result is `false'.

          (%i1) poly_grobner_equal([y+x,x-y],[x,y],[x,y]);
          (%o1)                         false

Function: poly_grobner_subsetp (<polylist1>, <polylist2>, <varlist>)

     `poly_grobner_subsetp' tests whether an ideal generated by
     <polylist1> is contained in the ideal generated by <polylist2>.
     For this test to always succeed, <polylist2> must be a Groebner
     basis.

Function: poly_grobner_member (<poly>, <polylist>, <varlist>)

     Returns `true' if a polynomial <poly> belongs to the ideal
     generated by the polynomial list <polylist>, which is assumed to
     be a Groebner basis. Returns `false' otherwise.

     `poly_grobner_member' tests whether a polynomial belongs to an
     ideal generated by a list of polynomials, which is assumed to be a
     Groebner basis. Equivalent to `normal_form' being 0.

Function: poly_ideal_saturation1 (<polylist>, <poly>, <varlist>)

     Returns the reduced Groebner basis of the saturation of the ideal

     I(polylist):poly^inf

     Geometrically, over an algebraically closed field, this is the set
     of polynomials in the ideal generated by <polylist> which do not
     identically vanish on the variety of <poly>.

Function: poly_ideal_saturation (<polylist1>, <polylist2>, <varlist>)

     Returns the reduced Groebner basis of the saturation of the ideal

     I(polylist1):I(polylist2)^inf

     Geometrically, over an algebraically closed field, this is the set
     of polynomials in the ideal generated by <polylist1> which do not
     identically vanish on the variety of <polylist2>.

Function: poly_ideal_polysaturation1 (<polylist1>, <polylist2>, <varlist>)

     <polylist2> ist a list of n polynomials `[poly1,...,polyn]'.
     Returns the reduced Groebner basis of the ideal

     I(polylist):poly1^inf:...:polyn^inf

     obtained by a sequence of successive saturations in the polynomials
     of the polynomial list <polylist2> of the ideal generated by the
     polynomial list <polylist1>.

Function: poly_ideal_polysaturation (<polylist>, <polylistlist>, <varlist>)

     <polylistlist> is a list of n list of polynomials
     `[polylist1,...,polylistn]'.  Returns the reduced Groebner basis
     of the saturation of the ideal

     I(polylist):I(polylist_1)^inf:...:I(polylist_n)^inf

Function: poly_saturation_extension (<poly>, <polylist>, <varlist1>, <varlist2>)

     `poly_saturation_extension' implements the famous Rabinowitz trick.

Function: poly_polysaturation_extension (<poly>, <polylist>, <varlist1>, <varlist2>)


58 impdiff


Functions and Variables for impdiff

58.1 Functions and Variables for impdiff

Function: implicit_derivative

          (<f>,<indvarlist>,<orderlist>,<depvar>)
     This subroutine computes implicit derivatives of multivariable
     functions.  <f> is an array function, the indexes are the
     derivative degree in the <indvarlist> order; <indvarlist> is the
     independent variable list; <orderlist> is the order desired; and
     <depvar> is the dependent variable.

     To use this function write first `load("impdiff")'.

59 interpol


Introduction to interpol
Functions and Variables for interpol

59.1 Introduction to interpol

Package `interpol' defines the Lagrangian, the linear and the cubic
splines methods for polynomial interpolation.

   For comments, bugs or suggestions, please contact me at <'mario AT
edu DOT xunta DOT es'>.

59.2 Functions and Variables for interpol

Function: lagrange (<points>)
Function: lagrange (<points>, <option>)

     Computes the polynomial interpolation by the Lagrangian method.
     Argument <points> must be either:

        * a two column matrix, `p:matrix([2,4],[5,6],[9,3])',

        * a list of pairs, `p: [[2,4],[5,6],[9,3]]',

        * a list of numbers, `p: [4,6,3]', in which case the abscissas
          will be assigned automatically to 1, 2, 3, etc.

     In the first two cases the pairs are ordered with respect to the
     first coordinate before making computations.

     With the <option> argument it is possible to select the name for
     the independent variable, which is `'x' by default; to define
     another one, write something like `varname='z'.

     Note that when working with high degree polynomials, floating
     point evaluations are unstable.

     Examples:

          (%i1) load(interpol)$
          (%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
          (%i3) lagrange(p);
                 (x - 7) (x - 6) (x - 3) (x - 1)
          (%o3)  -------------------------------
                               35
             (x - 8) (x - 6) (x - 3) (x - 1)
           - -------------------------------
                           12
             7 (x - 8) (x - 7) (x - 3) (x - 1)
           + ---------------------------------
                            30
             (x - 8) (x - 7) (x - 6) (x - 1)
           - -------------------------------
                           60
             (x - 8) (x - 7) (x - 6) (x - 3)
           + -------------------------------
                           84
          (%i4) f(x):=''%;
                         (x - 7) (x - 6) (x - 3) (x - 1)
          (%o4)  f(x) := -------------------------------
                                       35
             (x - 8) (x - 6) (x - 3) (x - 1)
           - -------------------------------
                           12
             7 (x - 8) (x - 7) (x - 3) (x - 1)
           + ---------------------------------
                            30
             (x - 8) (x - 7) (x - 6) (x - 1)
           - -------------------------------
                           60
             (x - 8) (x - 7) (x - 6) (x - 3)
           + -------------------------------
                           84
          (%i5) /* Evaluate the polynomial at some points */
                expand(map(f,[2.3,5/7,%pi]));
                                            4          3           2
                              919062  73 %pi    701 %pi    8957 %pi
          (%o5)  [- 1.567535, ------, ------- - -------- + ---------
                              84035     420       210         420
                                                       5288 %pi   186
                                                     - -------- + ---]
                                                         105       5
          (%i6) %,numer;
          (%o6) [- 1.567535, 10.9366573451538, 2.89319655125692]
          (%i7) load(draw)$  /* load draw package */
          (%i8) /* Plot the polynomial together with points */
                draw2d(
                  color      = red,
                  key        = "Lagrange polynomial",
                  explicit(f(x),x,0,10),
                  point_size = 3,
                  color      = blue,
                  key        = "Sample points",
                  points(p))$
          (%i9) /* Change variable name */
                lagrange(p, varname=w);
                 (w - 7) (w - 6) (w - 3) (w - 1)
          (%o9)  -------------------------------
                               35
             (w - 8) (w - 6) (w - 3) (w - 1)
           - -------------------------------
                           12
             7 (w - 8) (w - 7) (w - 3) (w - 1)
           + ---------------------------------
                            30
             (w - 8) (w - 7) (w - 6) (w - 1)
           - -------------------------------
                           60
             (w - 8) (w - 7) (w - 6) (w - 3)
           + -------------------------------
                           84

Function: charfun2 (<x>, <a>, <b>)

     Returns `true' if number <x> belongs to the interval [a, b), and
     `false' otherwise.

Function: linearinterpol (<points>)
Function: linearinterpol (<points>, <option>)

     Computes the polynomial interpolation by the linear method.
     Argument <points> must be either:

        * a two column matrix, `p:matrix([2,4],[5,6],[9,3])',

        * a list of pairs, `p: [[2,4],[5,6],[9,3]]',

        * a list of numbers, `p: [4,6,3]', in which case the abscissas
          will be assigned automatically to 1, 2, 3, etc.

     In the first two cases the pairs are ordered with respect to the
     first coordinate before making computations.

     With the <option> argument it is possible to select the name for
     the independent variable, which is `'x' by default; to define
     another one, write something like `varname='z'.

     Examples:
          (%i1) load(interpol)$
          (%i2) p: matrix([7,2],[8,3],[1,5],[3,2],[6,7])$
          (%i3) linearinterpol(p);
                  13   3 x
          (%o3)  (-- - ---) charfun2(x, minf, 3)
                  2     2
           + (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
              5 x
           + (--- - 3) charfun2(x, 3, 6)
               3

          (%i4) f(x):=''%;
                          13   3 x
          (%o4)  f(x) := (-- - ---) charfun2(x, minf, 3)
                          2     2
           + (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
              5 x
           + (--- - 3) charfun2(x, 3, 6)
               3
          (%i5)  /* Evaluate the polynomial at some points */
                 map(f,[7.3,25/7,%pi]);
                                      62  5 %pi
          (%o5)                 [2.3, --, ----- - 3]
                                      21    3
          (%i6) %,numer;
          (%o6)  [2.3, 2.952380952380953, 2.235987755982989]
          (%i7) load(draw)$  /* load draw package */
          (%i8)  /* Plot the polynomial together with points */
                 draw2d(
                   color      = red,
                   key        = "Linear interpolator",
                   explicit(f(x),x,-5,20),
                   point_size = 3,
                   color      = blue,
                   key        = "Sample points",
                   points(args(p)))$
          (%i9)  /* Change variable name */
                 linearinterpol(p, varname='s);
                 13   3 s
          (%o9) (-- - ---) charfun2(s, minf, 3)
                 2     2
           + (s - 5) charfun2(s, 7, inf) + (37 - 5 s) charfun2(s, 6, 7)
              5 s
           + (--- - 3) charfun2(s, 3, 6)
               3

Function: cspline (<points>)
Function: cspline (<points>, <option1>, <option2>, ...)

     Computes the polynomial interpolation by the cubic splines method.
     Argument <points> must be either:

        * a two column matrix, `p:matrix([2,4],[5,6],[9,3])',

        * a list of pairs, `p: [[2,4],[5,6],[9,3]]',

        * a list of numbers, `p: [4,6,3]', in which case the abscissas
          will be assigned automatically to 1, 2, 3, etc.

     In the first two cases the pairs are ordered with respect to the
     first coordinate before making computations.

     There are three options to fit specific needs:
        * `'d1', default `'unknown', is the first derivative at x_1; if
          it is `'unknown', the second derivative at x_1 is made equal
          to 0 (natural cubic spline); if it is equal to a number, the
          second derivative is calculated based on this number.

        * `'dn', default `'unknown', is the first derivative at x_n; if
          it is `'unknown', the second derivative at x_n is made equal
          to 0 (natural cubic spline); if it is equal to a number, the
          second derivative is calculated based on this number.

        * `'varname', default `'x', is the name of the independent
          variable.

     Examples:
          (%i1) load(interpol)$
          (%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
          (%i3) /* Unknown first derivatives at the extremes
                   is equivalent to natural cubic splines */
                cspline(p);
                        3         2
                  1159 x    1159 x    6091 x   8283
          (%o3)  (------- - ------- - ------ + ----) charfun2(x, minf, 3)
                   3288      1096      3288    1096
                      3         2
                2587 x    5174 x    494117 x   108928
           + (- ------- + ------- - -------- + ------) charfun2(x, 7, inf)
                 1644       137       1644      137
                    3          2
              4715 x    15209 x    579277 x   199575
           + (------- - -------- + -------- - ------) charfun2(x, 6, 7)
               1644       274        1644      274
                      3         2
                3287 x    2223 x    48275 x   9609
           + (- ------- + ------- - ------- + ----) charfun2(x, 3, 6)
                 4932       274      1644     274

          (%i4) f(x):=''%$
          (%i5) /* Some evaluations */
                map(f,[2.3,5/7,%pi]), numer;
          (%o5) [1.991460766423356, 5.823200187269903, 2.227405312429507]
          (%i6) load(draw)$  /* load draw package */
          (%i7) /* Plotting interpolating function */
                draw2d(
                  color      = red,
                  key        = "Cubic splines",
                  explicit(f(x),x,0,10),
                  point_size = 3,
                  color      = blue,
                  key        = "Sample points",
                  points(p))$
          (%i8) /* New call, but giving values at the derivatives */
                cspline(p,d1=0,dn=0);
                        3          2
                  1949 x    11437 x    17027 x   1247
          (%o8)  (------- - -------- + ------- + ----) charfun2(x, minf, 3)
                   2256       2256      2256     752
                      3          2
                1547 x    35581 x    68068 x   173546
           + (- ------- + -------- - ------- + ------) charfun2(x, 7, inf)
                  564       564        141      141
                   3          2
              607 x    35147 x    55706 x   38420
           + (------ - -------- + ------- - -----) charfun2(x, 6, 7)
               188       564        141      47
                      3         2
                3895 x    1807 x    5146 x   2148
           + (- ------- + ------- - ------ + ----) charfun2(x, 3, 6)
                 5076       188      141      47
          (%i8) /* Defining new interpolating function */
                g(x):=''%$
          (%i9) /* Plotting both functions together */
                draw2d(
                  color      = black,
                  key        = "Cubic splines (default)",
                  explicit(f(x),x,0,10),
                  color      = red,
                  key        = "Cubic splines (d1=0,dn=0)",
                  explicit(g(x),x,0,10),
                  point_size = 3,
                  color      = blue,
                  key        = "Sample points",
                  points(p))$

Function: ratinterpol (<points>, <numdeg>)
Function: ratinterpol (<points>, <numdeg>, <option1>, <option2>, ...)

     Generates a rational interpolator for data given by <points> and
     the degree of the numerator being equal to <numdeg>; the degree of
     the denominator is calculated automatically. Argument <points>
     must be either:

        * a two column matrix, `p:matrix([2,4],[5,6],[9,3])',

        * a list of pairs, `p: [[2,4],[5,6],[9,3]]',

        * a list of numbers, `p: [4,6,3]', in which case the abscissas
          will be assigned automatically to 1, 2, 3, etc.

     In the first two cases the pairs are ordered with respect to the
     first coordinate before making computations.

     There are two options to fit specific needs:
        * `'denterm', default `1', is the independent term of the
          polynomial in the denominator.

        * `'varname', default `'x', is the name of the independent
          variable.

     Examples:

          (%i1) load(interpol)$
          (%i2) load(draw)$
          (%i3) p:[[7.2,2.5],[8.5,2.1],[1.6,5.1],[3.4,2.4],[6.7,7.9]]$
          (%i4) for k:0 thru length(p)-1 do
                  draw2d(
                    explicit(ratinterpol(p,k),x,0,9),
                    point_size = 3,
                    points(p),
                    title = concat("Degree of numerator = ",k),
                    yrange=[0,10])$

60 lapack


Introduction to lapack
Functions and Variables for lapack

60.1 Introduction to lapack

`lapack' is a Common Lisp translation (via the program `f2c') of the
Fortran library LAPACK, as obtained from the SLATEC project.

60.2 Functions and Variables for lapack

Function: dgeev (<A>)
Function: dgeev (<A>, <right_p>, <left_p>)

     Computes the eigenvalues and, optionally, the eigenvectors of a
     matrix <A>.  All elements of <A> must be integer or floating point
     numbers.  <A> must be square (same number of rows and columns).
     <A> might or might not be symmetric.

     `dgeev(<A>)' computes only the eigenvalues of <A>.  `dgeev(<A>,
     <right_p>, <left_p>)' computes the eigenvalues of <A> and the
     right eigenvectors when <right_p> = `true' and the left
     eigenvectors when <left_p> = `true'.

     A list of three items is returned.  The first item is a list of
     the eigenvalues.  The second item is `false' or the matrix of
     right eigenvectors.  The third item is `false' or the matrix of
     left eigenvectors.

     The right eigenvector v(j) (the j-th column of the right
     eigenvector matrix) satisfies

     A . v(j) = lambda(j) . v(j)

     where lambda(j) is the corresponding eigenvalue.  The left
     eigenvector u(j) (the j-th column of the left eigenvector matrix)
     satisfies

     u(j)**H . A = lambda(j) . u(j)**H

     where u(j)**H denotes the conjugate transpose of u(j).  The Maxima
     function `ctranspose' computes the conjugate transpose.

     The computed eigenvectors are normalized to have Euclidean norm
     equal to 1, and largest component has imaginary part equal to zero.

     Example:

          (%i1) load (lapack)$
          (%i2) fpprintprec : 6;
          (%o2)                           6
          (%i3) M : matrix ([9.5, 1.75], [3.25, 10.45]);
                                   [ 9.5   1.75  ]
          (%o3)                    [             ]
                                   [ 3.25  10.45 ]
          (%i4) dgeev (M);
          (%o4)          [[7.54331, 12.4067], false, false]
          (%i5) [L, v, u] : dgeev (M, true, true);
                                     [ - .666642  - .515792 ]
          (%o5) [[7.54331, 12.4067], [                      ],
                                     [  .745378   - .856714 ]
                                                  [ - .856714  - .745378 ]
                                                  [                      ]]
                                                  [  .515792   - .666642 ]
          (%i6) D : apply (diag_matrix, L);
                                [ 7.54331     0    ]
          (%o6)                 [                  ]
                                [    0     12.4067 ]
          (%i7) M . v - v . D;
                          [      0.0       - 8.88178E-16 ]
          (%o7)           [                              ]
                          [ - 8.88178E-16       0.0      ]
          (%i8) transpose (u) . M - D . transpose (u);
                               [ 0.0  - 4.44089E-16 ]
          (%o8)                [                    ]
                               [ 0.0       0.0      ]

Function: dgeqrf (<A>)

     Computes the QR decomposition of the matrix <A>.  All elements of
     <A> must be integer or floating point numbers.  <A> may or may not
     have the same number of rows and columns.

     A list of two items is returned.  The first item is the matrix
     <Q>, which is a square, orthonormal matrix which has the same
     number of rows as <A>.  The second item is the matrix <R>, which
     is the same size as <A>, and which has all elements equal to zero
     below the diagonal.  The product `<Q> . <R>', where "." is the
     noncommutative multiplication operator, is equal to <A> (ignoring
     floating point round-off errors).

          (%i1) load (lapack) $
          (%i2) fpprintprec : 6 $
          (%i3) M : matrix ([1, -3.2, 8], [-11, 2.7, 5.9]) $
          (%i4) [q, r] : dgeqrf (M);
                 [ - .0905357  .995893  ]
          (%o4) [[                      ],
                 [  .995893    .0905357 ]
                                         [ - 11.0454   2.97863   5.15148 ]
                                         [                               ]]
                                         [     0      - 2.94241  8.50131 ]
          (%i5) q . r - M;
                   [ - 7.77156E-16   1.77636E-15   - 8.88178E-16 ]
          (%o5)    [                                             ]
                   [      0.0       - 1.33227E-15   8.88178E-16  ]
          (%i6) mat_norm (%, 1);
          (%o6)                      3.10862E-15

Function: dgesv (<A>, <b>)

     Computes the solution <x> of the linear equation <A> <x> = <b>,
     where <A> is a square matrix, and <b> is a matrix of the same
     number of rows as <A> and any number of columns.  The return value
     <x> is the same size as <b>.

     The elements of <A> and <b> must evaluate to real floating point
     numbers via `float'; thus elements may be any numeric type,
     symbolic numerical constants, or expressions which evaluate to
     floats.  The elements of <x> are always floating point numbers.
     All arithmetic is carried out as floating point operations.

     `dgesv' computes the solution via the LU decomposition of <A>.

     Examples:

     `dgesv' computes the solution of the linear equation <A> <x> = <b>.

          (%i1) A : matrix ([1, -2.5], [0.375, 5]);
                                         [   1    - 2.5 ]
          (%o1)                          [              ]
                                         [ 0.375    5   ]
          (%i2) b : matrix ([1.75], [-0.625]);
                                            [  1.75   ]
          (%o2)                             [         ]
                                            [ - 0.625 ]
          (%i3) x : dgesv (A, b);
                                      [  1.210526315789474  ]
          (%o3)                       [                     ]
                                      [ - 0.215789473684211 ]
          (%i4) dlange (inf_norm, b - A.x);
          (%o4)                                 0.0

     <b> is a matrix with the same number of rows as <A> and any number
     of columns.  <x> is the same size as <b>.

          (%i1) A : matrix ([1, -0.15], [1.82, 2]);
                                         [  1    - 0.15 ]
          (%o1)                          [              ]
                                         [ 1.82    2    ]
          (%i2) b : matrix ([3.7, 1, 8], [-2.3, 5, -3.9]);
                                        [  3.7   1    8   ]
          (%o2)                         [                 ]
                                        [ - 2.3  5  - 3.9 ]
          (%i3) x : dgesv (A, b);
                [  3.103827540695117  1.20985481742191    6.781786185657722 ]
          (%o3) [                                                           ]
                [ -3.974483062032557  1.399032116146062  -8.121425428948527 ]
          (%i4) dlange (inf_norm, b - A . x);
          (%o4)                       1.1102230246251565E-15

     The elements of <A> and <b> must evaluate to real floating point
     numbers.

          (%i1) A : matrix ([5, -%pi], [1b0, 11/17]);
                                         [   5    - %pi ]
                                         [              ]
          (%o1)                          [         11   ]
                                         [ 1.0b0   --   ]
                                         [         17   ]
          (%i2) b : matrix ([%e], [sin(1)]);
                                            [   %e   ]
          (%o2)                             [        ]
                                            [ sin(1) ]
          (%i3) x : dgesv (A, b);
                                       [ 0.690375643155986 ]
          (%o3)                        [                   ]
                                       [ 0.233510982552952 ]
          (%i4) dlange (inf_norm, b - A . x);
          (%o4)                        2.220446049250313E-16

Function: dgesvd (<A>)
Function: dgesvd (<A>, <left_p>, <right_p>)

     Computes the singular value decomposition (SVD) of a matrix <A>,
     comprising the singular values and, optionally, the left and right
     singular vectors.  All elements of <A> must be integer or floating
     point numbers.  <A> might or might not be square (same number of
     rows and columns).

     Let m be the number of rows, and n the number of columns of <A>.
     The singular value decomposition of <A> comprises three matrices,
     <U>, <Sigma>, and <V^T>, such that

     <A> = <U> . <Sigma> . <V>^T

     where <U> is an m-by-m unitary matrix, <Sigma> is an m-by-n
     diagonal matrix, and <V^T> is an n-by-n unitary matrix.

     Let sigma[i] be a diagonal element of Sigma, that is, <Sigma>[i,
     i] = <sigma>[i].  The elements sigma[i] are the so-called singular
     values of <A>; these are real and nonnegative, and returned in
     descending order.  The first min(m, n) columns of <U> and <V> are
     the left and right singular vectors of <A>.  Note that `dgesvd'
     returns the transpose of <V>, not <V> itself.

     `dgesvd(<A>)' computes only the singular values of <A>.
     `dgesvd(<A>, <left_p>, <right_p>)' computes the singular values of
     <A> and the left singular vectors when <left_p> = `true' and the
     right singular vectors when <right_p> = `true'.

     A list of three items is returned.  The first item is a list of
     the singular values.  The second item is `false' or the matrix of
     left singular vectors.  The third item is `false' or the matrix of
     right singular vectors.

     Example:

          (%i1) load (lapack)$
          (%i2) fpprintprec : 6;
          (%o2)                           6
          (%i3) M: matrix([1, 2, 3], [3.5, 0.5, 8], [-1, 2, -3], [4, 9, 7]);
                                  [  1    2    3  ]
                                  [               ]
                                  [ 3.5  0.5   8  ]
          (%o3)                   [               ]
                                  [ - 1   2   - 3 ]
                                  [               ]
                                  [  4    9    7  ]
          (%i4) dgesvd (M);
          (%o4)      [[14.4744, 6.38637, .452547], false, false]
          (%i5) [sigma, U, VT] : dgesvd (M, true, true);
          (%o5) [[14.4744, 6.38637, .452547],
          [ - .256731  .00816168   .959029    - .119523 ]
          [                                             ]
          [ - .526456   .672116   - .206236   - .478091 ]
          [                                             ],
          [  .107997   - .532278  - .0708315  - 0.83666 ]
          [                                             ]
          [ - .803287  - .514659  - .180867    .239046  ]
          [ - .374486  - .538209  - .755044 ]
          [                                 ]
          [  .130623   - .836799   0.5317   ]]
          [                                 ]
          [ - .917986   .100488    .383672  ]
          (%i6) m : length (U);
          (%o6)                           4
          (%i7) n : length (VT);
          (%o7)                           3
          (%i8) Sigma:
                  genmatrix(lambda ([i, j], if i=j then sigma[i] else 0),
                            m, n);
                            [ 14.4744     0        0    ]
                            [                           ]
                            [    0     6.38637     0    ]
          (%o8)             [                           ]
                            [    0        0     .452547 ]
                            [                           ]
                            [    0        0        0    ]
          (%i9) U . Sigma . VT - M;
                    [  1.11022E-15        0.0       1.77636E-15 ]
                    [                                           ]
                    [  1.33227E-15    1.66533E-15       0.0     ]
          (%o9)     [                                           ]
                    [ - 4.44089E-16  - 8.88178E-16  4.44089E-16 ]
                    [                                           ]
                    [  8.88178E-16    1.77636E-15   8.88178E-16 ]
          (%i10) transpose (U) . U;
                 [     1.0      5.55112E-17    2.498E-16     2.77556E-17  ]
                 [                                                        ]
                 [ 5.55112E-17      1.0       5.55112E-17    4.16334E-17  ]
          (%o10) [                                                        ]
                 [  2.498E-16   5.55112E-17       1.0       - 2.08167E-16 ]
                 [                                                        ]
                 [ 2.77556E-17  4.16334E-17  - 2.08167E-16       1.0      ]
          (%i11) VT . transpose (VT);
                    [      1.0           0.0      - 5.55112E-17 ]
                    [                                           ]
          (%o11)    [      0.0           1.0       5.55112E-17  ]
                    [                                           ]
                    [ - 5.55112E-17  5.55112E-17       1.0      ]

Function: dlange (<norm>, <A>)
Function: zlange (<norm>, <A>)

     Computes a norm or norm-like function of the matrix <A>.

    `max'
          Compute max(abs(A(i, j))) where i and j range over the rows
          and columns, respectively, of <A>.  Note that this function
          is not a proper matrix norm.

    `one_norm'
          Compute the L[1] norm of <A>, that is, the maximum of the sum
          of the absolute value of elements in each column.

    `inf_norm'
          Compute the L[inf] norm of <A>, that is, the maximum of the
          sum of the absolute value of elements in each row.

    `frobenius'
          Compute the Frobenius norm of <A>, that is, the square root
          of the sum of squares of the matrix elements.

Function: dgemm (<A>, <B>)
Function: dgemm (<A>, <B>, <options>)

     Compute the product of two matrices and optionally add the product
     to a third matrix.

     In the simplest form, `dgemm(<A>, <B>)' computes the product of
     the two real matrices, <A> and <B>.

     In the second form, `dgemm' computes the <alpha> * <A> * <B> +
     <beta> * <C> where <A>, <B>, <C> are real matrices of the
     appropriate sizes and <alpha> and <beta> are real numbers.
     Optionally, <A> and/or <B> can be transposed before computing the
     product.  The extra parameters are specifed by optional keyword
     arguments: The keyword arguments are optional and may be specified
     in any order.  They all take the form `key=val'.  The keyword
     arguments are:

    <C>
          The matrix <C> that should be added.  The default is `false',
          which means no matrix is added.

    <alpha>
          The product of <A> and <B> is multiplied by this value.  The
          default is 1.

    <beta>
          If a matrix <C> is given, this value multiplies <C> before it
          is added.  The default value is 0, which implies that <C> is
          not added, even if <C> is given.  Hence, be sure to specify a
          non-zero value for <beta>.

    <transpose_a>
          If `true', the transpose of <A> is used instead of <A> for
          the product.  The default is `false'.

    <transpose_b>
          If `true', the transpose of <B> is used instead of <B> for
          the product.  The default is `false'.

          (%i1) load (lapack)$
          (%i2) A : matrix([1,2,3],[4,5,6],[7,8,9]);
                                            [ 1  2  3 ]
                                            [         ]
          (%o2)                             [ 4  5  6 ]
                                            [         ]
                                            [ 7  8  9 ]
          (%i3) B : matrix([-1,-2,-3],[-4,-5,-6],[-7,-8,-9]);
                                         [ - 1  - 2  - 3 ]
                                         [               ]
          (%o3)                          [ - 4  - 5  - 6 ]
                                         [               ]
                                         [ - 7  - 8  - 9 ]
          (%i4) C : matrix([3,2,1],[6,5,4],[9,8,7]);
                                            [ 3  2  1 ]
                                            [         ]
          (%o4)                             [ 6  5  4 ]
                                            [         ]
                                            [ 9  8  7 ]
          (%i5) dgemm(A,B);
                                   [ - 30.0   - 36.0   - 42.0  ]
                                   [                           ]
          (%o5)                    [ - 66.0   - 81.0   - 96.0  ]
                                   [                           ]
                                   [ - 102.0  - 126.0  - 150.0 ]
          (%i6) A . B;
                                      [ - 30   - 36   - 42  ]
                                      [                     ]
          (%o6)                       [ - 66   - 81   - 96  ]
                                      [                     ]
                                      [ - 102  - 126  - 150 ]
          (%i7) dgemm(A,B,transpose_a=true);
                                   [ - 66.0  - 78.0   - 90.0  ]
                                   [                          ]
          (%o7)                    [ - 78.0  - 93.0   - 108.0 ]
                                   [                          ]
                                   [ - 90.0  - 108.0  - 126.0 ]
          (%i8) transpose(A) . B;
                                     [ - 66  - 78   - 90  ]
                                     [                    ]
          (%o8)                      [ - 78  - 93   - 108 ]
                                     [                    ]
                                     [ - 90  - 108  - 126 ]
          (%i9) dgemm(A,B,c=C,beta=1);
                                   [ - 27.0  - 34.0   - 41.0  ]
                                   [                          ]
          (%o9)                    [ - 60.0  - 76.0   - 92.0  ]
                                   [                          ]
                                   [ - 93.0  - 118.0  - 143.0 ]
          (%i10) A . B + C;
                                      [ - 27  - 34   - 41  ]
                                      [                    ]
          (%o10)                      [ - 60  - 76   - 92  ]
                                      [                    ]
                                      [ - 93  - 118  - 143 ]
          (%i11) dgemm(A,B,c=C,beta=1, alpha=-1);
                                      [ 33.0   38.0   43.0  ]
                                      [                     ]
          (%o11)                      [ 72.0   86.0   100.0 ]
                                      [                     ]
                                      [ 111.0  134.0  157.0 ]
          (%i12) -A . B + C;
                                         [ 33   38   43  ]
                                         [               ]
          (%o12)                         [ 72   86   100 ]
                                         [               ]
                                         [ 111  134  157 ]

61 lbfgs


Introduction to lbfgs
Functions and Variables for lbfgs

61.1 Introduction to lbfgs

`lbfgs' is an implementation of the L-BFGS algorithm [1] to solve
unconstrained minimization problems via a limited-memory quasi-Newton
(BFGS) algorithm.  It is called a limited-memory method because a
low-rank approximation of the Hessian matrix inverse is stored instead
of the entire Hessian inverse.  The program was originally written in
Fortran [2] by Jorge Nocedal, incorporating some functions originally
written by Jorge J. More' and David J. Thuente, and translated into
Lisp automatically via the program `f2cl'.  The Maxima package `lbfgs'
comprises the translated code plus an interface function which manages
some details.

   References:

   [1] D. Liu and J. Nocedal. "On the limited memory BFGS method for
large scale optimization". Mathematical Programming B 45:503-528 (1989)

   [2] `http://netlib.org/opt/lbfgs_um.shar'

61.2 Functions and Variables for lbfgs

Function: lbfgs (<FOM>, <X>, <X0>, <epsilon>, <iprint>)
Function: lbfgs ([<FOM>, <grad>] <X>, <X0>, <epsilon>, <iprint>)

     Finds an approximate solution of the unconstrained minimization of
     the figure of merit <FOM> over the list of variables <X>, starting
     from initial estimates <X0>, such that norm(grad(FOM)) <
     epsilon*max(1, norm(X)).

     <grad>, if present, is the gradient of <FOM> with respect to the
     variables <X>.  <grad> is a list, with one element for each
     element of <X>.  If not present, the gradient is computed
     automatically by symbolic differentiation.

     The algorithm applied is a limited-memory quasi-Newton (BFGS)
     algorithm [1].  It is called a limited-memory method because a
     low-rank approximation of the Hessian matrix inverse is stored
     instead of the entire Hessian inverse.  Each iteration of the
     algorithm is a line search, that is, a search along a ray in the
     variables <X>, with the search direction computed from the
     approximate Hessian inverse.  The FOM is always decreased by a
     successful line search.  Usually (but not always) the norm of the
     gradient of FOM also decreases.

     <iprint> controls progress messages printed by `lbfgs'.

    `iprint[1]'
          `<iprint>[1]' controls the frequency of progress messages.
         `iprint[1] < 0'
               No progress messages.

         `iprint[1] = 0'
               Messages at the first and last iterations.

         `iprint[1] > 0'
               Print a message every `<iprint>[1]' iterations.

    `iprint[2]'
          `<iprint>[2]' controls the verbosity of progress messages.
         `iprint[2] = 0'
               Print out iteration count, number of evaluations of
               <FOM>, value of <FOM>, norm of the gradient of <FOM>,
               and step length.

         `iprint[2] = 1'
               Same as `<iprint>[2] = 0', plus <X0> and the gradient of
               <FOM> evaluated at <X0>.

         `iprint[2] = 2'
               Same as `<iprint>[2] = 1', plus values of <X> at each
               iteration.

         `iprint[2] = 3'
               Same as `<iprint>[2] = 2', plus the gradient of <FOM> at
               each iteration.

     The columns printed by `lbfgs' are the following.

    `I'
          Number of iterations. It is incremented for each line search.

    `NFN'
          Number of evaluations of the figure of merit.

    `FUNC'
          Value of the figure of merit at the end of the most recent
          line search.

    `GNORM'
          Norm of the gradient of the figure of merit at the end of the
          most recent line search.

    `STEPLENGTH'
          An internal parameter of the search algorithm.

     Additional information concerning details of the algorithm are
     found in the comments of the original Fortran code [2].

     See also `lbfgs_nfeval_max' and `lbfgs_ncorrections'.

     References:

     [1] D. Liu and J. Nocedal. "On the limited memory BFGS method for
     large scale optimization". Mathematical Programming B 45:503-528
     (1989)

     [2] `http://netlib.org/opt/lbfgs_um.shar'

     Examples:

     The same FOM as computed by FGCOMPUTE in the program sdrive.f in
     the LBFGS package from Netlib.  Note that the variables in
     question are subscripted variables.  The FOM has an exact minimum
     equal to zero at u[k] = 1 for k = 1, ..., 8.

          (%i1) load (lbfgs);
          (%o1)   /usr/share/maxima/5.10.0cvs/share/lbfgs/lbfgs.mac
          (%i2) t1[j] := 1 - u[j];
          (%o2)                     t1  := 1 - u
                                      j         j
          (%i3) t2[j] := 10*(u[j + 1] - u[j]^2);
                                                    2
          (%o3)                t2  := 10 (u      - u )
                                 j         j + 1    j
          (%i4) n : 8;
          (%o4)                           8
          (%i5) FOM : sum (t1[2*j - 1]^2 + t2[2*j - 1]^2, j, 1, n/2);
                           2 2           2              2 2           2
          (%o5) 100 (u  - u )  + (1 - u )  + 100 (u  - u )  + (1 - u )
                      8    7           7           6    5           5
                               2 2           2              2 2           2
                  + 100 (u  - u )  + (1 - u )  + 100 (u  - u )  + (1 - u )
                          4    3           3           2    1           1
          (%i6) lbfgs (FOM, '[u[1],u[2],u[3],u[4],u[5],u[6],u[7],u[8]],
                 [-1.2, 1, -1.2, 1, -1.2, 1, -1.2, 1], 1e-3, [1, 0]);
          *************************************************
            N=    8   NUMBER OF CORRECTIONS=25
                 INITIAL VALUES
           F=  9.680000000000000D+01   GNORM=  4.657353755084532D+02
          *************************************************

           I NFN   FUNC                    GNORM                   STEPLENGTH

           1   3   1.651479526340304D+01   4.324359291335977D+00   7.926153934390631D-04
           2   4   1.650209316638371D+01   3.575788161060007D+00   1.000000000000000D+00
           3   5   1.645461701312851D+01   6.230869903601577D+00   1.000000000000000D+00
           4   6   1.636867301275588D+01   1.177589920974980D+01   1.000000000000000D+00
           5   7   1.612153014409201D+01   2.292797147151288D+01   1.000000000000000D+00
           6   8   1.569118407390628D+01   3.687447158775571D+01   1.000000000000000D+00
           7   9   1.510361958398942D+01   4.501931728123680D+01   1.000000000000000D+00
           8  10   1.391077875774294D+01   4.526061463810632D+01   1.000000000000000D+00
           9  11   1.165625686278198D+01   2.748348965356917D+01   1.000000000000000D+00
          10  12   9.859422687859137D+00   2.111494974231644D+01   1.000000000000000D+00
          11  13   7.815442521732281D+00   6.110762325766556D+00   1.000000000000000D+00
          12  15   7.346380905773160D+00   2.165281166714631D+01   1.285316401779533D-01
          13  16   6.330460634066370D+00   1.401220851762050D+01   1.000000000000000D+00
          14  17   5.238763939851439D+00   1.702473787613255D+01   1.000000000000000D+00
          15  18   3.754016790406701D+00   7.981845727704576D+00   1.000000000000000D+00
          16  20   3.001238402309352D+00   3.925482944716691D+00   2.333129631296807D-01
          17  22   2.794390709718290D+00   8.243329982546473D+00   2.503577283782332D-01
          18  23   2.563783562918759D+00   1.035413426521790D+01   1.000000000000000D+00
          19  24   2.019429976377856D+00   1.065187312346769D+01   1.000000000000000D+00
          20  25   1.428003167670903D+00   2.475962450826961D+00   1.000000000000000D+00
          21  27   1.197874264861340D+00   8.441707983493810D+00   4.303451060808756D-01
          22  28   9.023848941942773D-01   1.113189216635162D+01   1.000000000000000D+00
          23  29   5.508226405863770D-01   2.380830600326308D+00   1.000000000000000D+00
          24  31   3.902893258815567D-01   5.625595816584421D+00   4.834988416524465D-01
          25  32   3.207542206990315D-01   1.149444645416472D+01   1.000000000000000D+00
          26  33   1.874468266362791D-01   3.632482152880997D+00   1.000000000000000D+00
          27  34   9.575763380706598D-02   4.816497446154354D+00   1.000000000000000D+00
          28  35   4.085145107543406D-02   2.087009350166495D+00   1.000000000000000D+00
          29  36   1.931106001379290D-02   3.886818608498966D+00   1.000000000000000D+00
          30  37   6.894000721499670D-03   3.198505796342214D+00   1.000000000000000D+00
          31  38   1.443296033051864D-03   1.590265471025043D+00   1.000000000000000D+00
          32  39   1.571766603154336D-04   3.098257063980634D-01   1.000000000000000D+00
          33  40   1.288011776581970D-05   1.207784183577257D-02   1.000000000000000D+00
          34  41   1.806140173752971D-06   4.587890233385193D-02   1.000000000000000D+00
          35  42   1.769004645459358D-07   1.790537375052208D-02   1.000000000000000D+00
          36  43   3.312164100763217D-10   6.782068426119681D-04   1.000000000000000D+00

Variable: lbfgs_nfeval_max

     Default value: 100

     `lbfgs_nfeval_max' is the maximum number of evaluations of the
     figure of merit (FOM) in `lbfgs'.  When `lbfgs_nfeval_max' is
     reached, `lbfgs' returns the result of the last successful line
     search.

Variable: lbfgs_ncorrections

     Default value: 25

     `lbfgs_ncorrections' is the number of corrections applied to the
     approximate inverse Hessian matrix which is maintained by `lbfgs'.

62 lindstedt


Functions and Variables for lindstedt

62.1 Functions and Variables for lindstedt

Function: Lindstedt (<eq>,<pvar>,<torder>,<ic>)

     This is a first pass at a Lindstedt code.  It can solve problems
     with initial conditions entered, which can be arbitrary constants,
     (just not <%k1> and <%k2>) where the initial conditions on the
     perturbation equations are z[i]=0, z'[i]=0 for i>0. <ic> is the
     list of initial conditions.

     Problems occur when initial conditions are not given, as the
     constants in the perturbation equations are the same as the zero
     order equation solution.  Also, problems occur when the initial
     conditions for the perturbation equations are not z[i]=0, z'[i]=0
     for i>0, such as the Van der Pol equation.

     Example:
          (%i1) load("makeOrders")$

          (%i2) load("lindstedt")$

          (%i3) Lindstedt('diff(x,t,2)+x-(e*x^3)/6,e,2,[1,0]);
                    2
                   e  (cos(5 T) - 24 cos(3 T) + 23 cos(T))
          (%o3) [[[---------------------------------------
                                    36864
             e (cos(3 T) - cos(T))
           - --------------------- + cos(T)],
                      192
                    2
                 7 e    e
          T = (- ---- - -- + 1) t]]
                 3072   16

     To use this function write first `load("makeOrders")' and
     `load("lindstedt")'.

63 linearalgebra


Introduction to linearalgebra
Functions and Variables for linearalgebra

63.1 Introduction to linearalgebra

`linearalgebra' is a collection of functions for linear algebra.

   Example:

     (%i1) M : matrix ([1, 2], [1, 2]);
                                 [ 1  2 ]
     (%o1)                       [      ]
                                 [ 1  2 ]
     (%i2) nullspace (M);
                                    [  1  ]
                                    [     ]
     (%o2)                     span([   1 ])
                                    [ - - ]
                                    [   2 ]
     (%i3) columnspace (M);
                                     [ 1 ]
     (%o3)                      span([   ])
                                     [ 1 ]
     (%i4) ptriangularize (M - z*ident(2), z);
                              [ 1   2 - z   ]
     (%o4)                    [             ]
                              [           2 ]
                              [ 0  3 z - z  ]
     (%i5) M : matrix ([1, 2, 3], [4, 5, 6], [7, 8, 9]) - z*ident(3);
                          [ 1 - z    2      3   ]
                          [                     ]
     (%o5)                [   4    5 - z    6   ]
                          [                     ]
                          [   7      8    9 - z ]
     (%i6) MM : ptriangularize (M, z);
                   [ 4  5 - z            6            ]
                   [                                  ]
                   [                2                 ]
                   [     66        z    102 z   132   ]
                   [ 0   --      - -- + ----- + ---   ]
     (%o6)         [     49        7     49     49    ]
                   [                                  ]
                   [               3        2         ]
                   [           49 z    245 z    147 z ]
                   [ 0    0    ----- - ------ - ----- ]
                   [            264      88      44   ]
     (%i7) algebraic : true;
     (%o7)                         true
     (%i8) tellrat (MM [3, 3]);
                              3       2
     (%o8)                  [z  - 15 z  - 18 z]
     (%i9) MM : ratsimp (MM);
                    [ 4  5 - z           6           ]
                    [                                ]
                    [                2               ]
     (%o9)          [     66      7 z  - 102 z - 132 ]
                    [ 0   --    - ------------------ ]
                    [     49              49         ]
                    [                                ]
                    [ 0    0             0           ]
     (%i10) nullspace (MM);
                             [        1         ]
                             [                  ]
                             [   2              ]
                             [  z  - 14 z - 16  ]
                             [  --------------  ]
     (%o10)             span([        8         ])
                             [                  ]
                             [    2             ]
                             [   z  - 18 z - 12 ]
                             [ - -------------- ]
                             [         12       ]
     (%i11) M : matrix ([1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12],
                        [13, 14, 15, 16]);
                            [ 1   2   3   4  ]
                            [                ]
                            [ 5   6   7   8  ]
     (%o11)                 [                ]
                            [ 9   10  11  12 ]
                            [                ]
                            [ 13  14  15  16 ]
     (%i12) columnspace (M);
                                [ 1  ]  [ 2  ]
                                [    ]  [    ]
                                [ 5  ]  [ 6  ]
     (%o12)                span([    ], [    ])
                                [ 9  ]  [ 10 ]
                                [    ]  [    ]
                                [ 13 ]  [ 14 ]
     (%i13) apply ('orthogonal_complement, args (nullspace (transpose (M))));
                                [ 0 ]  [  1  ]
                                [   ]  [     ]
                                [ 1 ]  [  0  ]
     (%o13)                span([   ], [     ])
                                [ 2 ]  [ - 1 ]
                                [   ]  [     ]
                                [ 3 ]  [ - 2 ]

63.2 Functions and Variables for linearalgebra

Function: addmatrices (<f>, <M_1>, ..., <M_n>)

     Using the function <f> as the addition function, return the sum of
     the matrices <M_1>, ..., <M_n>.  The function <f> must accept any
     number of arguments (a Maxima nary function).

     Examples:

          (%i1) m1 : matrix([1,2],[3,4])$
          (%i2) m2 : matrix([7,8],[9,10])$
          (%i3) addmatrices('max,m1,m2);
          (%o3) matrix([7,8],[9,10])
          (%i4) addmatrices('max,m1,m2,5*m1);
          (%o4) matrix([7,10],[15,20])

Function: blockmatrixp (<M>)

     Return true if and only if <M> is a matrix and every entry of <M>
     is a matrix.

Function: columnop (<M>, <i>, <j>, <theta>)

     If <M> is a matrix, return the matrix that results from doing the
     column operation `C_i <- C_i - <theta> * C_j'. If <M> doesn't have
     a row <i> or <j>, signal an error.

Function: columnswap (<M>, <i>, <j>)

     If <M> is a matrix, swap columns <i> and <j>.  If <M> doesn't have
     a column <i> or <j>, signal an error.

Function: columnspace (<M>)

     If <M> is a matrix, return `span (v_1, ..., v_n)', where the set
     `{v_1, ..., v_n}' is a basis for the column space of <M>.  The span
     of the empty set is `{0}'.  Thus, when the column space has only
     one member, return `span ()'.

Function: copy (<e>)

     Return a copy of the Maxima expression <e>.  Although <e> can be
     any Maxima expression, the copy function is the most useful when
     <e> is either a list or a matrix; consider:

          (%i1) m : [1,[2,3]]$
          (%i2) mm : m$
          (%i3) mm[2][1] : x$
          (%i4) m;
          (%o4)                      [1,[x,3]]
          (%i5) mm;
          (%o5)                      [1,[x,3]]

     Let's try the same experiment, but this time let <mm> be a copy of
     <m>

          (%i6) m : [1,[2,3]]$
          (%i7) mm : copy(m)$
          (%i8) mm[2][1] : x$
          (%i9) m;
          (%o9)                     [1,[2,3]]
          (%i10) mm;
          (%o10)                    [1,[x,3]]

     This time, the assignment to <mm> does not change the value of <m>.

Function: cholesky (<M>)
Function: cholesky (<M>, <field>)

     Return the Cholesky factorization of the matrix selfadjoint (or
     hermitian) matrix <M>.  The second argument defaults to
     'generalring.' For a description of the possible values for
     <field>, see `lu_factor'.

Function: ctranspose (<M>)

     Return the complex conjugate transpose of the matrix <M>.  The
     function `ctranspose' uses `matrix_element_transpose' to transpose
     each matrix element.

Function: diag_matrix (<d_1>, <d_2>, ..., <d_n>)

     Return a diagonal matrix with diagonal entries <d_1>, <d_2>, ...,
     <d_n>.  When the diagonal entries are matrices, the zero entries
     of the returned matrix are zero matrices of the appropriate size;
     for example:

          (%i1) diag_matrix(diag_matrix(1,2),diag_matrix(3,4));

                                      [ [ 1  0 ]  [ 0  0 ] ]
                                      [ [      ]  [      ] ]
                                      [ [ 0  2 ]  [ 0  0 ] ]
          (%o1)                       [                    ]
                                      [ [ 0  0 ]  [ 3  0 ] ]
                                      [ [      ]  [      ] ]
                                      [ [ 0  0 ]  [ 0  4 ] ]
          (%i2) diag_matrix(p,q);

                                             [ p  0 ]
          (%o2)                              [      ]
                                             [ 0  q ]

Function: dotproduct (<u>, <v>)

     Return the dotproduct of vectors <u> and <v>.  This is the same as
     `conjugate (transpose (<u>)) .  <v>'.  The arguments <u> and <v>
     must be column vectors.

Function: eigens_by_jacobi (<A>)
Function: eigens_by_jacobi (<A>, <field_type>)

     Computes the eigenvalues and eigenvectors of <A> by the method of
     Jacobi rotations.  <A> must be a symmetric matrix (but it need not
     be positive definite nor positive semidefinite).  <field_type>
     indicates the computational field, either `floatfield' or
     `bigfloatfield'.  If <field_type> is not specified, it defaults to
     `floatfield'.

     The elements of <A> must be numbers or expressions which evaluate
     to numbers via `float' or `bfloat' (depending on <field_type>).

     Examples:

          (%i1) S: matrix([1/sqrt(2), 1/sqrt(2)],[-1/sqrt(2), 1/sqrt(2)]);
                               [     1         1    ]
                               [  -------   ------- ]
                               [  sqrt(2)   sqrt(2) ]
          (%o1)                [                    ]
                               [      1        1    ]
                               [ - -------  ------- ]
                               [   sqrt(2)  sqrt(2) ]
          (%i2) L : matrix ([sqrt(3), 0], [0, sqrt(5)]);
                                [ sqrt(3)     0    ]
          (%o2)                 [                  ]
                                [    0     sqrt(5) ]
          (%i3) M : S . L . transpose (S);
                      [ sqrt(5)   sqrt(3)  sqrt(5)   sqrt(3) ]
                      [ ------- + -------  ------- - ------- ]
                      [    2         2        2         2    ]
          (%o3)       [                                      ]
                      [ sqrt(5)   sqrt(3)  sqrt(5)   sqrt(3) ]
                      [ ------- - -------  ------- + ------- ]
                      [    2         2        2         2    ]
          (%i4) eigens_by_jacobi (M);
          The largest percent change was 0.1454972243679
          The largest percent change was 0.0
          number of sweeps: 2
          number of rotations: 1
          (%o4) [[1.732050807568877, 2.23606797749979],
                                  [  0.70710678118655   0.70710678118655 ]
                                  [                                      ]]
                                  [ - 0.70710678118655  0.70710678118655 ]
          (%i5) float ([[sqrt(3), sqrt(5)], S]);
          (%o5) [[1.732050807568877, 2.23606797749979],
                                  [  0.70710678118655   0.70710678118655 ]
                                  [                                      ]]
                                  [ - 0.70710678118655  0.70710678118655 ]
          (%i6) eigens_by_jacobi (M, bigfloatfield);
          The largest percent change was 1.454972243679028b-1
          The largest percent change was 0.0b0
          number of sweeps: 2
          number of rotations: 1
          (%o6) [[1.732050807568877b0, 2.23606797749979b0],
                          [  7.071067811865475b-1   7.071067811865475b-1 ]
                          [                                              ]]
                          [ - 7.071067811865475b-1  7.071067811865475b-1 ]

Function: get_lu_factors (<x>)

     When `<x> = lu_factor (<A>)', then `get_lu_factors' returns a list
     of the form `[P, L, U]', where <P> is a permutation matrix, <L> is
     lower triangular with ones on the diagonal, and <U> is upper
     triangular, and `<A> = <P> <L> <U>'.

Function: hankel (<col>)
Function: hankel (<col>, <row>)

     Return a Hankel matrix <H>.  The first column of <H> is <col>;
     except for the first entry, the last row of <H> is <row>.  The
     default for <row> is the zero vector with the same length as <col>.

Function: hessian (<f>, <x>)

     Returns the Hessian matrix of <f> with respect to the list of
     variables <x>.  The `(i, j)'-th element of the Hessian matrix is
     `diff(<f>, <x>[i], 1, <x>[j], 1)'.

     Examples:

          (%i1) hessian (x * sin (y), [x, y]);
                               [   0       cos(y)   ]
          (%o1)                [                    ]
                               [ cos(y)  - x sin(y) ]
          (%i2) depends (F, [a, b]);
          (%o2)                       [F(a, b)]
          (%i3) hessian (F, [a, b]);
                                  [   2      2   ]
                                  [  d F    d F  ]
                                  [  ---   ----- ]
                                  [    2   da db ]
                                  [  da          ]
          (%o3)                   [              ]
                                  [   2      2   ]
                                  [  d F    d F  ]
                                  [ -----   ---  ]
                                  [ da db     2  ]
                                  [         db   ]

Function: hilbert_matrix (<n>)

     Return the <n> by <n> Hilbert matrix.  When <n> isn't a positive
     integer, signal an error.

Function: identfor (<M>)
Function: identfor (<M>, <fld>)

     Return an identity matrix that has the same shape as the matrix
     <M>.  The diagonal entries of the identity matrix are the
     multiplicative identity of the field <fld>; the default for <fld>
     is <generalring>.

     The first argument <M> should be a square matrix or a non-matrix.
     When <M> is a matrix, each entry of <M> can be a square matrix -
     thus <M> can be a blocked Maxima matrix.  The matrix can be
     blocked to any (finite) depth.

     See also `zerofor'

Function: invert_by_lu (<M>, <(rng generalring)>)

     Invert a matrix <M> by using the LU factorization.  The LU
     factorization is done using the ring <rng>.

Function: jacobian (<f>, <x>)

     Returns the Jacobian matrix of the list of functions <f> with
     respect to the list of variables <x>.  The `(i, j)'-th element of
     the Jacobian matrix is `diff(<f>[i], <x>[j])'.

     Examples:

          (%i1) jacobian ([sin (u - v), sin (u * v)], [u, v]);
                            [ cos(v - u)  - cos(v - u) ]
          (%o1)             [                          ]
                            [ v cos(u v)   u cos(u v)  ]
          (%i2) depends ([F, G], [y, z]);
          (%o2)                  [F(y, z), G(y, z)]
          (%i3) jacobian ([F, G], [y, z]);
                                     [ dF  dF ]
                                     [ --  -- ]
                                     [ dy  dz ]
          (%o3)                      [        ]
                                     [ dG  dG ]
                                     [ --  -- ]
                                     [ dy  dz ]

Function: kronecker_product (<A>, <B>)

     Return the Kronecker product of the matrices <A> and <B>.

Function: listp (<e>, <p>)
Function: listp (<e>)

     Given an optional argument <p>, return `true' if <e> is a Maxima
     list and <p> evaluates to `true' for every list element.  When
     `listp' is not given the optional argument, return `true' if <e>
     is a Maxima list.  In all other cases, return `false'.

Function: locate_matrix_entry (<M>, <r_1>, <c_1>, <r_2>, <c_2>, <f>, <rel>)

     The first argument must be a matrix; the arguments <r_1> through
     <c_2> determine a sub-matrix of <M> that consists of rows <r_1>
     through <r_2> and columns <c_1> through <c_2>.

     Find a entry in the sub-matrix <M> that satisfies some property.
     Three cases:

     (1) `<rel> = 'bool' and <f> a predicate:

     Scan the sub-matrix from left to right then top to bottom, and
     return the index of the first entry that satisfies the predicate
     <f>.  If no matrix entry satisfies <f>, return `false'.

     (2) `<rel> = 'max' and <f> real-valued:

     Scan the sub-matrix looking for an entry that maximizes <f>.
     Return the index of a maximizing entry.

     (3) `<rel> = 'min' and <f> real-valued:

     Scan the sub-matrix looking for an entry that minimizes <f>.
     Return the index of a minimizing entry.

Function: lu_backsub (<M>, <b>)

     When `<M> = lu_factor (<A>, <field>)', then `lu_backsub (<M>,
     <b>)' solves the linear system `<A> <x> = <b>'.

Function: lu_factor (<M>, <field>)

     Return a list of the form `[<LU>, <perm>, <fld>]', or `[<LU>,
     <perm>, <fld>, <lower-cnd> <upper-cnd>]', where

     (1) The matrix <LU> contains the factorization of <M> in a packed
     form.      Packed form means three things: First, the rows of <LU>
     are permuted     according to the list <perm>.  If, for example,
     <perm> is the list     `[3,2,1]', the actual first row of the <LU>
     factorization is the     third row of the matrix <LU>.  Second,
     the lower triangular factor of     m is the lower triangular part
     of <LU> with the diagonal entries     replaced by all ones.
     Third, the upper triangular factor of <M> is the     upper
     triangular part of <LU>.

     (2) When the field is either `floatfield' or `complexfield', the
      numbers <lower-cnd> and <upper-cnd> are lower and upper bounds for
        the infinity norm condition number of <M>.  For all fields, the
        condition number might not be estimated; for such fields,
     `lu_factor'     returns a two item list.  Both the lower and upper
     bounds can differ from     their true values by arbitrarily large
     factors.  (See also `mat_cond'.)

     The argument <M> must be a square matrix.

     The optional argument <fld> must be a symbol that determines a
     ring or   field.  The pre-defined fields and rings are:

     (a) `generalring'      - the ring of Maxima expressions,

     (b) `floatfield'       - the field of floating point numbers of the
                                     type double,

     (c) `complexfield'     - the field of complex floating point
     numbers of                                  the type double,

     (d) `crering'          - the ring of Maxima CRE expressions,

     (e) `rationalfield'    - the field of rational numbers,

     (f) `runningerror'     - track the all floating point rounding
     errors,

     (g) `noncommutingring' - the ring of Maxima expressions where
                                multiplication is the non-commutative
     dot                                  operator.

     When the field is `floatfield', `complexfield', or `runningerror',
     the algorithm uses partial pivoting; for all other fields, rows
     are switched only when needed to avoid a zero pivot.

     Floating point addition arithmetic isn't associative, so the
     meaning of 'field' differs from the mathematical definition.

     A member of the field `runningerror' is a two member Maxima list
     of the form `[x,n]',where <x> is a floating point number and `n'
     is an integer.  The relative difference between the 'true' value
     of `x' and `x' is approximately bounded by the machine epsilon
     times `n'.  The running error bound drops some terms that of the
     order the square of the machine epsilon.

     There is no user-interface for defining a new field.  A user that
     is familiar with Common Lisp should be able to define a new field.
     To do this, a user must define functions for the arithmetic
     operations and functions for converting from the field
     representation to Maxima and back.  Additionally, for ordered
     fields (where partial pivoting will be used), a user must define
     functions for the magnitude and for comparing field members.
     After that all that remains is to define a Common Lisp structure
     `mring'.  The file `mring' has many examples.

     To compute the factorization, the first task is to convert each
     matrix entry to a member of the indicated field.  When conversion
     isn't possible, the factorization halts with an error message.
     Members of the field needn't be Maxima expressions.  Members of the
     `complexfield', for example, are Common Lisp complex numbers.  Thus
     after computing the factorization, the matrix entries must be
     converted to Maxima expressions.

     See also  `get_lu_factors'.

     Examples:

          (%i1) w[i,j] := random (1.0) + %i * random (1.0);
          (%o1)          w     := random(1.) + %i random(1.)
                          i, j
          (%i2) showtime : true$
          Evaluation took 0.00 seconds (0.00 elapsed)
          (%i3) M : genmatrix (w, 100, 100)$
          Evaluation took 7.40 seconds (8.23 elapsed)
          (%i4) lu_factor (M, complexfield)$
          Evaluation took 28.71 seconds (35.00 elapsed)
          (%i5) lu_factor (M, generalring)$
          Evaluation took 109.24 seconds (152.10 elapsed)
          (%i6) showtime : false$

          (%i7) M : matrix ([1 - z, 3], [3, 8 - z]);
                                  [ 1 - z    3   ]
          (%o7)                   [              ]
                                  [   3    8 - z ]
          (%i8) lu_factor (M, generalring);
                    [ 1 - z         3        ]
                    [                        ]
          (%o8)    [[   3            9       ], [1, 2], generalring]
                    [ -----  - z - ----- + 8 ]
                    [ 1 - z        1 - z     ]
          (%i9) get_lu_factors (%);
                            [   1    0 ]  [ 1 - z         3        ]
                  [ 1  0 ]  [          ]  [                        ]
          (%o9)  [[      ], [   3      ], [                9       ]]
                  [ 0  1 ]  [ -----  1 ]  [   0    - z - ----- + 8 ]
                            [ 1 - z    ]  [              1 - z     ]
          (%i10) %[1] . %[2] . %[3];
                                  [ 1 - z    3   ]
          (%o10)                  [              ]
                                  [   3    8 - z ]

Function: mat_cond (<M>, 1)
Function: mat_cond (<M>, inf)

     Return the <p>-norm matrix condition number of the matrix <m>.
     The allowed values for <p> are 1 and <inf>.  This function uses
     the LU factorization to invert the matrix <m>.  Thus the running
     time for `mat_cond' is proportional to the cube of the matrix
     size; `lu_factor' determines lower and upper bounds for the
     infinity norm condition number in time proportional to the square
     of the matrix size.

Function: mat_norm (<M>, 1)
Function: mat_norm (<M>, inf)
Function: mat_norm (<M>, frobenius)

     Return the matrix <p>-norm of the matrix <M>.  The allowed values
     for <p> are 1, `inf', and `frobenius' (the Frobenius matrix norm).
     The matrix <M> should be an unblocked matrix.

Function: matrixp (<e>, <p>)
Function: matrixp (<e>)

     Given an optional argument <p>, return `true' if <e> is a matrix
     and <p> evaluates to `true' for every matrix element.  When
     `matrixp' is not given an optional argument, return `true' if `e'
     is a matrix.  In all other cases, return `false'.

     See also `blockmatrixp'

Function: matrix_size (<M>)

     Return a two member list that gives the number of rows and
     columns, respectively of the matrix <M>.

Function: mat_fullunblocker (<M>)

     If <M> is a block matrix, unblock the matrix to all levels.  If
     <M> is a matrix, return <M>; otherwise, signal an error.

Function: mat_trace (<M>)

     Return the trace of the matrix <M>.  If <M> isn't a matrix, return
     a noun form.  When <M> is a block matrix, `mat_trace(M)' returns
     the same value as does `mat_trace(mat_unblocker(m))'.

Function: mat_unblocker (<M>)

     If <M> is a block matrix, unblock <M> one level.  If <M> is a
     matrix, `mat_unblocker (M)' returns <M>; otherwise, signal an
     error.

     Thus if each entry of <M> is matrix, `mat_unblocker (M)' returns an
     unblocked matrix, but if each entry of <M> is a block matrix,
     `mat_unblocker (M)' returns a block matrix with one less level of
     blocking.

     If you use block matrices, most likely you'll want to set
     `matrix_element_mult' to `"."' and `matrix_element_transpose' to
     `'transpose'.  See also `mat_fullunblocker'.

     Example:

          (%i1) A : matrix ([1, 2], [3, 4]);
                                      [ 1  2 ]
          (%o1)                       [      ]
                                      [ 3  4 ]
          (%i2) B : matrix ([7, 8], [9, 10]);
                                      [ 7  8  ]
          (%o2)                       [       ]
                                      [ 9  10 ]
          (%i3) matrix ([A, B]);
                               [ [ 1  2 ]  [ 7  8  ] ]
          (%o3)                [ [      ]  [       ] ]
                               [ [ 3  4 ]  [ 9  10 ] ]
          (%i4) mat_unblocker (%);
                                   [ 1  2  7  8  ]
          (%o4)                    [             ]
                                   [ 3  4  9  10 ]

Function: nullspace (<M>)

     If <M> is a matrix, return `span (v_1, ..., v_n)', where the set
     `{v_1, ..., v_n}' is a basis for the nullspace of <M>.  The span of
     the empty set is `{0}'.  Thus, when the nullspace has only one
     member, return `span ()'.

Function: nullity (<M>)

     If <M> is a matrix, return the dimension of the nullspace of <M>.

Function: orthogonal_complement (<v_1>, ..., <v_n>)

     Return `span (u_1, ..., u_m)', where the set `{u_1, ..., u_m}' is a
     basis for the orthogonal complement of the set `(v_1, ..., v_n)'.

     Each vector <v_1> through <v_n> must be a column vector.

Function: polynomialp (<p>, <L>, <coeffp>, <exponp>)
Function: polynomialp (<p>, <L>, <coeffp>)
Function: polynomialp (<p>, <L>)

     Return `true' if <p> is a polynomial in the variables in the list
     <L>.  The predicate <coeffp> must evaluate to `true' for each
     coefficient, and the predicate <exponp> must evaluate to `true'
     for all exponents of the variables in <L>.  If you want to use a
     non-default value for <exponp>, you must supply <coeffp> with a
     value even if you want to use the default for <coeffp>.

     The command `polynomialp (<p>, <L>, <coeffp>)' is equivalent to
     `polynomialp (<p>, <L>, <coeffp>, 'nonnegintegerp)' and the
     command `polynomialp (<p>, <L>)' is equivalent to `polynomialp
     (<p>, L<,> 'constantp, 'nonnegintegerp)'.

     The polynomial needn't be expanded:

          (%i1) polynomialp ((x + 1)*(x + 2), [x]);
          (%o1)                         true
          (%i2) polynomialp ((x + 1)*(x + 2)^a, [x]);
          (%o2)                         false

     An example using non-default values for coeffp and exponp:

          (%i1) polynomialp ((x + 1)*(x + 2)^(3/2), [x], numberp, numberp);
          (%o1)                         true
          (%i2) polynomialp ((x^(1/2) + 1)*(x + 2)^(3/2), [x], numberp,
                                                                  numberp);
          (%o2)                         true

     Polynomials with two variables:

          (%i1) polynomialp (x^2 + 5*x*y + y^2, [x]);
          (%o1)                         false
          (%i2) polynomialp (x^2 + 5*x*y + y^2, [x, y]);
          (%o2)                         true

Function: polytocompanion (<p>, <x>)

     If <p> is a polynomial in <x>, return the companion matrix of <p>.
     For a monic polynomial <p> of degree <n>, we have `<p> = (-1)^<n>
     charpoly (polytocompanion (<p>, <x>))'.

     When <p> isn't a polynomial in <x>, signal an error.

Function: ptriangularize (<M>, <v>)

     If <M> is a matrix with each entry a polynomial in <v>, return a
     matrix <M2> such that

     (1) <M2> is upper triangular,

     (2) `<M2> = <E_n> ... <E_1> <M>', where <E_1> through <E_n> are
     elementary matrices whose entries are polynomials in <v>,

     (3) `|det (<M>)| = |det (<M2>)|',

     Note: This function doesn't check that every entry is a polynomial
     in <v>.

Function: rowop (<M>, <i>, <j>, <theta>)

     If <M> is a matrix, return the matrix that results from doing the
     row operation `R_i <- R_i - theta * R_j'.  If <M> doesn't have a
     row <i> or <j>, signal an error.

Function: rank (<M>)

     Return the rank of that matrix <M>.  The rank is the dimension of
     the column space.  Example:

          (%i1) rank(matrix([1,2],[2,4]));
          (%o1)                                  1
          (%i2) rank(matrix([1,b],[c,d]));
          Proviso:  {d - b c # 0}
          (%o2)                                  2

Function: rowswap (<M>, <i>, <j>)

     If <M> is a matrix, swap rows <i> and <j>.  If <M> doesn't have a
     row <i> or <j>, signal an error.

Function: toeplitz (<col>)
Function: toeplitz (<col>, <row>)

     Return a Toeplitz matrix <T>.  The first first column of <T> is
     <col>; except for the first entry, the first row of <T> is <row>.
     The default for <row> is complex conjugate of <col>.  Example:

          (%i1)  toeplitz([1,2,3],[x,y,z]);
                                            [ 1  y  z ]
                                            [         ]
          (%o1)                             [ 2  1  y ]
                                            [         ]
                                            [ 3  2  1 ]
          (%i2)  toeplitz([1,1+%i]);

                                        [   1     1 - %I ]
          (%o2)                         [                ]
                                        [ %I + 1    1    ]

Function: vandermonde_matrix ([<x_1>, ..., <x_n>])

     Return a <n> by <n> matrix whose <i>-th row is `[1, <x_i>,
     <x_i>^2, ... <x_i>^(<n>-1)]'.

Function: zerofor (<M>)
Function: zerofor (<M>, <fld>)

     Return a zero  matrix that has the same shape as the matrix <M>.
     Every entry of the zero matrix is the additive identity of the
     field <fld>; the default for <fld> is <generalring>.

     The first argument <M> should be a square matrix or a non-matrix.
     When <M> is a matrix, each entry of <M> can be a square matrix -
     thus <M> can be a blocked Maxima matrix.  The matrix can be
     blocked to any (finite) depth.

     See also `identfor'

Function: zeromatrixp (<M>)

     If <M> is not a block matrix, return `true' if `is (equal (<e>,
     0))' is true for each element <e> of the matrix <M>.  If <M> is a
     block matrix, return `true' if `zeromatrixp' evaluates to `true'
     for each element of <e>.

64 lsquares


Introduction to lsquares
Functions and Variables for lsquares

64.1 Introduction to lsquares

`lsquares' is a collection of functions to implement the method of
least squares to estimate parameters for a model from numerical data.

64.2 Functions and Variables for lsquares

Function: lsquares_estimates (<D>, <x>, <e>, <a>)
Function: lsquares_estimates (<D>, <x>, <e>, <a>, initial = <L>, tol = <t>)

     Estimate parameters <a> to best fit the equation <e> in the
     variables <x> and <a> to the data <D>, as determined by the method
     of least squares.  `lsquares_estimates' first seeks an exact
     solution, and if that fails, then seeks an approximate solution.

     The return value is a list of lists of equations of the form `[a =
     ..., b = ..., c = ...]'.  Each element of the list is a distinct,
     equivalent minimum of the mean square error.

     The data <D> must be a matrix.  Each row is one datum (which may
     be called a `record' or `case' in some contexts), and each column
     contains the values of one variable across all data.  The list of
     variables <x> gives a name for each column of <D>, even the
     columns which do not enter the analysis.  The list of parameters
     <a> gives the names of the parameters for which estimates are
     sought.  The equation <e> is an expression or equation in the
     variables <x> and <a>; if <e> is not an equation, it is treated
     the same as `<e> = 0'.

     Additional arguments to `lsquares_estimates' are specified as
     equations and passed on verbatim to the function `lbfgs' which is
     called to find estimates by a numerical method when an exact
     result is not found.

     If some exact solution can be found (via `solve'), the data <D>
     may contain non-numeric values.  However, if no exact solution is
     found, each element of <D> must have a numeric value.  This
     includes numeric constants such as `%pi' and `%e' as well as
     literal numbers (integers, rationals, ordinary floats, and
     bigfloats).  Numerical calculations are carried out with ordinary
     floating-point arithmetic, so all other kinds of numbers are
     converted to ordinary floats for calculations.

     `load(lsquares)' loads this function.

     See also `lsquares_estimates_exact',
     `lsquares_estimates_approximate',
     `lsquares_mse', `lsquares_residuals', and `lsquares_residual_mse'.

     Examples:

     A problem for which an exact solution is found.

          (%i1) load (lsquares)$
          (%i2) M : matrix (
                  [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                            [ 1  1  1 ]
                                            [         ]
                                            [ 3       ]
                                            [ -  1  2 ]
                                            [ 2       ]
                                            [         ]
          (%o2)                             [ 9       ]
                                            [ -  2  1 ]
                                            [ 4       ]
                                            [         ]
                                            [ 3  2  2 ]
                                            [         ]
                                            [ 2  2  1 ]
          (%i3) lsquares_estimates (
                   M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
                                   59        27      10921        107
          (%o3)            [[A = - --, B = - --, C = -----, D = - ---]]
                                   16        16      1024         32

     A problem for which no exact solution is found, so
     `lsquares_estimates' resorts to numerical approximation.

          (%i1) load (lsquares)$
          (%i2) M : matrix ([1, 1], [2, 7/4], [3, 11/4], [4, 13/4]);
                                             [ 1  1  ]
                                             [       ]
                                             [    7  ]
                                             [ 2  -  ]
                                             [    4  ]
                                             [       ]
          (%o2)                              [    11 ]
                                             [ 3  -- ]
                                             [    4  ]
                                             [       ]
                                             [    13 ]
                                             [ 4  -- ]
                                             [    4  ]
          (%i3) lsquares_estimates (
            M, [x,y], y=a*x^b+c, [a,b,c], initial=[3,3,3], iprint=[-1,0]);
          (%o3) [[a = 1.387365874920637, b = .7110956639593767,
                                                  c = - .4142705622439105]]

Function: lsquares_estimates_exact (<MSE>, <a>)

     Estimate parameters <a> to minimize the mean square error <MSE>,
     by constructing a system of equations and attempting to solve them
     symbolically via `solve'.  The mean square error is an expression
     in the parameters <a>, such as that returned by `lsquares_mse'.

     The return value is a list of lists of equations of the form `[a =
     ..., b = ..., c = ...]'.  The return value may contain zero, one,
     or two or more elements.  If two or more elements are returned,
     each represents a distinct, equivalent minimum of the mean square
     error.

     See also `lsquares_estimates', `lsquares_estimates_approximate',
     `lsquares_mse', `lsquares_residuals', and `lsquares_residual_mse'.

     Example:

          (%i1) load (lsquares)$
          (%i2) M : matrix (
                   [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                     [ 1  1  1 ]
                                     [         ]
                                     [ 3       ]
                                     [ -  1  2 ]
                                     [ 2       ]
                                     [         ]
          (%o2)                      [ 9       ]
                                     [ -  2  1 ]
                                     [ 4       ]
                                     [         ]
                                     [ 3  2  2 ]
                                     [         ]
                                     [ 2  2  1 ]
          (%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
                     5
                    ====
                    \                 2                         2
                     >    ((D + M    )  - C - M     B - M     A)
                    /            i, 1          i, 3      i, 2
                    ====
                    i = 1
          (%o3)     ---------------------------------------------
                                          5
          (%i4) lsquares_estimates_exact (mse, [A, B, C, D]);
                            59        27      10921        107
          (%o4)     [[A = - --, B = - --, C = -----, D = - ---]]
                            16        16      1024         32

Function: lsquares_estimates_approximate (<MSE>, <a>, initial =

          <L>, tol = <t>)
     Estimate parameters <a> to minimize the mean square error <MSE>,
     via the numerical minimization function `lbfgs'.  The mean square
     error is an expression in the parameters <a>, such as that
     returned by `lsquares_mse'.

     The solution returned by `lsquares_estimates_approximate' is a
     local (perhaps global) minimum of the mean square error.  For
     consistency with `lsquares_estimates_exact', the return value is a
     nested list which contains one element, namely a list of equations
     of the form `[a = ..., b = ..., c = ...]'.

     Additional arguments to `lsquares_estimates_approximate' are
     specified as equations and passed on verbatim to the function
     `lbfgs'.

     <MSE> must evaluate to a number when the parameters are assigned
     numeric values.  This requires that the data from which <MSE> was
     constructed comprise only numeric constants such as `%pi' and `%e'
     and literal numbers (integers, rationals, ordinary floats, and
     bigfloats).  Numerical calculations are carried out with ordinary
     floating-point arithmetic, so all other kinds of numbers are
     converted to ordinary floats for calculations.

     `load(lsquares)' loads this function.

     See also `lsquares_estimates', `lsquares_estimates_exact',
     `lsquares_mse',
     `lsquares_residuals', and `lsquares_residual_mse'.

     Example:

          (%i1) load (lsquares)$
          (%i2) M : matrix (
                   [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                     [ 1  1  1 ]
                                     [         ]
                                     [ 3       ]
                                     [ -  1  2 ]
                                     [ 2       ]
                                     [         ]
          (%o2)                      [ 9       ]
                                     [ -  2  1 ]
                                     [ 4       ]
                                     [         ]
                                     [ 3  2  2 ]
                                     [         ]
                                     [ 2  2  1 ]
          (%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
                     5
                    ====
                    \                 2                         2
                     >    ((D + M    )  - C - M     B - M     A)
                    /            i, 1          i, 3      i, 2
                    ====
                    i = 1
          (%o3)     ---------------------------------------------
                                          5
          (%i4) lsquares_estimates_approximate (
                        mse, [A, B, C, D], iprint = [-1, 0]);
          (%o4) [[A = - 3.67850494740174, B = - 1.683070351177813,
                           C = 10.63469950148635, D = - 3.340357993175206]]

Function: lsquares_mse (<D>, <x>, <e>)

     Returns the mean square error (MSE), a summation expression, for
     the equation <e> in the variables <x>, with data <D>.

     The MSE is defined as:

                              n
                             ====
                         1   \                        2
                         -    >    (lhs(e ) - rhs(e ))
                         n   /           i         i
                             ====
                             i = 1

     where <n> is the number of data and `<e>[i]' is the equation <e>
     evaluated with the variables in <x> assigned values from the
     `i'-th datum, `<D>[i]'.

     `load(lsquares)' loads this function.

     Example:

          (%i1) load (lsquares)$
          (%i2) M : matrix (
                   [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                     [ 1  1  1 ]
                                     [         ]
                                     [ 3       ]
                                     [ -  1  2 ]
                                     [ 2       ]
                                     [         ]
          (%o2)                      [ 9       ]
                                     [ -  2  1 ]
                                     [ 4       ]
                                     [         ]
                                     [ 3  2  2 ]
                                     [         ]
                                     [ 2  2  1 ]

          (%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
                     5
                    ====
                    \                 2                         2
                     >    ((D + M    )  - C - M     B - M     A)
                    /            i, 1          i, 3      i, 2
                    ====
                    i = 1
          (%o3)     ---------------------------------------------
                                          5

          (%i4) diff (mse, D);
                   5
                  ====
                  \                             2
                4  >    (D + M    ) ((D + M    )  - C - M     B - M     A)
                  /           i, 1         i, 1          i, 3      i, 2
                  ====
                  i = 1
          (%o4) ----------------------------------------------------------
                                            5

          (%i5) ''mse, nouns;
                         2                 2         9 2               2
          (%o5) (((D + 3)  - C - 2 B - 2 A)  + ((D + -)  - C - B - 2 A)
                                                     4
                     2               2         3 2               2
           + ((D + 2)  - C - B - 2 A)  + ((D + -)  - C - 2 B - A)
                                               2
                     2             2
           + ((D + 1)  - C - B - A) )/5

Function: lsquares_residuals (<D>, <x>, <e>, <a>)

     Returns the residuals for the equation <e> with specified
     parameters <a> and data <D>.

     <D> is a matrix, <x> is a list of variables, <e> is an equation or
     general expression; if not an equation, <e> is treated as if it
     were `<e> = 0'.  <a> is a list of equations which specify values
     for any free parameters in <e> aside from <x>.

     The residuals are defined as:

                                  lhs(e ) - rhs(e )
                                       i         i

     where `<e>[i]' is the equation <e> evaluated with the variables in
     <x> assigned values from the `i'-th datum, `<D>[i]', and assigning
     any remaining free variables from <a>.

     `load(lsquares)' loads this function.

     Example:

          (%i1) load (lsquares)$
          (%i2) M : matrix (
                   [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                            [ 1  1  1 ]
                                            [         ]
                                            [ 3       ]
                                            [ -  1  2 ]
                                            [ 2       ]
                                            [         ]
          (%o2)                             [ 9       ]
                                            [ -  2  1 ]
                                            [ 4       ]
                                            [         ]
                                            [ 3  2  2 ]
                                            [         ]
                                            [ 2  2  1 ]
          (%i3) a : lsquares_estimates (
                    M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
                                   59        27      10921        107
          (%o3)            [[A = - --, B = - --, C = -----, D = - ---]]
                                   16        16      1024         32
          (%i4) lsquares_residuals (
                    M, [z,x,y], (z+D)^2 = A*x+B*y+C, first(a));
                                      13    13    13  13  13
          (%o4)                      [--, - --, - --, --, --]
                                      64    64    32  64  64

Function: lsquares_residual_mse (<D>, <x>, <e>, <a>)

     Returns the residual mean square error (MSE) for the equation <e>
     with specified parameters <a> and data <D>.

     The residual MSE is defined as:

                              n
                             ====
                         1   \                        2
                         -    >    (lhs(e ) - rhs(e ))
                         n   /           i         i
                             ====
                             i = 1

     where `<e>[i]' is the equation <e> evaluated with the variables in
     <x> assigned values from the `i'-th datum, `<D>[i]', and assigning
     any remaining free variables from <a>.

     `load(lsquares)' loads this function.

     Example:

          (%i1) load (lsquares)$
          (%i2) M : matrix (
                   [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                     [ 1  1  1 ]
                                     [         ]
                                     [ 3       ]
                                     [ -  1  2 ]
                                     [ 2       ]
                                     [         ]
          (%o2)                      [ 9       ]
                                     [ -  2  1 ]
                                     [ 4       ]
                                     [         ]
                                     [ 3  2  2 ]
                                     [         ]
                                     [ 2  2  1 ]
          (%i3) a : lsquares_estimates (
                       M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);

                            59        27      10921        107
          (%o3)     [[A = - --, B = - --, C = -----, D = - ---]]
                            16        16      1024         32
          (%i4) lsquares_residual_mse (
                       M, [z,x,y], (z + D)^2 = A*x + B*y + C, first (a));
                                        169
          (%o4)                         ----
                                        2560

Function: plsquares (<Mat>,<VarList>,<depvars>)
Function: plsquares (<Mat>,<VarList>,<depvars>,<maxexpon>)
Function: plsquares

          (<Mat>,<VarList>,<depvars>,<maxexpon>,<maxdegree>)
     Multivariable polynomial adjustment of a data table by the "least
     squares" method. <Mat> is a matrix containing the data, <VarList>
     is a list of variable names (one for each Mat column, but use "-"
     instead of varnames to ignore Mat columns), <depvars> is the name
     of a dependent variable or a list with one or more names of
     dependent variables (which names should be in <VarList>),
     <maxexpon> is the optional maximum exponent for each independent
     variable (1 by default), and <maxdegree> is the optional maximum
     polynomial degree (<maxexpon> by default); note that the sum of
     exponents of each term must be equal or smaller than <maxdegree>,
     and if `maxdgree = 0' then no limit is applied.

     If <depvars> is the name of a dependent variable (not in a list),
     `plsquares' returns the adjusted polynomial. If <depvars> is a
     list of one or more dependent variables, `plsquares' returns a
     list with the adjusted polynomial(s). The Coefficients of
     Determination  are displayed in order to inform about the goodness
     of fit, which ranges from 0 (no correlation) to 1 (exact
     correlation). These values are also stored in the global variable
     <DETCOEF> (a list if <depvars> is a list).

     A simple example of multivariable linear adjustment:
          (%i1) load("plsquares")$

          (%i2) plsquares(matrix([1,2,0],[3,5,4],[4,7,9],[5,8,10]),
                          [x,y,z],z);
               Determination Coefficient for z = .9897039897039897
                                 11 y - 9 x - 14
          (%o2)              z = ---------------
                                        3

     The same example without degree restrictions:
          (%i3) plsquares(matrix([1,2,0],[3,5,4],[4,7,9],[5,8,10]),
                          [x,y,z],z,1,0);
               Determination Coefficient for z = 1.0
                              x y + 23 y - 29 x - 19
          (%o3)           z = ----------------------
                                        6

     How many diagonals does a N-sides polygon have? What polynomial
     degree should be used?
          (%i4) plsquares(matrix([3,0],[4,2],[5,5],[6,9],[7,14],[8,20]),
                          [N,diagonals],diagonals,5);
               Determination Coefficient for diagonals = 1.0
                                          2
                                         N  - 3 N
          (%o4)              diagonals = --------
                                            2
          (%i5) ev(%, N=9);   /* Testing for a 9 sides polygon */
          (%o5)                 diagonals = 27

     How many ways do we have to put two queens without they are
     threatened into a n x n chessboard?
          (%i6) plsquares(matrix([0,0],[1,0],[2,0],[3,8],[4,44]),
                          [n,positions],[positions],4);
               Determination Coefficient for [positions] = [1.0]
                                   4       3      2
                                3 n  - 10 n  + 9 n  - 2 n
          (%o6)    [positions = -------------------------]
                                            6
          (%i7) ev(%[1], n=8); /* Testing for a (8 x 8) chessboard */
          (%o7)                positions = 1288

     An example with six dependent variables:
          (%i8) mtrx:matrix([0,0,0,0,0,1,1,1],[0,1,0,1,1,1,0,0],
                            [1,0,0,1,1,1,0,0],[1,1,1,1,0,0,0,1])$
          (%i8) plsquares(mtrx,[a,b,_And,_Or,_Xor,_Nand,_Nor,_Nxor],
                               [_And,_Or,_Xor,_Nand,_Nor,_Nxor],1,0);
                Determination Coefficient for
          [_And, _Or, _Xor, _Nand, _Nor, _Nxor] =
          [1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
          (%o2) [_And = a b, _Or = - a b + b + a,
          _Xor = - 2 a b + b + a, _Nand = 1 - a b,
          _Nor = a b - b - a + 1, _Nxor = 2 a b - b - a + 1]

     To use this function write first `load("lsquares")'.

65 minpack


Introduction to minpack
Functions and Variables for minpack

65.1 Introduction to minpack

`Minpack' is a Common Lisp translation (via `f2cl') of the Fortran
library MINPACK, as obtained from Netlib.

65.2 Functions and Variables for minpack

Function: minpack_lsquares (<flist>, <varlist>, <guess> [, <tolerance>, <jacobian>])

     Compute the point that minimizes the sum of the squares of the
     functions in the list <flist>.  The variables are in the list
     <varlist>.  An initial guess of the optimum point must be provided
     in <guess>.

     The optional keyword arguments, <tolerance> and <jacobian> provide
     some control over the algorithm.  <tolerance> is the estimated
     relative error desired in the sum of squares.  <jacobian> can be
     used to specify the Jacobian.  If <jacobian> is not given or is
     `true' (the default), the Jacobian is computed from <flist>.  If
     <jacobian> is `false', a numerical approximation is used.

     `minpack_lsquares' returns a list.  The first item is the
     estimated solution; the second is the sum of squares, and the third
     indicates the success of the algorithm.  The possible values are

    `0'
          improper input parameters.

    `1'
          algorithm estimates that the relative error in the sum of
          squares is at most `tolerance'.

    `2'
          algorithm estimates that the relative error between x and the
          solution is at most `tolerance'.

    `3'
          conditions for info = 1 and info = 2 both hold.

    `4'
          fvec is orthogonal to the columns of the jacobian to machine
          precision.

    `5'
          number of calls to fcn with iflag = 1 has reached 100*(n+1).

    `6'
          tol is too small. no further reduction in the sum of squares
          is possible.

    `7'
          tol is too small. no further improvement in the approximate
          solution x is possible.

          /* Problem 6: Powell singular function */
          (%i1) powell(x1,x2,x3,x4) :=
                   [x1+10*x2, sqrt(5)*(x3-x4), (x2-2*x3)^2,
                        sqrt(10)*(x1-x4)^2]$
          (%i2) minpack_lsquares(powell(x1,x2,x3,x4), [x1,x2,x3,x4],
                                 [3,-1,0,1]);
          (%o2) [[1.652117596168394e-17, - 1.652117596168393e-18,
                  2.643388153869468e-18, 2.643388153869468e-18],
                 6.109327859207777e-34, 4]

          /* Same problem but use numerical approximation to Jacobian */
          (%i3) minpack_lsquares(powell(x1,x2,x3,x4), [x1,x2,x3,x4],
                                 [3,-1,0,1], jacobian = false);
          (%o3) [[5.060282149485331e-11, - 5.060282149491206e-12,
                  2.179447843547218e-11, 2.179447843547218e-11],
                 3.534491794847031e-21, 5]

Function: minpack_solve (<flist>, <varlist>, <guess> [, <tolerance>, <jacobian>])

     Solve a system of `n' equations in `n' unknowns.  The `n'
     equations are given in the list <flist>, and the unknowns are in
     <varlist>.  An initial guess of the solution must be provided in
     <guess>.

     The optional keyword arguments, <tolerance> and <jacobian> provide
     some control over the algorithm.  <tolerance> is the estimated
     relative error desired in the sum of squares.  <jacobian> can be
     used to specify the Jacobian.  If <jacobian> is not given or is
     `true' (the default), the Jacobian is computed from <flist>.  If
     <jacobian> is `false', a numerical approximation is used.

     `minpack_solve' returns a list.  The first item is the estimated
     solution; the second is the sum of squares, and the third
     indicates the success of the algorithm.  The possible values are

    `0'
          improper input parameters.

    `1'
          algorithm estimates that the relative error in the solution is
          at most `tolerance'.

    `2'
          number of calls to fcn with iflag = 1 has reached 100*(n+1).

    `3'
          tol is too small. no further reduction in the sum of squares
          is possible.

    `4'
          Iteration is not making good progress.

66 makeOrders


Functions and Variables for makeOrders

66.1 Functions and Variables for makeOrders

Function: makeOrders (<indvarlist>,<orderlist>)

     Returns a list of all powers for a polynomial up to and including
     the arguments.

          (%i1) load("makeOrders")$

          (%i2) makeOrders([a,b],[2,3]);
          (%o2) [[0, 0], [0, 1], [0, 2], [0, 3], [1, 0], [1, 1],
                      [1, 2], [1, 3], [2, 0], [2, 1], [2, 2], [2, 3]]
          (%i3) expand((1+a+a^2)*(1+b+b^2+b^3));
                 2  3      3    3    2  2      2    2    2
          (%o3) a  b  + a b  + b  + a  b  + a b  + b  + a  b + a b
                                                            2
                                                     + b + a  + a + 1
     where `[0, 1]' is associated with the term b and `[2, 3]' with a^2
     b^3.

     To use this function write first `load("makeOrders")'.

67 mnewton


Introduction to mnewton
Functions and Variables for mnewton

67.1 Introduction to mnewton

`mnewton' is an implementation of Newton's method for solving nonlinear
equations in one or more variables.

67.2 Functions and Variables for mnewton

Option variable: newtonepsilon

     Default value: `10.0^(-fpprec/2)'

     Precision to determine when the `mnewton' function has converged
     towards the solution.  If `newtonepsilon' is a bigfloat, then
     `mnewton' computations are done with bigfloats.  See also
     `mnewton'.

Option variable: newtonmaxiter

     Default value: `50'

     Maximum number of iterations to stop the `mnewton' function if it
     does not converge or if it converges too slowly.

     See also `mnewton'.

Function: mnewton (<FuncList>,<VarList>,<GuessList>)

     Multiple nonlinear functions solution using the Newton method.
     <FuncList> is the list of functions to solve, <VarList> is the
     list of variable names, and <GuessList> is the list of initial
     approximations.

     The solution is returned in the same format that `solve()' returns.
     If the solution is not found, `[]' is returned.

     This function is controlled by global variables `newtonepsilon' and
     `newtonmaxiter'.

          (%i1) load("mnewton")$

          (%i2) mnewton([x1+3*log(x1)-x2^2, 2*x1^2-x1*x2-5*x1+1],
                        [x1, x2], [5, 5]);
          (%o2) [[x1 = 3.756834008012769, x2 = 2.779849592817897]]
          (%i3) mnewton([2*a^a-5],[a],[1]);
          (%o3)             [[a = 1.70927556786144]]
          (%i4) mnewton([2*3^u-v/u-5, u+2^v-4], [u, v], [2, 2]);
          (%o4) [[u = 1.066618389595407, v = 1.552564766841786]]

     The variable `newtonepsilon' controls the precision of the
     approximations.  It also controls if computations are performed
     with floats or bigfloats.

          (%i1) load(mnewton)$

          (%i2) (fpprec : 25, newtonepsilon : bfloat(10^(-fpprec+5)))$

          (%i3) mnewton([2*3^u-v/u-5, u+2^v-4], [u, v], [2, 2]);
          (%o3) [[u = 1.066618389595406772591173b0,
                                         v = 1.552564766841786450100418b0]]

     To use this function write first `load("mnewton")'.  See also
     `newtonepsilon' and `newtonmaxiter'.

68 numericalio


Introduction to numericalio
Functions and Variables for plain-text input and output
Functions and Variables for binary input and output

68.1 Introduction to numericalio

`numericalio' is a collection of functions to read and write files and
streams.  Functions for plain-text input and output can read and write
numbers (integer, float, or bigfloat), symbols, and strings.  Functions
for binary input and output can read and write only floating-point
numbers.

   If there already exists a list, matrix, or array object to store
input data, `numericalio' input functions can write data into that
object.  Otherwise, `numericalio' can guess, to some degree, the
structure of an object to store the data, and return that object.

68.2 Functions and Variables for plain-text input and output

Function: read_matrix (<S>)
Function: read_matrix (<S>, <M>)
Function: read_matrix (<S>, <separator_flag>)
Function: read_matrix (<S>, <M>, <separator_flag>)

     `read_matrix(<S>)' reads the source <S> and returns its entire
     content as a matrix.  The size of the matrix is inferred from the
     input data; each line of the file becomes one row of the matrix.
     If some lines have different lengths, `read_matrix' complains.

     `read_matrix(<S>, <M>)' read the source <S> into the matrix <M>,
     until <M> is full or the source is exhausted.  Input data are read
     into the matrix in row-major order; the input need not have the
     same number of rows and columns as <M>.

     The source <S> may be a file name or a stream.

     The recognized values of <separator_flag> are `comma', `pipe',
     `semicolon', and `space'.  If <separator_flag> is not specified,
     the file is assumed space-delimited.

Function: read_array (<S>, <A>)
Function: read_array (<S>, <A>, <separator_flag>)

     Reads the source <S> into the array <A>, until <A> is full or the
     source is exhausted.  Input data are read into the array in
     row-major order; the input need not conform to the dimensions of
     <A>.

     The source <S> may be a file name or a stream.

     The recognized values of <separator_flag> are `comma', `pipe',
     `semicolon', and `space'.  If <separator_flag> is not specified,
     the file is assumed space-delimited.

Function: read_hashed_array (<S>, <A>)
Function: read_hashed_array (<S>, <A>, <separator_flag>)

     Reads the source <S> and returns its entire content as a hashed
     array.  The source <S> may be a file name or a stream.

     `read_hashed_array' treats the first item on each line as a hash
     key, and associates the remainder of the line (as a list) with the
     key.  For example, the line `567 12 17 32 55' is equivalent to
     `A[567]: [12, 17, 32, 55]$'.  Lines need not have the same numbers
     of elements.

     The recognized values of <separator_flag> are `comma', `pipe',
     `semicolon', and `space'.  If <separator_flag> is not specified,
     the file is assumed space-delimited.

Function: read_nested_list (<S>)
Function: read_nested_list (<S>, <separator_flag>)

     Reads the source <S> and returns its entire content as a nested
     list.  The source <S> may be a file name or a stream.

     `read_nested_list' returns a list which has a sublist for each
     line of input. Lines need not have the same numbers of elements.
     Empty lines are not ignored: an empty line yields an empty sublist.

     The recognized values of <separator_flag> are `comma', `pipe',
     `semicolon', and `space'.  If <separator_flag> is not specified,
     the file is assumed space-delimited.

Function: read_list (<S>)
Function: read_list (<S>, <L>)
Function: read_list (<S>, <separator_flag>)
Function: read_list (<S>, <L>, <separator_flag>)

     `read_list(<S>)' reads the source <S> and returns its entire
     content as a flat list.

     `read_list(<S>, <L>)' reads the source <S> into the list <L>,
     until <L> is full or the source is exhausted.

     The source <S> may be a file name or a stream.

     The recognized values of <separator_flag> are `comma', `pipe',
     `semicolon', and `space'.  If <separator_flag> is not specified,
     the file is assumed space-delimited.

Function: write_data (<X>, <D>)
Function: write_data (<X>, <D>, <separator_flag>)

     Writes the object <X> to the destination <D>.

     `write_data' writes a matrix in row-major order, with one line per
     row.

     `write_data' writes an array created by `array' or `make_array' in
     row-major order, with a new line at the end of every slab.
     Higher-dimensional slabs are separated by additional new lines.

     `write_data' writes a hashed array with each key followed by its
     associated list on one line.

     `write_data' writes a nested list with each sublist on one line.

     `write_data' writes a flat list all on one line.

     The destination <D> may be a file name or a stream.  When the
     destination is a file name, the global variable
     `file_output_append' governs whether the output file is appended
     or truncated.  When the destination is a stream, no special action
     is taken by `write_data' after all the data are written; in
     particular, the stream remains open.

     The recognized values of <separator_flag> are `comma', `pipe',
     `semicolon', `space', and `tab'.  If <separator_flag> is not
     specified, the file is assumed space-delimited.

68.3 Functions and Variables for binary input and output

Function: assume_external_byte_order (<byte_order_flag>)

     Tells `numericalio' the byte order for reading and writing binary
     data.  Two values of <byte_order_flag> are recognized: `lsb' which
     indicates least-significant byte first, also called little-endian
     byte order; and `msb' which indicates most-significant byte first,
     also called big-endian byte order.

     If not specified, `numericalio' assumes the external byte order is
     most-significant byte first.

Function: openr_binary (<file_name>)

     Returns an input stream of 8-bit unsigned bytes to read the file
     named by <file_name>.

Function: openw_binary (<file_name>)

     Returns an output stream of 8-bit unsigned bytes to write the file
     named by <file_name>.

Function: opena_binary (<file_name>)

     Returns an output stream of 8-bit unsigned bytes to append the
     file named by <file_name>.

Function: read_binary_matrix (<S>, <M>)

     Reads binary 8-byte floating point numbers from the source <S>
     into the matrix <M> until <M> is full, or the source is exhausted.
     Elements of <M> are read in row-major order.

     The source <S> may be a file name or a stream.

     The byte order in elements of the source is specified by
     `assume_external_byte_order'.

Function: read_binary_array (<S>, <A>)

     Reads binary 8-byte floating point numbers from the source <S>
     into the array <A> until <A> is full, or the source is exhausted.
     <A> must be an array created by `array' or `make_array'.  Elements
     of <A> are read in row-major order.

     The source <S> may be a file name or a stream.

     The byte order in elements of the source is specified by
     `assume_external_byte_order'.

Function: read_binary_list (<S>)
Function: read_binary_list (<S>, <L>)

     `read_binary_list(<S>)' reads the entire content of the source <S>
     as a sequence of binary 8-byte floating point numbers, and returns
     it as a list.  The source <S> may be a file name or a stream.

     `read_binary_list(<S>, <L>)' reads 8-byte binary floating point
     numbers from the source <S> until the list <L> is full, or the
     source is exhausted.

     The byte order in elements of the source is specified by
     `assume_external_byte_order'.

Function: write_binary_data (<X>, <D>)

     Writes the object <X>, comprising binary 8-byte IEEE 754
     floating-point numbers, to the destination <D>.  Other kinds of
     numbers are coerced to 8-byte floats.  `write_binary_data' cannot
     write non-numeric data.

     The object <X> may be a list, a nested list, a matrix, or an array
     created by `array' or `make_array'; <X> cannot be an undeclared
     array or any other type of object.  `write_binary_data' writes
     nested lists, matrices, and arrays in row-major order.

     The destination <D> may be a file name or a stream.  When the
     destination is a file name, the global variable
     `file_output_append' governs whether the output file is appended
     or truncated.  When the destination is a stream, no special action
     is taken by `write_binary_data' after all the data are written; in
     particular, the stream remains open.

     The byte order in elements of the destination is specified by
     `assume_external_byte_order'.

69 opsubst


Functions and Variables for opsubst

69.1 Functions and Variables for opsubst

Function: opsubst (<f>,<g>,<e>)
Function: opsubst (<g>=<f>,<e>)
Function: opsubst ([<g1>=<f1>,<g2>=<f2>,..., <gn>=<fn>],<e>)

     The function `opsubst' is similar to the function `subst', except
     that `opsubst' only makes substitutions for the operators in an
     expression. In general, When <f> is an operator in the expression
     <e>, substitute <g> for <f> in the expression <e>.

     To determine the operator, `opsubst' sets `inflag' to true. This
     means `opsubst' substitutes for the internal, not the displayed,
     operator in the expression.

     Examples:
          (%i1) load (opsubst)$

          (%i2) opsubst(f,g,g(g(x)));
          (%o2)                     f(f(x))
          (%i3) opsubst(f,g,g(g));
          (%o3)                       f(g)
          (%i4) opsubst(f,g[x],g[x](z));
          (%o4)                       f(z)
          (%i5) opsubst(g[x],f, f(z));
          (%o5)                      g (z)
                                      x
          (%i6) opsubst(tan, sin, sin(sin));
          (%o6)                     tan(sin)
          (%i7) opsubst([f=g,g=h],f(x));
          (%o7)                       h(x)

     Internally, Maxima does not use the unary negation, division, or
     the subtraction operators; thus:
          (%i8) opsubst("+","-",a-b);
          (%o8)                     a - b
          (%i9) opsubst("f","-",-a);
          (%o9)                      - a
          (%i10) opsubst("^^","/",a/b);
                                       a
          (%o10)                       -
                                       b

     The internal representation of -a*b is *(-1,a,b); thus
          (%i11) opsubst("[","*", -a*b);
          (%o11)                  [- 1, a, b]

     When either operator isn't a Maxima symbol, generally some other
     function will signal an error:
          (%i12) opsubst(a+b,f, f(x));

          Improper name or value in functional position:
          b + a
           -- an error.  Quitting.  To debug this try debugmode(true);

     However, subscripted operators are allowed:
          (%i13) opsubst(g[5],f, f(x));
          (%o13)                     g (x)
                                      5

     To use this function write first `load("opsubst")'.

70 orthopoly


Introduction to orthogonal polynomials
Functions and Variables for orthogonal polynomials

70.1 Introduction to orthogonal polynomials

`orthopoly' is a package for symbolic and numerical evaluation of
several kinds of orthogonal polynomials, including Chebyshev, Laguerre,
Hermite, Jacobi, Legendre, and ultraspherical (Gegenbauer) polynomials.
Additionally, `orthopoly' includes support for the spherical Bessel,
spherical Hankel, and spherical harmonic functions.

   For the most part, `orthopoly' follows the conventions of Abramowitz
and Stegun Handbook of Mathematical Functions, Chapter 22 (10th
printing, December 1972); additionally, we use Gradshteyn and Ryzhik,
Table of Integrals, Series, and Products (1980 corrected and enlarged
edition), and Eugen Merzbacher Quantum Mechanics (2nd edition, 1970).

   Barton Willis of the University of Nebraska at Kearney (UNK) wrote
the `orthopoly' package and its documentation. The package is released
under the GNU General Public License (GPL).

70.2 Functions and Variables for orthogonal polynomials

Function: assoc_legendre_p (<n>, <m>, <x>)

     The associated Legendre function of the first kind of degree <n>
     and order <m>.

     Reference: Abramowitz and Stegun, equations 22.5.37, page 779,
     8.6.6 (second equation), page 334, and 8.2.5, page 333.

Function: assoc_legendre_q (<n>, <m>, <x>)

     The associated Legendre function of the second kind of degree <n>
     and order <m>.

     Reference: Abramowitz and Stegun, equation 8.5.3 and 8.1.8.

Function: chebyshev_t (<n>, <x>)

     The Chebyshev function of the first kind.

     Reference: Abramowitz and Stegun, equation 22.5.47, page 779.

Function: chebyshev_u (<n>, <x>)

     The Chebyshev function of the second kind.

     Reference: Abramowitz and Stegun, equation 22.5.48, page 779.

Function: gen_laguerre (<n>, <a>, <x>)

     The generalized Laguerre polynomial of degree <n>.

     Reference: Abramowitz and Stegun, equation 22.5.54, page 780.

Function: hermite (<n>, <x>)

     The Hermite polynomial.

     Reference: Abramowitz and Stegun, equation 22.5.55, page 780.

Function: intervalp (<e>)

     Return `true' if the input is an interval and return false if it
     isn't.

Function: jacobi_p (<n>, <a>, <b>, <x>)

     The Jacobi polynomial.

     The Jacobi polynomials are actually defined for all <a> and <b>;
     however, the Jacobi polynomial weight `(1 - <x>)^<a> (1 +
     <x>)^<b>' isn't integrable for `<a> <= -1' or `<b> <= -1'.

     Reference: Abramowitz and Stegun, equation 22.5.42, page 779.

Function: laguerre (<n>, <x>)

     The Laguerre polynomial.

     Reference: Abramowitz and Stegun, equations 22.5.16 and 22.5.54,
     page 780.

Function: legendre_p (<n>, <x>)

     The Legendre polynomial of the first kind.

     Reference: Abramowitz and Stegun, equations 22.5.50 and 22.5.51,
     page 779.

Function: legendre_q (<n>, <x>)

     The Legendre polynomial of the first kind.

     Reference: Abramowitz and Stegun, equations 8.5.3 and 8.1.8.

Function: orthopoly_recur (<f>, <args>)

     Returns a recursion relation for the orthogonal function family
     <f> with arguments <args>. The recursion is with respect to the
     polynomial degree.

          (%i1) orthopoly_recur (legendre_p, [n, x]);
                          (2 n - 1) P     (x) x + (1 - n) P     (x)
                                     n - 1                 n - 2
          (%o1)   P (x) = -----------------------------------------
                   n                          n

     The second argument to `orthopoly_recur' must be a list with the
     correct number of arguments for the function <f>; if it isn't,
     Maxima signals an error.

          (%i1) orthopoly_recur (jacobi_p, [n, x]);

          Function jacobi_p needs 4 arguments, instead it received 2
           -- an error.  Quitting.  To debug this try debugmode(true);

     Additionally, when <f> isn't the name of one of the families of
     orthogonal polynomials, an error is signalled.

          (%i1) orthopoly_recur (foo, [n, x]);

          A recursion relation for foo isn't known to Maxima
           -- an error.  Quitting.  To debug this try debugmode(true);

Variable: orthopoly_returns_intervals

     Default value: `true'

     When `orthopoly_returns_intervals' is `true', floating point
     results are returned in the form `interval (<c>, <r>)', where <c>
     is the center of an interval and <r> is its radius. The center can
     be a complex number; in that case, the interval is a disk in the
     complex plane.

Function: orthopoly_weight (<f>, <args>)

     Returns a three element list; the first element is the formula of
     the weight for the orthogonal polynomial family <f> with arguments
     given by the list <args>; the second and third elements give the
     lower and upper endpoints of the interval of orthogonality. For
     example,

          (%i1) w : orthopoly_weight (hermite, [n, x]);
                                      2
                                   - x
          (%o1)                 [%e    , - inf, inf]
          (%i2) integrate(w[1]*hermite(3, x)*hermite(2, x), x, w[2], w[3]);
          (%o2)                           0

     The main variable of <f> must be a symbol; if it isn't, Maxima
     signals an error.

Function: pochhammer (<n>, <x>)

     The Pochhammer symbol. For nonnegative integers <n> with `<n> <=
     pochhammer_max_index', the expression `pochhammer (<x>, <n>)'
     evaluates to the product `<x> (<x> + 1) (<x> + 2) ... (<x> + n -
     1)' when `<n> > 0' and to 1 when `<n> = 0'. For negative <n>,
     `pochhammer (<x>, <n>)' is defined as `(-1)^<n> / pochhammer (1 -
     <x>, -<n>)'.  Thus

          (%i1) pochhammer (x, 3);
          (%o1)                   x (x + 1) (x + 2)
          (%i2) pochhammer (x, -3);
                                           1
          (%o2)               - -----------------------
                                (1 - x) (2 - x) (3 - x)

     To convert a Pochhammer symbol into a quotient of gamma functions,
     (see Abramowitz and Stegun, equation 6.1.22) use `makegamma'; for
     example

          (%i1) makegamma (pochhammer (x, n));
                                    gamma(x + n)
          (%o1)                     ------------
                                      gamma(x)

     When <n> exceeds `pochhammer_max_index' or when <n> is symbolic,
     `pochhammer' returns a noun form.

          (%i1) pochhammer (x, n);
          (%o1)                         (x)
                                           n

Variable: pochhammer_max_index

     Default value: 100

     `pochhammer (<n>, <x>)' expands to a product if and only if `<n>
     <= pochhammer_max_index'.

     Examples:

          (%i1) pochhammer (x, 3), pochhammer_max_index : 3;
          (%o1)                   x (x + 1) (x + 2)
          (%i2) pochhammer (x, 4), pochhammer_max_index : 3;
          (%o2)                         (x)
                                           4

     Reference: Abramowitz and Stegun, equation 6.1.16, page 256.

Function: spherical_bessel_j (<n>, <x>)

     The spherical Bessel function of the first kind.

     Reference: Abramowitz and Stegun, equations 10.1.8, page 437 and
     10.1.15, page 439.

Function: spherical_bessel_y (<n>, <x>)

     The spherical Bessel function of the second kind.

     Reference: Abramowitz and Stegun, equations 10.1.9, page 437 and
     10.1.15, page 439.

Function: spherical_hankel1 (<n>, <x>)

     The spherical Hankel function of the first kind.

     Reference: Abramowitz and Stegun, equation 10.1.36, page 439.

Function: spherical_hankel2 (<n>, <x>)

     The spherical Hankel function of the second kind.

     Reference: Abramowitz and Stegun, equation 10.1.17, page 439.

Function: spherical_harmonic (<n>, <m>, <x>, <y>)

     The spherical harmonic function.

     Reference: Merzbacher 9.64.

Function: unit_step (<x>)

     The left-continuous unit step function; thus `unit_step (<x>)'
     vanishes for `<x> <= 0' and equals 1 for `<x> > 0'.

     If you want a unit step function that takes on the value 1/2 at
     zero, use `(1 + signum (<x>))/2'.

Function: ultraspherical (<n>, <a>, <x>)

     The ultraspherical polynomial (also known as the Gegenbauer
     polynomial).

     Reference: Abramowitz and Stegun, equation 22.5.46, page 779.

71 plotdf


Introduction to plotdf
Functions and Variables for plotdf

71.1 Introduction to plotdf

The function `plotdf' creates a plot of the direction field (also
called slope field) for a first-order Ordinary Differential Equation
(ODE) or a system of two autonomous first-order ODE's.

   Plotdf requires Xmaxima. It can be used from the console or any other
interface to Maxima, but the resulting file will be sent to Xmaxima for
plotting. Please make sure you have installed Xmaxima before trying to
use plotdf.

   To plot the direction field of a single ODE, the ODE must be written
in the form:
            dy
            -- = F(x,y)
            dx

   and the function <F> should be given as the argument for `plotdf'.
If the independent and dependent variables are not <x>, and <y>, as in
the equation above, then those two variables should be named explicitly
in a list given as an argument to the plotdf command (see the examples).

   To plot the direction field of a set of two autonomous ODE's, they
must be written in the form
            dx             dy
            -- = G(x,y)    -- = F(x,y)
            dt             dt

   and the argument for `plotdf' should be a list with the two
functions <G> and <F>, in that order; namely, the first expression in
the list will be taken to be the time derivative of the variable
represented on the horizontal axis, and the second expression will be
the time derivative of the variable represented on the vertical axis.
Those two variables do not have to be <x> and <y>, but if they are not,
then the second argument given to plotdf must be another list naming
the two variables, first the one on the horizontal axis and then the
one on the vertical axis.

   If only one ODE is given, `plotdf' will implicitly admit `x=t', and
`G(x,y)=1', transforming the non-autonomous equation into a system of
two autonomous equations.

71.2 Functions and Variables for plotdf

Function: plotdf (<dydx>, ...options...)
Function: plotdf (<dvdu>, `['<u>,<v>`]', ...options...)
Function: plotdf (`['<dxdt>,<dydt>`]', ...options...)
Function: plotdf (`['<dudt>,<dvdt>`]', `['<u>,<v>`]', ...options...)

     Displays a direction field in two dimensions <x> and <y>.

     <dydx>, <dxdt> and <dydt> are expressions that depend on <x> and
     <y>. <dvdu>, <dudt> and <dvdt> are expressions that depend on <u>
     and <v>. In addition to those two variables, the expressions can
     also depend on a set of parameters, with numerical values given
     with the `parameters' option (the option syntax is given below),
     or with a range of allowed values specified by a <sliders> option.

     Several other options can be given within the command, or selected
     in the menu. Integral curves can be obtained by clicking on the
     plot, or with the option `trajectory_at'. The direction of the
     integration can be controlled with the `direction' option, which
     can have values of _forward_, _backward_ or _both_. The number of
     integration steps is given by `nsteps' and the time interval
     between them is set up with the `tstep' option. The Adams Moulton
     method is used for the integration; it is also possible to switch
     to an adaptive Runge-Kutta 4th order method.

     Plot window menu:

     The menu in the plot window has the following options: _Zoom_, will
     change the behavior of the mouse so that it will allow you to zoom
     in on a region of the plot by clicking with the left button. Each
     click near a point magnifies the plot, keeping the center at the
     point where you clicked. Holding the <Shift> key while clicking,
     zooms out to the previous magnification. To resume computing
     trajectories when you click on a point, select _Integrate_ from
     the menu.

     The option _Config_ in the menu can be used to change the ODE(s) in
     use and various other settings. After configuration changes are
     made, the menu option _Replot_ should be selected, to activate the
     new settings.  If a pair of coordinates are entered in the field
     _Trajectory at_ in the _Config_ dialog menu, and the <enter> key
     is pressed, a new integral curve will be shown, in addition to the
     ones already shown. When _Replot_ is selected, only the last
     integral curve entered will be shown.

     Holding the right mouse button down while the cursor is moved, can
     be used to drag the plot sideways or up and down. Additional
     parameters such as the number of steps, the initial value of <t>
     and the x and y centers and radii, may be set in the Config menu.

     A copy of the plot can be saved as a postscript file, using the
     menu option _Save_.

     Plot options:

     The `plotdf' command may include several commands, each command is
     a list of two or more items.  The first item is the name of the
     option, and the remainder comprises the value or values assigned
     to the option.

     The options which are recognized by `plotdf' are the following:

        * "tstep" defines the length of the increments on the
          independent variable <t>, used to compute an integral curve.
          If only one expression <dydx> is given to `plotdf', the <x>
          variable will be directly proportional to <t>.  The default
          value is 0.1.

        * "nsteps" defines the number of steps of length `tstep' that
          will be used for the independent variable, to compute an
          integral curve.  The default value is 100.

        * "direction" defines the direction of the independent variable
          that will be followed to compute an integral curve. Possible
          values are `forward', to make the independent variable
          increase `nsteps' times, with increments `tstep', `backward',
          to make the independent variable decrease, or `both' that
          will lead to an integral curve that extends `nsteps' forward,
          and `nsteps' backward. The keywords `right' and `left' can be
          used as synonyms for `forward' and `backward'.  The default
          value is `both'.

        * "tinitial" defines the initial value of variable <t> used to
          compute integral curves. Since the differential equations are
          autonomous, that setting will only appear in the plot of the
          curves as functions of <t>.  The default value is 0.

        * "versus_t" is used to create a second plot window, with a
          plot of an integral curve, as two functions <x>, <y>, of the
          independent variable <t>. If `versus_t' is given any value
          different from 0, the second plot window will be displayed.
          The second plot window includes another menu, similar to the
          menu of the main plot window.  The default value is 0.

        * "trajectory_at" defines the coordinates <xinitial> and
          <yinitial> for the starting point of an integral curve.  The
          option is empty by default.

        * "parameters" defines a list of parameters, and their
          numerical values, used in the definition of the differential
          equations. The name and values of the parameters must be
          given in a string with a comma-separated sequence of pairs
          `name=value'.

        * "sliders" defines a list of parameters that will be changed
          interactively using slider buttons, and the range of
          variation of those parameters. The names and ranges of the
          parameters must be given in a string with a comma-separated
          sequence of elements `name=min:max'

        * "xfun" defines a string with semi-colon-separated sequence of
          functions of <x> to be displayed, on top of the direction
          field.  Those functions will be parsed by Tcl and not by
          Maxima.

        * "x" should be followed by two numbers, which will set up the
          minimum and maximum values shown on the horizontal axis. If
          the variable on the horizontal axis is not <x>, then this
          option should have the name of the variable on the horizontal
          axis.  The default horizontal range is from -10 to 10.

        * "y" should be followed by two numbers, which will set up the
          minimum and maximum values shown on the vertical axis. If the
          variable on the vertical axis is not <y>, then this option
          should have the name of the variable on the vertical axis.
          The default vertical range is from -10 to 10.

     *Examples:*

        * To show the direction field of the differential equation y' =
          exp(-x) + y and the solution that goes through (2, -0.1):
               (%i1) plotdf(exp(-x)+y,[trajectory_at,2,-0.1])$

        * To obtain the direction field for the equation diff(y,x) = x
          - y^2 and the solution with initial condition y(-1) = 3, we
          can use the command:
               (%i1) plotdf(x-y^2,[xfun,"sqrt(x);-sqrt(x)"],
                        [trajectory_at,-1,3], [direction,forward],
                        [y,-5,5], [x,-4,16])$

          The graph also shows the function y = sqrt(x).

        * The following example shows the direction field of a harmonic
          oscillator, defined by the two equations dz/dt = v and dv/dt
          = -k*z/m, and the integral curve through (z,v) = (6,0), with
          a slider that will allow you to change the value of m
          interactively (k is fixed at 2):
               (%i1) plotdf([v,-k*z/m], [z,v], [parameters,"m=2,k=2"],
                          [sliders,"m=1:5"], [trajectory_at,6,0])$

        * To plot the direction field of the Duffing equation,
          m*x"+c*x'+k*x+b*x^3 = 0, we introduce the variable y=x' and
          use:
               (%i1) plotdf([y,-(k*x + c*y + b*x^3)/m],
                            [parameters,"k=-1,m=1.0,c=0,b=1"],
                            [sliders,"k=-2:2,m=-1:1"],[tstep,0.1])$

        * The direction field for a damped pendulum, including the
          solution for the given initial conditions, with a slider that
          can be used to change the value of the mass m, and with a
          plot of the two state variables as a function of time:

               (%i1) plotdf([w,-g*sin(a)/l - b*w/m/l], [a,w],
                       [parameters,"g=9.8,l=0.5,m=0.3,b=0.05"],
                       [trajectory_at,1.05,-9],[tstep,0.01],
                       [a,-10,2], [w,-14,14], [direction,forward],
                       [nsteps,300], [sliders,"m=0.1:1"], [versus_t,1])$

72 romberg


Functions and Variables for romberg

72.1 Functions and Variables for romberg

Function: romberg (<expr>, <x>, <a>, <b>)
Function: romberg (<F>, <a>, <b>)

     Computes a numerical integration by Romberg's method.

     `romberg(<expr>, <x>, <a>, <b>)' returns an estimate of the
     integral `integrate(<expr>, <x>, <a>, <b>)'.  <expr> must be an
     expression which evaluates to a floating point value when <x> is
     bound to a floating point value.

     `romberg(<F>, <a>, <b>)' returns an estimate of the integral
     `integrate(<F>(x), x, <a>, <b>)' where `x' represents the unnamed,
     sole argument of <F>; the actual argument is not named `x'.  <F>
     must be a Maxima or Lisp function which returns a floating point
     value when the argument is a floating point value.  <F> may name a
     translated or compiled Maxima function.

     The accuracy of `romberg' is governed by the global variables
     `rombergabs' and `rombergtol'.  `romberg' terminates successfully
     when the absolute difference between successive approximations is
     less than `rombergabs', or the relative difference in successive
     approximations is less than `rombergtol'.  Thus when `rombergabs'
     is 0.0 (the default) only the relative error test has any effect
     on `romberg'.

     `romberg' halves the stepsize at most `rombergit' times before it
     gives up; the maximum number of function evaluations is therefore
     `2^rombergit'.  If the error criterion established by `rombergabs'
     and `rombergtol' is not satisfied, `romberg' prints an error
     message.  `romberg' always makes at least `rombergmin' iterations;
     this is a heuristic intended to prevent spurious termination when
     the integrand is oscillatory.

     `romberg' repeatedly evaluates the integrand after binding the
     variable of integration to a specific value (and not before).
     This evaluation policy makes it possible to nest calls to
     `romberg', to compute multidimensional integrals.  However, the
     error calculations do not take the errors of nested integrations
     into account, so errors may be underestimated.  Also, methods
     devised especially for multidimensional problems may yield the
     same accuracy with fewer function evaluations.

     `load(romberg)' loads this function.

     See also `QUADPACK', a collection of numerical integration
     functions.

     Examples:

     A 1-dimensional integration.

          (%i1) load (romberg);
          (%o1)    /usr/share/maxima/5.11.0/share/numeric/romberg.lisp
          (%i2) f(x) := 1/((x - 1)^2 + 1/100) + 1/((x - 2)^2 + 1/1000)
                        + 1/((x - 3)^2 + 1/200);
                              1                 1                1
          (%o2) f(x) := -------------- + --------------- + --------------
                               2    1           2    1            2    1
                        (x - 1)  + ---   (x - 2)  + ----   (x - 3)  + ---
                                   100              1000              200
          (%i3) rombergtol : 1e-6;
          (%o3)                 9.9999999999999995E-7
          (%i4) rombergit : 15;
          (%o4)                          15
          (%i5) estimate : romberg (f(x), x, -5, 5);
          (%o5)                   173.6730736617464
          (%i6) exact : integrate (f(x), x, -5, 5);
          (%o6) 10 sqrt(10) atan(70 sqrt(10))
           + 10 sqrt(10) atan(30 sqrt(10)) + 10 sqrt(2) atan(80 sqrt(2))
           + 10 sqrt(2) atan(20 sqrt(2)) + 10 atan(60) + 10 atan(40)
          (%i7) abs (estimate - exact) / exact, numer;
          (%o7)                7.5527060865060088E-11

     A 2-dimensional integration, implemented by nested calls to
     `romberg'.

          (%i1) load (romberg);
          (%o1)    /usr/share/maxima/5.11.0/share/numeric/romberg.lisp
          (%i2) g(x, y) := x*y / (x + y);
                                              x y
          (%o2)                   g(x, y) := -----
                                             x + y
          (%i3) rombergtol : 1e-6;
          (%o3)                 9.9999999999999995E-7
          (%i4) estimate : romberg (romberg (g(x, y), y, 0, x/2), x, 1, 3);
          (%o4)                   0.81930239628356
          (%i5) assume (x > 0);
          (%o5)                        [x > 0]
          (%i6) integrate (integrate (g(x, y), y, 0, x/2), x, 1, 3);
                                                    3
                                              2 log(-) - 1
                              9                     2        9
          (%o6)       - 9 log(-) + 9 log(3) + ------------ + -
                              2                    6         2
          (%i7) exact : radcan (%);
                              26 log(3) - 26 log(2) - 13
          (%o7)             - --------------------------
                                          3
          (%i8) abs (estimate - exact) / exact, numer;
          (%o8)                1.3711979871851024E-10

Option variable: rombergabs

     Default value: 0.0

     The accuracy of `romberg' is governed by the global variables
     `rombergabs' and `rombergtol'.  `romberg' terminates successfully
     when the absolute difference between successive approximations is
     less than `rombergabs', or the relative difference in successive
     approximations is less than `rombergtol'.  Thus when `rombergabs'
     is 0.0 (the default) only the relative error test has any effect
     on `romberg'.

     See also `rombergit' and `rombergmin'.

Option variable: rombergit

     Default value: 11

     `romberg' halves the stepsize at most `rombergit' times before it
     gives up; the maximum number of function evaluations is therefore
     `2^rombergit'.  `romberg' always makes at least `rombergmin'
     iterations; this is a heuristic intended to prevent spurious
     termination when the integrand is oscillatory.

     See also `rombergabs' and `rombergtol'.

Option variable: rombergmin

     Default value: 0

     `romberg' always makes at least `rombergmin' iterations; this is a
     heuristic intended to prevent spurious termination when the
     integrand is oscillatory.

     See also `rombergit', `rombergabs', and `rombergtol'.

Option variable: rombergtol

     Default value: 1e-4

     The accuracy of `romberg' is governed by the global variables
     `rombergabs' and `rombergtol'.  `romberg' terminates successfully
     when the absolute difference between successive approximations is
     less than `rombergabs', or the relative difference in successive
     approximations is less than `rombergtol'.  Thus when `rombergabs'
     is 0.0 (the default) only the relative error test has any effect
     on `romberg'.

     See also `rombergit' and `rombergmin'.

73 simplex


Introduction to simplex
Functions and Variables for simplex

73.1 Introduction to simplex

`simplex' is a package for linear optimization using the simplex
algorithm.

   Example:

     (%i1) load("simplex")$
     (%i2) minimize_lp(x+y, [3*x+2*y>2, x+4*y>3]);
                       9        7       1
     (%o2)            [--, [y = --, x = -]]
                       10       10      5

73.2 Functions and Variables for simplex

Option variable: epsilon_lp

     Default value: `10^-8'

     Epsilon used for numerical computations in `linear_program'.

     See also: `linear_program'.

Function: linear_program (<A>, <b>, <c>)

     `linear_program' is an implementation of the simplex algorithm.
     `linear_program(A, b, c)' computes a vector <x> for which `c.x' is
     minimum possible among vectors for which `A.x = b' and `x >= 0'.
     Argument <A> is a matrix and arguments <b> and <c> are lists.

     `linear_program' returns a list which contains the minimizing
     vector <x> and the minimum value `c.x'. If the problem is not
     bounded, it returns "Problem not bounded!" and if the problem is
     not feasible, it returns "Problem not feasible!".

     To use this function first load the `simplex' package with
     `load(simplex);'.

     Example:

          (%i2) A: matrix([1,1,-1,0], [2,-3,0,-1], [4,-5,0,0])$
          (%i3) b: [1,1,6]$
          (%i4) c: [1,-2,0,0]$
          (%i5) linear_program(A, b, c);
                             13     19        3
          (%o5)            [[--, 4, --, 0], - -]
                             2      2         2

     See also: `minimize_lp', `scale_lp', and `epsilon_lp'.

Function: maximize_lp (<obj>, <cond>, [<pos>])

     Maximizes linear objective function <obj> subject to some linear
     constraints <cond>. See `minimize_lp' for detailed description of
     arguments and return value.

     See also: `minimize_lp'.

Function: minimize_lp (<obj>, <cond>, [<pos>])

     Minimizes a linear objective function <obj> subject to some linear
     constraints <cond>. <cond> a list of linear equations or
     inequalities. In strict inequalities `>' is replaced by `>=' and
     `<' by `<='. The optional argument <pos> is a list of decision
     variables which are assumed to be positive.

     If the minimum exists, `minimize_lp' returns a list which contains
     the minimum value of the objective function and a list of decision
     variable values for which the minimum is attained. If the problem
     is not bounded, `minimize_lp' returns "Problem not bounded!" and
     if the problem is not feasible, it returns "Ploblem not feasible!".

     The decision variables are not assumed to be nonegative by
     default. If all decision variables are nonegative, set
     `nonegative_lp' to `true'.  If only some of decision variables are
     positive, list them in the optional argument <pos> (note that this
     is more efficient than adding constraints).

     `minimize_lp' uses the simplex algorithm which is implemented in
     maxima `linear_program' function.

     To use this function first load the `simplex' package with
     `load(simplex);'.

     Examples:

          (%i1) minimize_lp(x+y, [3*x+y=0, x+2*y>2]);
                                4       6        2
          (%o1)                [-, [y = -, x = - -]]
                                5       5        5
          (%i2) minimize_lp(x+y, [3*x+y>0, x+2*y>2]), nonegative_lp=true;
          (%o2)                [1, [y = 1, x = 0]]
          (%i3) minimize_lp(x+y, [3*x+y=0, x+2*y>2]), nonegative_lp=true;
          (%o3)                Problem not feasible!
          (%i4) minimize_lp(x+y, [3*x+y>0]);
          (%o4)                Problem not bounded!

     See also: `maximize_lp', `nonegative_lp', `epsilon_lp'.

Option variable: nonegative_lp

     Default value: `false'

     If `nonegative_lp' is true all decision variables to `minimize_lp'
     and `maximize_lp' are assumed to be positive.

     See also: `minimize_lp'.

74 simplification


Introduction to simplification
Package absimp
Package facexp
Package functs
Package ineq
Package rducon
Package scifac
Package sqdnst

74.1 Introduction to simplification

The directory `maxima/share/simplification' contains several scripts
which implement simplification rules and functions, and also some
functions not related to simplification.

74.2 Package absimp

The `absimp' package contains pattern-matching rules that extend the
built-in simplification rules for the `abs' and `signum' functions.
`absimp' respects relations established with the built-in `assume'
function and by declarations such as  `modedeclare (m, even, n, odd)'
for even or odd integers.

   `absimp' defines `unitramp' and `unitstep' functions in terms of
`abs' and `signum'.

   `load (absimp)' loads this package.  `demo (absimp)' shows a
demonstration of this package.

   Examples:

     (%i1) load (absimp)$
     (%i2) (abs (x))^2;
                                            2
     (%o2)                                 x
     (%i3) diff (abs (x), x);
                                           x
     (%o3)                               ------
                                         abs(x)
     (%i4) cosh (abs (x));
     (%o4)                               cosh(x)

74.3 Package facexp

The `facexp' package contains several related  functions that provide
the user with the ability to structure expressions by controlled
expansion.   This capability  is especially  useful when  the expression
contains variables that have physical meaning, because it is  often true
that the most economical form  of such an expression can be  obtained by
fully expanding the expression with respect to those variables, and then
factoring their coefficients.  While it is  true that this  procedure is
not difficult to carry out using standard Maxima  functions, additional
fine-tuning may also  be desirable, and  these finishing touches  can be
more  difficult to  apply.

   The  function `facsum'  and its  related forms provide a convenient
means for controlling the structure  of expressions in this way.
Another function, `collectterms', can be used to add  two or more
expressions that have already been simplified to this form, without
resimplifying the whole expression again.  This function may be useful
when the expressions are very large.

   `load (facexp)' loads this package.  `demo (facexp)' shows a
demonstration of this package.
Function: facsum (<expr>, <arg_1>, ..., <arg_n>)

     Returns  a form  of <expr>  which depends  on the arguments
     <arg_1>, ..., <arg_n>.  The arguments can be any form suitable for
     `ratvars', or they can be lists  of such  forms.  If  the
     arguments  are not  lists, then  the form returned is  fully
     expanded with respect  to the arguments,  and the coefficients of
     the arguments are factored.  These  coefficients are free of the
     arguments, except perhaps in a non-rational sense.

     If any of the arguments are  lists, then all such lists are
     combined into  a  single  list,   and  instead  of  calling
     `factor'   on  the coefficients  of  the  arguments,  `facsum'
     calls  itself   on  these coefficients, using  this newly
     constructed  single list as  the new argument list  for this
     recursive  call.  This  process can  be  repeated to arbitrary
     depth by nesting the desired elements in lists.

     It is possible that one may wish to `facsum' with respect  to more
     complicated subexpressions,  such as  `log (x + y)'.  Such
     arguments are also  permissible.

     Occasionally the user may wish to obtain any of the  above forms
     for expressions which are specified only by their leading
     operators.  For example, one may wish  to `facsum' with respect to
     all  `log''s.  In this situation, one may  include among the
     arguments either  the specific `log''s which are to be treated in
     this way, or  alternatively, either the expression  `operator
     (log)' or `'operator (log)'.   If one  wished to `facsum' the
     expression <expr> with respect to the operators <op_1>, ...,
     <op_n>, one   would  evaluate  `facsum (<expr>, operator (<op_1>,
     ..., <op_n>))'.  The `operator' form may also appear inside list
     arguments.

     In  addition,  the  setting  of  the  switches   `facsum_combine'
     and `nextlayerfactor' may affect the result of `facsum'.

Global variable: nextlayerfactor

     Default value: `false'

     When `nextlayerfactor' is `true', recursive calls  of `facsum' are
     applied  to  the  factors  of  the  factored  form   of  the
     coefficients of the arguments.

     When  `false', `facsum' is applied to each coefficient as a whole
     whenever recusive calls to  `facsum' occur.

     Inclusion   of   the  atom `nextlayerfactor' in  the argument
     list of `facsum'  has the  effect of `nextlayerfactor: true', but
     for the next level of the expression only.  Since
     `nextlayerfactor' is  always bound to  either `true' or  `false',
     it must be presented single-quoted whenever it appears in the
     argument list of `facsum'.

Global variable: facsum_combine

     Default value: `true'

     `facsum_combine' controls the form  of the final result  returned
     by `facsum'  when  its  argument  is  a  quotient  of
     polynomials.   If `facsum_combine' is `false'  then the form will
     be returned as  a fully expanded  sum  as described  above,  but
     if  `true',  then  the expression returned is a ratio of
     polynomials, with each polynomial in the form described above.

     The `true' setting of this switch is useful when one wants to
     `facsum' both  the numerator and  denominator of  a rational
     expression,  but  does not  want  the denominator  to  be
     multiplied through the terms of the numerator.

Function: factorfacsum (<expr>, <arg_1>, ... <arg_n>)

     Returns a  form of <expr>  which is obtained by calling  `facsum'
     on the factors  of <expr> with <arg_1>, ... <arg_n> as arguments.
     If any of the factors of <expr> is raised to a  power, both the
     factor and the exponent will be processed in this way.

Function: collectterms (<expr>, <arg_1>, ..., <arg_n>)

     If several expressions have been simplified  with the following
     functions `facsum', `factorfacsum', `factenexpand', `facexpten' or
     `factorfacexpten', and they are to be added together, it may be
     desirable to combine them using the function  `collecterms'.
     `collecterms' can take as arguments all of the arguments that can
     be given to these other associated functions with the exception of
     `nextlayerfactor', which has no effect on `collectterms'.  The
     advantage of `collectterms' is that it returns a form  similar to
     `facsum', but since it is adding forms that have already been
     processed by `facsum', it does not need to repeat that effort.
     This capability is especially useful when the expressions to be
     summed are very large.

74.4 Package functs

Function: rempart (<expr>, <n>)

     Removes part <n> from the expression <expr>.

     If <n> is a list of the form `[<l>, <m>]' then parts <l> thru <m>
     are removed.

     To use this function write first `load(functs)'.

Function: wronskian ([<f_1>, ..., <f_n>], <x>)

     Returns the Wronskian matrix of the list of expressions [<f_1>,
     ..., <f_n>] in the variable <x>.  The determinant of the Wronskian
     matrix is the Wronskian determinant of the list of expressions.

     To use `wronskian', first `load(functs)'. Example:

          (%i1) load(functs)$
          (%i2) wronskian([f(x), g(x)],x);
          (%o2) matrix([f(x),g(x)],['diff(f(x),x,1),'diff(g(x),x,1)])

Function: tracematrix (<M>)

     Returns the trace (sum of the diagonal elements) of matrix <M>.

     To use this function write first `load(functs)'.

Function: rational (`z')

     Multiplies numerator and denominator of <z> by the complex
     conjugate of denominator, thus rationalizing the denominator.
     Returns canonical rational expression (CRE) form if given one,
     else returns general form.

     To use this function write first `load(functs)'.

Function: logand (`x',`y')

     Returns logical (bit-wise) "and" of arguments x and y.

     To use this function write first `load(functs)'.

Function: logor (`x',`y')

     Returns logical (bit-wise) "or" of arguments x and y.

     To use this function write first `load(functs)'.

Function: logxor (`x',`y')

     Returns logical (bit-wise) exclusive-or of arguments x and y.

     To use this function write first `load(functs)'.

Function: nonzeroandfreeof (<x>, <expr>)

     Returns `true' if <expr> is nonzero and `freeof (<x>, <expr>)'
     returns `true'.  Returns `false' otherwise.

     To use this function write first `load(functs)'.

Function: linear (<expr>, <x>)

     When <expr> is an expression linear in variable <x>, `linear'
     returns `<a>*<x> + <b>' where <a> is nonzero, and <a> and <b> are
     free of <x>.  Otherwise, `linear' returns <expr>.

     To use this function write first `load(functs)'.

Function: gcdivide (<p>, <q>)

     When the option variable `takegcd' is `true' which is the default,
     `gcdivide' divides the polynomials <p> and <q> by their greatest
     common divisor and returns the ratio of the results.  `gcdivde'
     calls the function `ezgcd'  to divide the polynomials by the
     greatest common divisor.

     When `takegcd' is `false', `gcdivide' returns the ratio `<p>/<q>'.

     To use this function write first `load(functs)'.

     See also `ezgcd',  `gcd',  `gcdex',  and `poly_gcd'.

     Example:

          (%i1) load(functs)$

          (%i2) p1:6*x^3+19*x^2+19*x+6;
                                  3       2
          (%o2)                6 x  + 19 x  + 19 x + 6
          (%i3) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                            5       4       3       2
          (%o3)          6 x  + 13 x  + 12 x  + 13 x  + 6 x
          (%i4) gcdivide(p1, p2);
                                       x + 1
          (%o4)                        ------
                                        3
                                       x  + x
          (%i5) takegcd:false;
          (%o5)                         false
          (%i6) gcdivide(p1, p2);
                                 3       2
                              6 x  + 19 x  + 19 x + 6
          (%o6)          ----------------------------------
                            5       4       3       2
                         6 x  + 13 x  + 12 x  + 13 x  + 6 x
          (%i7) ratsimp(%);
                                       x + 1
          (%o7)                        ------
                                        3
                                       x  + x

Function: arithmetic (<a>, <d>, <n>)

     Returns the <n>-th term of the arithmetic series `<a>, <a> + <d>,
     <a> + 2*<d>, ..., <a> + (<n> - 1)*<d>'.

     To use this function write first `load(functs)'.

Function: geometric (<a>, <r>, <n>)

     Returns the <n>-th term of the geometric series `<a>, <a>*<r>,
     <a>*<r>^2, ..., <a>*<r>^(<n> - 1)'.

     To use this function write first `load(functs)'.

Function: harmonic (<a>, <b>, <c>, <n>)

     Returns the <n>-th term of the harmonic series `<a>/<b>, <a>/(<b>
     + <c>), <a>/(<b> + 2*<c>), ..., <a>/(<b> + (<n> - 1)*<c>)'.

     To use this function write first `load(functs)'.

Function: arithsum (<a>, <d>, <n>)

     Returns the sum of the arithmetic series from 1 to <n>.

     To use this function write first `load(functs)'.

Function: geosum (<a>, <r>, <n>)

     Returns the sum of the geometric series from 1 to <n>.  If <n> is
     infinity (`inf') then a sum is finite only if the absolute value
     of <r> is less than 1.

     To use this function write first `load(functs)'.

Function: gaussprob (<x>)

     Returns the Gaussian probability function `%e^(-<x>^2/2) /
     sqrt(2*%pi)'.

     To use this function write first `load(functs)'.

Function: gd (<x>)

     Returns the Gudermannian function `2*atan(%e^x)-%pi/2'.

     To use this function write first `load(functs)'.

Function: agd (<x>)

     Returns the inverse Gudermannian function `log (tan (%pi/4 +
     x/2)))'.

     To use this function write first `load(functs)'.

Function: vers (<x>)

     Returns the versed sine `1 - cos (x)'.

     To use this function write first `load(functs)'.

Function: covers (<x>)

     Returns the coversed sine `1 - sin (<x>)'.

     To use this function write first `load(functs)'.

Function: exsec (<x>)

     Returns the exsecant `sec (<x>) - 1'.

     To use this function write first `load(functs)'.

Function: hav (<x>)

     Returns the haversine `(1 - cos(x))/2'.

     To use this function write first `load(functs)'.

Function: combination (<n>, <r>)

     Returns the number of combinations of <n> objects taken <r> at a
     time.

     To use this function write first `load(functs)'.

Function: permutation (<n>, <r>)

     Returns the number of permutations of <r> objects selected from a
     set of <n> objects.

     To use this function write first `load(functs)'.

74.5 Package ineq

The `ineq' package contains simplification rules for inequalities.

   Example session:

     (%i1) load(ineq)$
     Warning: Putting rules on '+' or '*' is inefficient, and may not work.
     Warning: Putting rules on '+' or '*' is inefficient, and may not work.
     Warning: Putting rules on '+' or '*' is inefficient, and may not work.
     Warning: Putting rules on '+' or '*' is inefficient, and may not work.
     Warning: Putting rules on '+' or '*' is inefficient, and may not work.
     Warning: Putting rules on '+' or '*' is inefficient, and may not work.
     Warning: Putting rules on '+' or '*' is inefficient, and may not work.
     Warning: Putting rules on '+' or '*' is inefficient, and may not work.
     (%i2) a>=4;  /* a sample inequality */
     (%o2)                               a >= 4
     (%i3) (b>c)+%; /* add a second, strict inequality */
     (%o3)                            b + a > c + 4
     (%i4) 7*(x<y); /* multiply by a positive number */
     (%o4)                              7 x < 7 y
     (%i5) -2*(x>=3*z); /* multiply by a negative number */
     (%o5)                           - 2 x <= - 6 z
     (%i6) (1+a^2)*(1/(1+a^2)<=1); /* Maxima knows that 1+a^2 > 0 */
                                             2
     (%o6)                             1 <= a  + 1
     (%i7) assume(x>0)$ x*(2<3); /* assuming x>0 */
     (%o7)                              2 x < 3 x
     (%i8) a>=b; /* another inequality */
     (%o8)                               a >= b
     (%i9) 3+%; /* add something */
     (%o9)                           a + 3 >= b + 3
     (%i10) %-3; /* subtract it out */
     (%o10)                              a >= b
     (%i11) a>=c-b; /* yet another inequality */
     (%o11)                            a >= c - b
     (%i12) b+%; /* add b to both sides */
     (%o12)                            b + a >= c
     (%i13) %-c; /* subtract c from both sides */
     (%o13)                         - c + b + a >= 0
     (%i14) -%;  /* multiply by -1 */
     (%o14)                          c - b - a <= 0
     (%i15) (z-1)^2>-2*z; /* determining truth of assertion */
                                           2
     (%o15)                         (z - 1)  > - 2 z
     (%i16) expand(%)+2*z; /* expand this and add 2*z to both sides */
                                        2
     (%o16)                            z  + 1 > 0
     (%i17) %,pred;
     (%o17)                               true

   Be careful about using parentheses around the inequalities: when the
user types in `(A > B) + (C = 5)' the result is `A + C > B + 5', but `A
> B + C = 5' is a syntax error, and `(A > B + C) = 5' is something else
entirely.

   Do `disprule (all)' to see a complete listing of the rule
definitions.

   The user will be queried if Maxima is unable to decide the sign of a
quantity multiplying an inequality.

   The most common mis-feature is illustrated by:

     (%i1) eq: a > b;
     (%o1)                              a > b
     (%i2) 2*eq;
     (%o2)                            2 (a > b)
     (%i3) % - eq;
     (%o3)                              a > b

   Another problem is 0 times an inequality; the default to have this
turn into 0 has been left alone. However, if you type
`X*<some_inequality>' and Maxima asks about the sign of `X' and you
respond `zero' (or `z'), the program returns `X*<some_inequality>' and
not use the information that `X' is 0. You should do `ev (%, x: 0)' in
such a case, as the database will only be used for comparison purposes
in decisions, and not for the purpose of evaluating `X'.

   The user may note a slower response when this package is loaded, as
the simplifier is forced to examine more rules than without the
package, so you might wish to remove the rules after making use of
them. Do `kill (rules)' to eliminate all of the rules (including any
that you might have defined); or you may be more selective by killing
only some of them; or use `remrule' on a specific rule.

   Note that if you load this package after defining your own rules you
will clobber your rules that have the same name. The rules in this
package are: `*rule1', ..., `*rule8', `+rule1', ..., `+rule18', and you
must enclose the rulename in quotes to refer to it, as in `remrule
("+", "+rule1")' to specifically remove the first rule on `"+"' or
`disprule ("*rule2")' to display the definition of the second
multiplicative rule.

74.6 Package rducon

Function: reduce_consts (<expr>)

     Replaces constant subexpressions of <expr> with constructed
     constant atoms, saving the definition of all these constructed
     constants in the list of equations `const_eqns', and returning the
     modified <expr>.  Those parts of <expr> are constant which return
     `true' when operated on by the function `constantp'.  Hence,
     before invoking `reduce_consts', one should do

          declare ([<objects to be given the constant property>], constant)$

     to set up a database of the constant quantities occurring in your
     expressions.

     If you are planning to generate Fortran output after these symbolic
     calculations, one of the first code sections should be the
     calculation of all constants.  To generate this code segment, do

          map ('fortran, const_eqns)$

     Variables besides `const_eqns' which affect `reduce_consts' are:

     `const_prefix' (default value: `xx') is the string of characters
     used to prefix all symbols generated by `reduce_consts' to
     represent constant subexpressions.

     `const_counter' (default value: 1) is the integer index used to
     generate unique symbols to represent each constant subexpression
     found by `reduce_consts'.

     `load (rducon)' loads this function.  `demo (rducon)' shows a
     demonstration of this function.

74.7 Package scifac

Function: gcfac (<expr>)

     `gcfac' is a factoring function that attempts to apply the same
     heuristics which scientists apply in trying to make expressions
     simpler.  `gcfac' is limited to monomial-type factoring.  For a
     sum, `gcfac' does the following:

       1. Factors over the integers.

       2. Factors out the largest powers of terms occurring as
          coefficients, regardless of the complexity of the terms.

       3. Uses (1) and (2) in factoring adjacent pairs of terms.

       4. Repeatedly and recursively applies these techniques until the
          expression no longer changes.

     Item (3) does not necessarily do an optimal job of pairwise
     factoring because of the combinatorially-difficult nature of
     finding which of all possible rearrangements of the pairs yields
     the most compact pair-factored result.

     `load (scifac)' loads this function.  `demo (scifac)' shows a
     demonstration of this function.

74.8 Package sqdnst

Function: sqrtdenest (<expr>)

     Denests `sqrt' of simple, numerical, binomial surds, where
     possible.  E.g.

          (%i1) load (sqdnst)$
          (%i2) sqrt(sqrt(3)/2+1)/sqrt(11*sqrt(2)-12);
                                              sqrt(3)
                                         sqrt(------- + 1)
                                                 2
          (%o2)                        ---------------------
                                       sqrt(11 sqrt(2) - 12)
          (%i3) sqrtdenest(%);
                                            sqrt(3)   1
                                            ------- + -
                                               2      2
          (%o3)                            -------------
                                              1/4    3/4
                                           3 2    - 2

     Sometimes it helps to apply `sqrtdenest' more than once, on such as
     `(19601-13860 sqrt(2))^(7/4)'.

     `load (sqdnst)' loads this function.

75 solve_rec


Introduction to solve_rec
Functions and Variables for solve_rec

75.1 Introduction to solve_rec

`solve_rec' is a package for solving linear recurrences with polynomial
coefficients.

   A demo is available with `demo(solve_rec);'.

   Example:

     (%i1) load("solve_rec")$
     (%i2) solve_rec((n+4)*s[n+2] + s[n+1] - (n+1)*s[n], s[n]);
                                         n
                      %k  (2 n + 3) (- 1)          %k
                        1                            2
     (%o2)       s  = -------------------- + ---------------
                  n     (n + 1) (n + 2)      (n + 1) (n + 2)

75.2 Functions and Variables for solve_rec

Function: reduce_order (<rec>, <sol>, <var>)

     Reduces the order of linear recurrence <rec> when a particular
     solution <sol> is known. The reduced reccurence can be used to get
     other solutions.

     Example:

          (%i3) rec: x[n+2] = x[n+1] + x[n]/n;
                                                x
                                                 n
          (%o3)               x      = x      + --
                               n + 2    n + 1   n
          (%i4) solve_rec(rec, x[n]);
          WARNING: found some hypergeometrical solutions!
          (%o4)                    x  = %k  n
                                    n     1
          (%i5) reduce_order(rec, n, x[n]);
          (%t5)                    x  = n %z
                                    n       n

                                     n - 1
                                     ====
                                     \
          (%t6)                %z  =  >     %u
                                 n   /        %j
                                     ====
                                     %j = 0

          (%o6)             (- n - 2) %u     - %u
                                        n + 1     n
          (%i6) solve_rec((n+2)*%u[n+1] + %u[n], %u[n]);
                                               n
                                      %k  (- 1)
                                        1
          (%o6)                 %u  = ----------
                                  n    (n + 1)!

          So the general solution is

                       n - 1
                       ====        j
                       \      (- 1)
                 %k  n  >    -------- + %k  n
                   2   /     (j + 1)!     1
                       ====
                       j = 0

Option variable: simplify_products

     Default value: `true'

     If `simplify_products' is `true', `solve_rec' will try to simplify
     products in result.

     See also: `solve_rec'.

Function: simplify_sum (<expr>)

     Tries to simplify all sums appearing in <expr> to a closed form.

     To use this function first load the `simplify_sum' package with
     `load(simplify_sum)'.

     Example:

          (%i1) load("simplify_sum")$
          (%i2) sum(binomial(n+k,k)/2^k,k,1,n)+sum(binomial(2*n,2*k),k,1,n);
                  n                          n
                 ====                       ====
                 \     binomial(n + k, k)   \
          (%o2)   >    ------------------ +  >    binomial(2 n, 2 k)
                 /              k           /
                 ====          2            ====
                 k = 1                      k = 1
          (%i3) simplify_sum(%);

                                   2 n - 1    n
          (%o3)                   2        + 2  - 2

Function: solve_rec (<eqn>, <var>, [<init>])

     Solves for hypergeometrical solutions to linear recurrence <eqn>
     with polynomials coefficient in variable <var>. Optional arguments
     <init> are initial conditions.

     `solve_rec' can solve linear recurrences with constant
     coefficients, finds hypergeometrical solutions to homogeneous
     linear recurrences with polynomial coefficients, rational
     solutions to linear recurrences with polynomial coefficients and
     can solve Ricatti type recurrences.

     Note that the running time of the algorithm used to find
     hypergeometrical solutions is exponential in the degree of the
     leading and trailing coefficient.

     To use this function first load the `solve_rec' package with
     `load(solve_rec);'.

     Example of linear recurrence with constant coefficients:

          (%i2) solve_rec(a[n]=a[n-1]+a[n-2]+n/2^n, a[n]);
                                  n          n
                     (sqrt(5) - 1)  %k  (- 1)
                                      1           n
          (%o2) a  = ------------------------- - ----
                 n               n                  n
                                2                5 2
                                                          n
                                             (sqrt(5) + 1)  %k
                                                              2    2
                                           + ------------------ - ----
                                                      n              n
                                                     2            5 2

     Example of linear recurrence with polynomial coefficients:

          (%i7) 2*x*(x+1)*y[x] - (x^2+3*x-2)*y[x+1] + (x-1)*y[x+2];
                                   2
          (%o7) (x - 1) y      - (x  + 3 x - 2) y      + 2 x (x + 1) y
                         x + 2                   x + 1                x
          (%i8) solve_rec(%, y[x], y[1]=1, y[3]=3);
                                        x
                                     3 2    x!
          (%o9)                 y  = ---- - --
                                 x    4     2

     Example of Ricatti type recurrence:

          (%i2) x*y[x+1]*y[x] - y[x+1]/(x+2) + y[x]/(x-1) = 0;
                                      y         y
                                       x + 1     x
          (%o2)         x y  y      - ------ + ----- = 0
                           x  x + 1   x + 2    x - 1
          (%i3) solve_rec(%, y[x], y[3]=5)$
          (%i4) ratsimp(minfactorial(factcomb(%)));
                                             3
                                         30 x  - 30 x
          (%o4) y  = - -------------------------------------------------
                 x        6      5       4       3       2
                       5 x  - 3 x  - 25 x  + 15 x  + 20 x  - 12 x - 1584

     See also: `solve_rec_rat', `simplify_products', and
     `product_use_gamma'.

Function: solve_rec_rat (<eqn>, <var>, [<init>])

     Solves for rational solutions to linear recurrences. See solve_rec
     for description of arguments.

     To use this function first load the `solve_rec' package with
     `load(solve_rec);'.

     Example:

          (%i1) (x+4)*a[x+3] + (x+3)*a[x+2] - x*a[x+1] + (x^2-1)*a[x];
          (%o1)  (x + 4) a      + (x + 3) a      - x a
                          x + 3            x + 2      x + 1
                                                             2
                                                         + (x  - 1) a
                                                                      x
          (%i2) solve_rec_rat(% = (x+2)/(x+1), a[x]);
                                 1
          (%o2)      a  = ---------------
                      x   (x - 1) (x + 1)

     See also: `solve_rec'.

Option variable: product_use_gamma

     Default value: `true'

     When simplifying products, `solve_rec' introduces gamma function
     into the expression if `product_use_gamma' is `true'.

     See also: `simplify_products', `solve_rec'.

Function: summand_to_rec (<summand>, <k>, <n>)
Function: summand_to_rec (<summand>, [<k>, <lo>, <hi>], <n>)

     Returns the recurrence sattisfied by the sum

               hi
              ====
              \
               >     summand
              /
              ====
            k = lo

     where summand is hypergeometrical in <k> and <n>. If <lo> and <hi>
     are omited, they are assumed to be `lo = -inf' and `hi = inf'.

     To use this function first load the `simplify_sum' package with
     `load(simplify_sum)'.

     Example:

          (%i1) load("simplify_sum")$
          (%i2) summand: binom(n,k);
          (%o2)                           binomial(n, k)
          (%i3) summand_to_rec(summand,k,n);
          (%o3)                      2 sm  - sm      = 0
                                         n     n + 1
          (%i7) summand: binom(n, k)/(k+1);
                                          binomial(n, k)
          (%o7)                           --------------
                                              k + 1
          (%i8) summand_to_rec(summand, [k, 0, n], n);
          (%o8)               2 (n + 1) sm  - (n + 2) sm      = - 1
                                          n             n + 1

76 stats


Introduction to stats
Functions and Variables for inference_result
Functions and Variables for stats
Functions and Variables for special distributions

76.1 Introduction to stats

Package `stats' contains a set of classical statistical inference and
hypothesis testing procedures.

   All these functions return an `inference_result' Maxima object which
contains the necessary results for population inferences and decision
making.

   Global variable `stats_numer' controls whether results are given in
floating point or symbolic and rational format; its default value is
`true' and results are returned in floating point format.

   Package `descriptive' contains some utilities to manipulate data
structures (lists and matrices); for example, to extract subsamples. It
also contains some examples on how to use package `numericalio' to read
data from plain text files. See `descriptive' and `numericalio' for
more details.

   Package `stats' loads packages `descriptive', `distrib' and
`inference_result'.

   For comments, bugs or suggestions, please contact the author at

   <'mario AT edu DOT xunta DOT es'>.

76.2 Functions and Variables for inference_result

Function: inference_result (<title>, <values>, <numbers>)

     Constructs an `inference_result' object of the type returned by the
     stats functions. Argument <title> is a string with the name of the
     procedure; <values> is a list with elements of the form `symbol =
     value' and <numbers> is a list with positive integer numbers
     ranging from one to `length(<values>)', indicating which values
     will be shown by default.

     Example:

     This is a simple example showing results concerning a rectangle.
     The title of this object is the string `"Rectangle"', it stores
     five results, named `'base', `'height', `'diagonal', `'area', and
     `'perimeter', but only the first, second, fifth, and fourth will
     be displayed. The `'diagonal' is stored in this object, but it is
     not displayed; to access its value, make use of function
     `take_inference'.

          (%i1) load(inference_result)$
          (%i2) b: 3$ h: 2$
          (%i3) inference_result("Rectangle",
                                  ['base=b,
                                   'height=h,
                                   'diagonal=sqrt(b^2+h^2),
                                   'area=b*h,
                                   'perimeter=2*(b+h)],
                                  [1,2,5,4] );
                                  |   Rectangle
                                  |
                                  |    base = 3
                                  |
          (%o3)                   |   height = 2
                                  |
                                  | perimeter = 10
                                  |
                                  |    area = 6
          (%i4) take_inference('diagonal,%);
          (%o4)                        sqrt(13)

     See also `take_inference'.

Function: inferencep (<obj>)

     Returns `true' or `false', depending on whether <obj> is an
     `inference_result' object or not.

Function: items_inference (<obj>)

     Returns a list with the names of the items stored in <obj>, which
     must be an `inference_result' object.

     Example:

     The `inference_result' object stores two values, named `'pi' and
     `'e', but only the second is displayed. The `items_inference'
     function returns the names of all items, no matter they are
     displayed or not.

          (%i1) load(inference_result)$
          (%i2) inference_result("Hi", ['pi=%pi,'e=%e],[2]);
                                      |   Hi
          (%o2)                       |
                                      | e = %e
          (%i3) items_inference(%);
          (%o3)                        [pi, e]

Function: take_inference (<n>, <obj>)
Function: take_inference (<name>, <obj>)
Function: take_inference (<list>, <obj>)

     Returns the <n>-th value stored in <obj> if <n> is a positive
     integer, or the item named <name> if this is the name of an item.
     If the first argument is a list of numbers and/or symbols,
     function `take_inference' returns a list with the corresponding
     results.

     Example:

     Given an `inference_result' object, function `take_inference' is
     called in order to extract some information stored in it.

          (%i1) load(inference_result)$
          (%i2) b: 3$ h: 2$
          (%i3) sol: inference_result("Rectangle",
                                      ['base=b,
                                       'height=h,
                                       'diagonal=sqrt(b^2+h^2),
                                       'area=b*h,
                                       'perimeter=2*(b+h)],
                                      [1,2,5,4] );
                                  |   Rectangle
                                  |
                                  |    base = 3
                                  |
          (%o3)                   |   height = 2
                                  |
                                  | perimeter = 10
                                  |
                                  |    area = 6
          (%i4) take_inference('base,sol);
          (%o4)                           3
          (%i5) take_inference(5,sol);
          (%o5)                          10
          (%i6) take_inference([1,'diagonal],sol);
          (%o6)                     [3, sqrt(13)]
          (%i7) take_inference(items_inference(sol),sol);
          (%o7)                [3, 2, sqrt(13), 6, 10]

     See also `inference_result' and `take_inference'.

76.3 Functions and Variables for stats

Option variable: stats_numer

     Default value: `true'

     If `stats_numer' is `true', inference statistical functions return
     their results in floating point numbers. If it is `false', results
     are given in symbolic and rational format.

Function: test_mean (<x>)
Function: test_mean (<x>, <options> ...)

     This is the mean <t>-test. Argument <x> is a list or a column
     matrix containing a one dimensional sample. It also performs an
     asymptotic test based on the Central Limit Theorem if option
     `'asymptotic' is `true'.

     Options:

        * `'mean', default `0', is the mean value to be checked.

        * `'alternative', default `'twosided', is the alternative
          hypothesis; valid values are: `'twosided', `'greater' and
          `'less'.

        * `'dev', default `'unknown', this is the value of the standard
          deviation when it is known; valid values are: `'unknown' or a
          positive expression.

        * `'conflevel', default `95/100', confidence level for the
          confidence interval; it must be an expression which takes a
          value in (0,1).

        * `'asymptotic', default `false', indicates whether it performs
          an exact <t>-test or an asymptotic one based on the Central
          Limit Theorem; valid values are `true' and `false'.

Function: test_means_difference (<x1>, <x2>)
Function: test_means_difference (<x1>, <x2>, <options> ...)

     This is the difference of means <t>-test for two samples.
     Arguments <x1> and <x2> are lists or column matrices containing
     two independent samples. In case of different unknown variances
     (see options `'dev1', `'dev2' and `'varequal' bellow), the degrees
     of freedom are computed by means of the Welch approximation.  It
     also performs an asymptotic test based on the Central Limit
     Theorem if option `'asymptotic' is set to `true'.

     Options:

        *        * `'alternative', default `'twosided', is the
          alternative hypothesis; valid values are: `'twosided',
          `'greater' and `'less'.

        * `'dev1', default `'unknown', this is the value of the
          standard deviation of the <x1> sample when it is known; valid
          values are: `'unknown' or a positive expression.

        * `'dev2', default `'unknown', this is the value of the
          standard deviation of the <x2> sample when it is known; valid
          values are: `'unknown' or a positive expression.

        * `'varequal', default `false', whether variances should be
          considered to be equal or not; this option takes effect only
          when `'dev1' and/or `'dev2' are  `'unknown'.

        * `'conflevel', default `95/100', confidence level for the
          confidence interval; it must be an expression which takes a
          value in (0,1).

        * `'asymptotic', default `false', indicates whether it performs
          an exact <t>-test or an asymptotic one based on the Central
          Limit Theorem; valid values are `true' and `false'.

Function: test_variance (<x>)
Function: test_variance (<x>, <options>, ...)

     This is the variance <chi^2>-test. Argument <x> is a list or a
     column matrix containing a one dimensional sample taken from a
     normal population.

     Options:

        * `'mean', default `'unknown', is the population's mean, when
          it is known.

        * `'alternative', default `'twosided', is the alternative
          hypothesis; valid values are: `'twosided', `'greater' and
          `'less'.

        * `'variance', default `1', this is the variance value
          (positive) to be checked.

        * `'conflevel', default `95/100', confidence level for the
          confidence interval; it must be an expression which takes a
          value in (0,1).

Function: test_variance_ratio (<x1>, <x2>)
Function: test_variance_ratio (<x1>, <x2>, <options> ...)

     This is the variance ratio <F>-test for two normal populations.
     Arguments <x1> and <x2> are lists or column matrices containing
     two independent samples.

     Options:

        * `'alternative', default `'twosided', is the alternative
          hypothesis; valid values are: `'twosided', `'greater' and
          `'less'.

        * `'mean1', default `'unknown', when it is known, this is the
          mean of the population from which <x1> was taken.

        * `'mean2', default `'unknown', when it is known, this is the
          mean of the population from which <x2> was taken.

        * `'conflevel', default `95/100', confidence level for the
          confidence interval of the ratio; it must be an expression
          which takes a value in (0,1).

Function: test_proportion (<x>, <n>)
Function: test_proportion (<x>, <n>, <options> ...)

     Inferences on a proportion. Argument <x> is the number of successes
     in <n> trials in a Bernoulli experiment with unknown probability.

     Options:

        * `'proportion', default `1/2', is the value of the proportion
          to be checked.

        * `'alternative', default `'twosided', is the alternative
          hypothesis; valid values are: `'twosided', `'greater' and
          `'less'.

        * `'conflevel', default `95/100', confidence level for the
          confidence interval; it must be an expression which takes a
          value in (0,1).

        * `'asymptotic', default `false', indicates whether it performs
          an exact test based on the binomial distribution, or an
          asymptotic one based on the Central Limit Theorem; valid
          values are `true' and `false'.

        * `'correct', default `true', indicates whether Yates
          correction is applied or not.

Function: test_proportions_difference (<x1>, <n1>, <x2>, <n2>)
Function: test_proportions_difference (<x1>, <n1>, <x2>, <n2>, <options> ...)

     Inferences on the difference of two proportions. Argument <x1> is
     the number of successes in <n1> trials in a Bernoulli experiment
     in the first population, and <x2> and <n2> are the corresponding
     values in the second population. Samples are independent and the
     test is asymptotic.

     Options:

        * `'alternative', default `'twosided', is the alternative
          hypothesis; valid values are: `'twosided' (`p1 # p2'),
          `'greater' (`p1 > p2') and `'less' (`p1 < p2').

        * `'conflevel', default `95/100', confidence level for the
          confidence interval; it must be an expression which takes a
          value in (0,1).

        * `'correct', default `true', indicates whether Yates
          correction is applied or not.

Function: test_sign (<x>)
Function: test_sign (<x>, <options> ...)

     This is the non parametric sign test for the median of a
     continuous population.  Argument <x> is a list or a column matrix
     containing a one dimensional sample.

     Options:

        * `'alternative', default `'twosided', is the alternative
          hypothesis; valid values are: `'twosided', `'greater' and
          `'less'.

        * `'median', default `0', is the median value to be checked.

Function: test_signed_rank (<x>)
Function: test_signed_rank (<x>, <options> ...)

     This is the Wilcoxon signed rank test to make inferences about the
     median of a continuous population. Argument <x> is a list or a
     column matrix containing a one dimensional sample. Performs normal
     approximation if the sample size is greater than 20, or if there
     are zeroes or ties.

     See also `pdf_rank_test' and `cdf_rank_test'.

     Options:

        * `'median', default `0', is the median value to be checked.

        * `'alternative', default `'twosided', is the alternative
          hypothesis; valid values are: `'twosided', `'greater' and
          `'less'.

Function: test_rank_sum (<x1>, <x2>)
Function: test_rank_sum (<x1>, <x2>, <option>)

     This is the Wilcoxon-Mann-Whitney test for comparing the medians
     of two continuous populations. The first two arguments <x1> and
     <x2> are lists or column matrices with the data of two independent
     samples. Performs normal approximation if any of the sample sizes
     is greater than 10, or if there are ties.

     Option:

        * `'alternative', default `'twosided', is the alternative
          hypothesis; valid values are: `'twosided', `'greater' and
          `'less'.

Function: test_normality (<x>)

     Shapiro-Wilk test for normality. Argument <x> is a list of
     numbers, and sample size must be greater than 2 and less or equal
     than 5000, otherwise, function `test_normality' signals an error
     message.

     Reference:

     [1] Algorithm AS R94, Applied Statistics (1995), vol.44, no.4,
     547-551

     The output of function `test_normality' is an `inference_result'
     Maxima object with the following results:

       1. `'statistic': value of the <W> statistic.

       2. `'p_value': p-value under normal assumption.

Function: simple_linear_regression (<x>)
Function: simple_linear_regression (<x> <option>)

     Simple linear regression, y_i=a+b x_i+e_i, where e_i are N(0,sigma)
     independent random variables. Argument <x> must be a two column
     matrix or a list of pairs.

     Options:

        * `'conflevel', default `95/100', confidence level for the
          confidence interval; it must be an expression which takes a
          value in (0,1).

        * `'regressor', default `'x', name of the independent variable.

76.4 Functions and Variables for special distributions

Function: pdf_signed_rank (<x>, <n>)

     Probability density function of the exact distribution of the
     signed rank statistic. Argument <x> is a real number and <n> a
     positive integer.

     See also `test_signed_rank'.

Function: cdf_signed_rank (<x>, <n>)

     Cumulative density function of the exact distribution of the
     signed rank statistic. Argument <x> is a real number and <n> a
     positive integer.

     See also `test_signed_rank'.

Function: pdf_rank_sum (<x>, <n>, <m>)

     Probability density function of the exact distribution of the rank
     sum statistic. Argument <x> is a real number and <n> and <m> are
     both positive integers.

     See also `test_rank_sum'.

Function: cdf_rank_sum (<x>, <n>, <m>)

     Cumulative density function of the exact distribution of the rank
     sum statistic. Argument <x> is a real number and <n> and <m> are
     both positive integers.

     See also `test_rank_sum'.

77 stirling

Functions and Variables for stirling

77.1 Functions and Variables for stirling

Function: stirling (<z>,<n>)
Function: stirling (<z>,<n>,<pred>)

     Replace `gamma(x)' with the O(1/x^(2n-1)) Stirling formula. when
     <n> isn't a nonnegative integer, signal an error. With the
     optional third argument `pred', the Stirling formula is applied
     only when `pred' is true.

     Reference: Abramowitz & Stegun, " Handbook of mathematical
     functions", 6.1.40.

     Examples:
          (%i1) load (stirling)$

          (%i2) stirling(gamma(%alpha+x)/gamma(x),1);
                 1/2 - x             x + %alpha - 1/2
          (%o2) x        (x + %alpha)
                                             1           1
                                      --------------- - ---- - %alpha
                                      12 (x + %alpha)   12 x
                                    %e
          (%i3) taylor(%,x,inf,1);
                              %alpha       2    %alpha
                    %alpha   x       %alpha  - x       %alpha
          (%o3)/T/ x       + -------------------------------- + . . .
                                           2 x
          (%i4) map('factor,%);
                                                 %alpha - 1
                   %alpha   (%alpha - 1) %alpha x
          (%o4)   x       + -------------------------------
                                            2

     The function `stirling' knows the difference between the variable
     'gamma' and the function gamma:

          (%i5) stirling(gamma + gamma(x),0);
                                              x - 1/2   - x
          (%o5)    gamma + sqrt(2) sqrt(%pi) x        %e
          (%i6) stirling(gamma(y) + gamma(x),0);
                                   y - 1/2   - y
          (%o6) sqrt(2) sqrt(%pi) y        %e
                                                        x - 1/2   - x
                                   + sqrt(2) sqrt(%pi) x        %e

     To apply the Stirling formula only to terms that involve the
     variable `k', use an optional third argument; for example
          (%i7) makegamma(pochhammer(a,k)/pochhammer(b,k));
          (%o7) (gamma(b)*gamma(k+a))/(gamma(a)*gamma(k+b))
          (%i8) stirling(%,1, lambda([s], not(freeof(k,s))));
          (%o8) (%e^(b-a)*gamma(b)*(k+a)^(k+a-1/2)*(k+b)^(-k-b+1/2))/gamma(a)
     The terms `gamma(a)' and `gamma(b)' are free of `k', so the
     Stirling formula was not applied to these two terms.

     To use this function write first `load("stirling")'.

78 stringproc


Introduction to string processing
Functions and Variables for input and output
Functions and Variables for characters
Functions and Variables for strings

78.1 Introduction to string processing

`stringproc.lisp' enlarges Maximas capabilities of working with strings
and adds some useful functions for file in/output.

   For questions and bugs please mail to van.nek at arcor.de .

   In Maxima a string is easily constructed by typing "text".
`stringp' tests for strings.

     (%i1) m: "text";
     (%o1)                         text
     (%i2) stringp(m);
     (%o2)                         true

   Characters are represented as strings of length 1.  These are not
Lisp characters.  Tests can be done with `charp' (respectively `lcharp'
and conversion from Lisp to Maxima characters with `cunlisp').

     (%i1) c: "e";
     (%o1)                           e
     (%i2) [charp(c),lcharp(c)];
     (%o2)                     [true, false]
     (%i3) supcase(c);
     (%o3)                           E
     (%i4) charp(%);
     (%o4)                         true

   All functions in `stringproc.lisp' that return characters, return
Maxima-characters.  Due to the fact, that the introduced characters are
strings of length 1, you can use a lot of string functions also for
characters.  As seen, `supcase' is one example.

   It is important to know, that the first character in a Maxima-string
is at position 1.  This is designed due to the fact that the first
element in a Maxima-list is at position 1 too.  See definitions of
`charat' and `charlist' for examples.

   In applications string-functions are often used when working with
files.  You will find some useful stream- and print-functions in
`stringproc.lisp'.  The following example shows some of the here
introduced functions at work.

   Example:

   `openw' returns an output stream to a file, `printf' then allows
formatted writing to this file. See `printf' for details.

     (%i1) s: openw("E:/file.txt");
     (%o1)                    #<output stream E:/file.txt>
     (%i2) for n:0 thru 10 do printf( s, "~d ", fib(n) );
     (%o2)                                done
     (%i3) printf( s, "~%~d ~f ~a ~a ~f ~e ~a~%",
                   42,1.234,sqrt(2),%pi,1.0e-2,1.0e-2,1.0b-2 );
     (%o3)                                false
     (%i4) close(s);
     (%o4)                                true

   After closing the stream you can open it again, this time with input
direction.  `readline' returns the entire line as one string. The
`stringproc' package now offers a lot of functions for manipulating
strings. Tokenizing can be done by `split' or `tokens'.

     (%i5) s: openr("E:/file.txt");
     (%o5)                     #<input stream E:/file.txt>
     (%i6) readline(s);
     (%o6)                     0 1 1 2 3 5 8 13 21 34 55
     (%i7) line: readline(s);
     (%o7)               42 1.234 sqrt(2) %pi 0.01 1.0E-2 1.0b-2
     (%i8) list: tokens(line);
     (%o8)           [42, 1.234, sqrt(2), %pi, 0.01, 1.0E-2, 1.0b-2]
     (%i9) map( parse_string, list );
     (%o9)            [42, 1.234, sqrt(2), %pi, 0.01, 0.01, 1.0b-2]
     (%i10) float(%);
     (%o10) [42.0, 1.234, 1.414213562373095, 3.141592653589793, 0.01,
                                                          0.01, 0.01]
     (%i11) readline(s);
     (%o11)                               false
     (%i12) close(s)$

   `readline' returns `false' when the end of file occurs.

78.2 Functions and Variables for input and output

Example:

     (%i1) s: openw("E:/file.txt");
     (%o1)                     #<output stream E:/file.txt>
     (%i2) control:
     "~2tAn atom: ~20t~a~%~2tand a list: ~20t~{~r ~}~%~2t\
     and an integer: ~20t~d~%"$
     (%i3) printf( s,control, 'true,[1,2,3],42 )$
     (%o3)                                false
     (%i4) close(s);
     (%o4)                                true
     (%i5) s: openr("E:/file.txt");
     (%o5)                     #<input stream E:/file.txt>
     (%i6) while stringp( tmp:readline(s) ) do print(tmp)$
       An atom:          true
       and a list:       one two three
       and an integer:   42
     (%i7) close(s)$
Function: close (<stream>)

     Closes <stream> and returns `true' if <stream> had been open.

Function: flength (<stream>)

     Returns the number of elements in <stream>.

Function: fposition (<stream>)
Function: fposition (<stream>, <pos>)

     Returns the current position in <stream>, if <pos> is not used.
     If <pos> is used, `fposition' sets the position in <stream>.
     <pos> has to be a positive number, the first element in <stream>
     is in position 1.

Function: freshline ()
Function: freshline (<stream>)

     Writes a new line (to <stream>), if the position is not at the
     beginning of a line.  See also `newline'.

Function: newline ()
Function: newline (<stream>)

     Writes a new line (to <stream>).  See `sprint' for an example of
     using `newline()'.  Note that there are some cases, where
     `newline()' does not work as expected.

Function: opena (<file>)

     Returns an output stream to <file>.  If an existing file is
     opened, `opena' appends elements at the end of file.

Function: openr (<file>)

     Returns an input stream to <file>.  If <file> does not exist, it
     will be created.

Function: openw (<file>)

     Returns an output stream to <file>.  If <file> does not exist, it
     will be created.  If an existing file is opened, `openw'
     destructively modifies <file>.

Function: printf (<dest>, <string>)
Function: printf (<dest>, <string>, <expr_1>, ..., <expr_n>)

     Makes the Common Lisp function FORMAT available in Maxima.  (From
     gcl.info: "format produces formatted output by outputting the
     characters of control-string string and observing that a tilde
     introduces a directive.  The character after the tilde, possibly
     preceded by prefix parameters and modifiers, specifies what kind
     of formatting is desired.  Most directives use one or more
     elements of args to create their output.")

     The following description and the examples may give an idea of
     using `printf'.  See a Lisp reference for more information.

             ~%       new line
             ~&       fresh line
             ~t       tab
             ~$       monetary
             ~d       decimal integer
             ~b       binary integer
             ~o       octal integer
             ~x       hexadecimal integer
             ~br      base-b integer
             ~r       spell an integer
             ~p       plural
             ~f       floating point
             ~e       scientific notation
             ~g       ~f or ~e, depending upon magnitude
             ~h       bigfloat
             ~a       uses Maxima function string
             ~s       like ~a, but output enclosed in "double quotes"
             ~~       ~
             ~<       justification, ~> terminates
             ~(       case conversion, ~) terminates
             ~[       selection, ~] terminates
             ~{       iteration, ~} terminates

     Note that the selection directive `~[' is zero-indexed.  Also note
     that the directive ~* is not supported.

          (%i1) printf( false, "~a ~a ~4f ~a ~@r",
                        "String",sym,bound,sqrt(12),144), bound = 1.234;
          (%o1)                 String sym 1.23 2*sqrt(3) CXLIV
          (%i2) printf( false,"~{~a ~}",["one",2,"THREE"] );
          (%o2)                          one 2 THREE
          (%i3) printf(true,"~{~{~9,1f ~}~%~}",mat ),
                    mat = args(matrix([1.1,2,3.33],[4,5,6],[7,8.88,9]))$
                1.1       2.0       3.3
                4.0       5.0       6.0
                7.0       8.9       9.0
          (%i4) control: "~:(~r~) bird~p ~[is~;are~] singing."$
          (%i5) printf( false,control, n,n,if n=1 then 0 else 1 ), n=2;
          (%o5)                    Two birds are singing.

     If <dest> is a stream or `true', then `printf' returns `false'.
     Otherwise, `printf' returns a string containing the output.

Function: readline (<stream>)

     Returns a string containing the characters from the current
     position in <stream> up to the end of the line or <false> if the
     end of the file is encountered.

Function: sprint (<expr_1>, ..., <expr_n>)

     Evaluates and displays its arguments one after the other `on a
     line' starting at the leftmost position.  The numbers are printed
     with the '-' right next to the number, and it disregards line
     length.  `newline()', which will be autoloaded from
     `stringproc.lisp' might be useful, if you whish to place
     intermediate line breaking.

     Example:

          (%i1) for n:0 thru 19 do sprint( fib(n) )$
          0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
          (%i2) for n:0 thru 22 do (
                   sprint(fib(n)), if mod(n,10)=9 then newline() )$
          0 1 1 2 3 5 8 13 21 34
          55 89 144 233 377 610 987 1597 2584 4181
          6765 10946 17711

78.3 Functions and Variables for characters

Function: alphacharp (<char>)

     Returns `true' if <char> is an alphabetic character.

Function: alphanumericp (<char>)

     Returns `true' if <char> is an alphabetic character or a digit.

Function: ascii (<int>)

     Returns the character corresponding to the ASCII number <int>.  (
     -1 < int < 256 )

          (%i1) for n from 0 thru 255 do (
             tmp: ascii(n), if alphacharp(tmp) then sprint(tmp),
                if n=96 then newline() )$
          A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
          a b c d e f g h i j k l m n o p q r s t u v w x y z

Function: cequal (<char_1>, <char_2>)

     Returns `true' if <char_1> and <char_2> are the same.

Function: cequalignore (<char_1>, <char_2>)

     Like `cequal' but ignores case.

Function: cgreaterp (<char_1>, <char_2>)

     Returns `true' if the ASCII number of <char_1> is greater than the
     number of <char_2>.

Function: cgreaterpignore (<char_1>, <char_2>)

     Like `cgreaterp' but ignores case.

Function: charp (<obj>)

     Returns `true' if <obj> is a Maxima-character.  See introduction
     for example.

Function: cint (<char>)

     Returns the ASCII number of <char>.

Function: clessp (<char_1>, <char_2>)

     Returns `true' if the ASCII number of <char_1> is less than the
     number of <char_2>.

Function: clesspignore (<char_1>, <char_2>)

     Like `clessp' but ignores case.

Function: constituent (<char>)

     Returns `true' if <char> is a graphic character and not the space
     character.  A graphic character is a character one can see, plus
     the space character.  (`constituent' is defined by Paul Graham,
     ANSI Common Lisp, 1996, page 67.)

          (%i1) for n from 0 thru 255 do (
          tmp: ascii(n), if constituent(tmp) then sprint(tmp) )$
          ! " #  %  ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B
          C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c
          d e f g h i j k l m n o p q r s t u v w x y z { | } ~

Function: cunlisp (<lisp_char>)

     Converts a Lisp-character into a Maxima-character.  (You won't
     need it.)

Function: digitcharp (<char>)

     Returns `true' if <char> is a digit.

Function: lcharp (<obj>)

     Returns `true' if <obj> is a Lisp-character.  (You won't need it.)

Function: lowercasep (<char>)

     Returns `true' if <char> is a lowercase character.

Variable: newline

     The newline character.

Variable: space

     The space character.

Variable: tab

     The tab character.

Function: uppercasep (<char>)

     Returns `true' if <char> is an uppercase character.

78.4 Functions and Variables for strings

Function: stringp (<obj>)

     Returns `true' if <obj> is a string.  See introduction for example.

Function: charat (<string>, <n>)

     Returns the <n>-th character of <string>.  The first character in
     <string> is returned with <n> = 1.

          (%i1) charat("Lisp",1);
          (%o1)                           L

Function: charlist (<string>)

     Returns the list of all characters in <string>.

          (%i1) charlist("Lisp");
          (%o1)                     [L, i, s, p]
          (%i2) %[1];
          (%o2)                           L

Function: eval_string (<str>)

     Parse the string <str> as a Maxima expression and evaluate it.
     The string <str> may or may not have a terminator (dollar sign `$'
     or semicolon `;').  Only the first expression is parsed and
     evaluated, if there is more than one.

     Complain if <str> is not a string.

     Examples:

          (%i1) eval_string ("foo: 42; bar: foo^2 + baz");
          (%o1)                       42
          (%i2) eval_string ("(foo: 42, bar: foo^2 + baz)");
          (%o2)                   baz + 1764

     See also `parse_string'.

Function: parse_string (<str>)

     Parse the string <str> as a Maxima expression (do not evaluate it).
     The string <str> may or may not have a terminator (dollar sign `$'
     or semicolon `;').  Only the first expression is parsed, if there
     is more than one.

     Complain if <str> is not a string.

     Examples:

          (%i1) parse_string ("foo: 42; bar: foo^2 + baz");
          (%o1)                    foo : 42
          (%i2) parse_string ("(foo: 42, bar: foo^2 + baz)");
                                             2
          (%o2)          (foo : 42, bar : foo  + baz)

     See also `eval_string'.

Function: scopy (<string>)

     Returns a copy of <string> as a new string.

Function: sdowncase (<string>)
Function: sdowncase (<string>, <start>)
Function: sdowncase (<string>, <start>, <end>)

     Like `supcase', but uppercase characters are converted to
     lowercase.

Function: sequal (<string_1>, <string_2>)

     Returns `true' if <string_1> and <string_2> are the same length
     and contain the same characters.

Function: sequalignore (<string_1>, <string_2>)

     Like `sequal' but ignores case.

Function: sexplode (<string>)

     `sexplode' is an alias for function `charlist'.

Function: simplode (<list>)
Function: simplode (<list>, <delim>)

     `simplode' takes a list of expressions and concatenates them into
     a string.  If no delimiter <delim> is specified, `simplode' uses
     no delimiter.  <delim> can be any string.

          (%i1) simplode(["xx[",3,"]:",expand((x+y)^3)]);
          (%o1)             xx[3]:y^3+3*x*y^2+3*x^2*y+x^3
          (%i2) simplode( sexplode("stars")," * " );
          (%o2)                   s * t * a * r * s
          (%i3) simplode( ["One","more","coffee."]," " );
          (%o3)                   One more coffee.

Function: sinsert (<seq>, <string>, <pos>)

     Returns a string that is a concatenation of `substring (<string>,
     1, <pos> - 1)', the string <seq> and `substring (<string>, <pos>)'.
     Note that the first character in <string> is in position 1.

          (%i1) s: "A submarine."$
          (%i2) concat( substring(s,1,3),"yellow ",substring(s,3) );
          (%o2)                  A yellow submarine.
          (%i3) sinsert("hollow ",s,3);
          (%o3)                  A hollow submarine.

Function: sinvertcase (<string>)
Function: sinvertcase (<string>, <start>)
Function: sinvertcase (<string>, <start>, <end>)

     Returns <string> except that each character from position <start>
     to <end> is inverted.  If <end> is not given, all characters from
     <start> to the end of <string> are replaced.

          (%i1) sinvertcase("sInvertCase");
          (%o1)                      SiNVERTcASE

Function: slength (<string>)

     Returns the number of characters in <string>.

Function: smake (<num>, <char>)

     Returns a new string with a number of <num> characters <char>.

          (%i1) smake(3,"w");
          (%o1)                          www

Function: smismatch (<string_1>, <string_2>)
Function: smismatch (<string_1>, <string_2>, <test>)

     Returns the position of the first character of <string_1> at which
     <string_1> and <string_2> differ or `false'.  Default test
     function for matching is `sequal'.  If `smismatch' should ignore
     case, use `sequalignore' as test.

          (%i1) smismatch("seven","seventh");
          (%o1)                           6

Function: split (<string>)
Function: split (<string>, <delim>)
Function: split (<string>, <delim>, <multiple>)

     Returns the list of all tokens in <string>.  Each token is an
     unparsed string.  `split' uses <delim> as delimiter.  If <delim>
     is not given, the space character is the default delimiter.
     <multiple> is a boolean variable with `true' by default.  Multiple
     delimiters are read as one.  This is useful if tabs are saved as
     multiple space characters.  If <multiple> is set to `false', each
     delimiter is noted.

          (%i1) split("1.2   2.3   3.4   4.5");
          (%o1)                 [1.2, 2.3, 3.4, 4.5]
          (%i2) split("first;;third;fourth",";",false);
          (%o2)               [first, , third, fourth]

Function: sposition (<char>, <string>)

     Returns the position of the first character in <string> which
     matches <char>.  The first character in <string> is in position 1.
     For matching characters ignoring case see `ssearch'.

Function: sremove (<seq>, <string>)
Function: sremove (<seq>, <string>, <test>)
Function: sremove (<seq>, <string>, <test>, <start>)
Function: sremove (<seq>, <string>, <test>, <start>, <end>)

     Returns a string like <string> but without all substrings matching
     <seq>.  Default test function for matching is `sequal'.  If
     `sremove' should ignore case while searching for <seq>, use
     `sequalignore' as test.  Use <start> and <end> to limit searching.
     Note that the first character in <string> is in position 1.

          (%i1) sremove("n't","I don't like coffee.");
          (%o1)                   I do like coffee.
          (%i2) sremove ("DO ",%,'sequalignore);
          (%o2)                    I like coffee.

Function: sremovefirst (<seq>, <string>)
Function: sremovefirst (<seq>, <string>, <test>)
Function: sremovefirst (<seq>, <string>, <test>, <start>)
Function: sremovefirst (<seq>, <string>, <test>, <start>, <end>)

     Like `sremove' except that only the first substring that matches
     `seq' is removed.

Function: sreverse (<string>)

     Returns a string with all the characters of <string> in reverse
     order.

Function: ssearch (<seq>, <string>)
Function: ssearch (<seq>, <string>, <test>)
Function: ssearch (<seq>, <string>, <test>, <start>)
Function: ssearch (<seq>, <string>, <test>, <start>, <end>)

     Returns the position of the first substring of <string> that
     matches the string <seq>.  Default test function for matching is
     `sequal'.  If `ssearch' should ignore case, use `sequalignore' as
     test.  Use <start> and <end> to limit searching.  Note that the
     first character in <string> is in position 1.

          (%i1) ssearch("~s","~{~S ~}~%",'sequalignore);
          (%o1)                                  4

Function: ssort (<string>)
Function: ssort (<string>, <test>)

     Returns a string that contains all characters from <string> in an
     order such there are no two successive characters <c> and <d> such
     that `test (<c>, <d>)' is `false' and `test (<d>, <c>)' is `true'.
     Default test function for sorting is <clessp>.  The set of test
     functions is `{clessp, clesspignore, cgreaterp, cgreaterpignore,
     cequal, cequalignore}'.

          (%i1) ssort("I don't like Mondays.");
          (%o1)                    '.IMaddeiklnnoosty
          (%i2) ssort("I don't like Mondays.",'cgreaterpignore);
          (%o2)                 ytsoonnMlkIiedda.'

Function: ssubst (<new>, <old>, <string>)
Function: ssubst (<new>, <old>, <string>, <test>)
Function: ssubst (<new>, <old>, <string>, <test>, <start>)
Function: ssubst (<new>, <old>, <string>, <test>, <start>, <end>)

     Returns a string like <string> except that all substrings matching
     <old> are replaced by <new>.  <old> and <new> need not to be of
     the same length.  Default test function for matching is `sequal'.
     If `ssubst' should ignore case while searching for old, use
     `sequalignore' as test.  Use <start> and <end> to limit searching.
     Note that the first character in <string> is in position 1.

          (%i1) ssubst("like","hate","I hate Thai food. I hate green tea.");
          (%o1)          I like Thai food. I like green tea.
          (%i2) ssubst("Indian","thai",%,'sequalignore,8,12);
          (%o2)         I like Indian food. I like green tea.

Function: ssubstfirst (<new>, <old>, <string>)
Function: ssubstfirst (<new>, <old>, <string>, <test>)
Function: ssubstfirst (<new>, <old>, <string>, <test>, <start>)
Function: ssubstfirst (<new>, <old>, <string>, <test>, <start>, <end>)

     Like `subst' except that only the first substring that matches
     <old> is replaced.

Function: strim (<seq>,<string>)

     Returns a string like <string>, but with all characters that
     appear in <seq> removed from both ends.

          (%i1) "/* comment */"$
          (%i2) strim(" /*",%);
          (%o2)                        comment
          (%i3) slength(%);
          (%o3)                           7

Function: striml (<seq>, <string>)

     Like `strim' except that only the left end of <string> is trimmed.

Function: strimr (<seq>, <string>)

     Like `strim' except that only the right end of string is trimmed.

Function: substring (<string>, <start>)
Function: substring (<string>, <start>, <end>)

     Returns the substring of <string> beginning at position <start>
     and ending at position <end>.  The character at position <end> is
     not included.  If <end> is not given, the substring contains the
     rest of the string.  Note that the first character in <string> is
     in position 1.

          (%i1) substring("substring",4);
          (%o1)                        string
          (%i2) substring(%,4,6);
          (%o2)                          in

Function: supcase (<string>)
Function: supcase (<string>, <start>)
Function: supcase (<string>, <start>, <end>)

     Returns <string> except that lowercase characters from position
     <start> to <end> are replaced by the corresponding uppercase ones.
     If <end> is not given, all lowercase characters from <start> to
     the end of <string> are replaced.

          (%i1) supcase("english",1,2);
          (%o1)                        English

Function: tokens (<string>)
Function: tokens (<string>, <test>)

     Returns a list of tokens, which have been extracted from <string>.
     The tokens are substrings whose characters satisfy a certain test
     function.  If test is not given, <constituent> is used as the
     default test.  `{constituent, alphacharp, digitcharp, lowercasep,
     uppercasep, charp, characterp, alphanumericp}' is the set of test
     functions.  (The Lisp-version of `tokens' is written by Paul
     Graham. ANSI Common Lisp, 1996, page 67.)

          (%i1) tokens("24 October 2005");
          (%o1)                  [24, October, 2005]
          (%i2) tokens("05-10-24",'digitcharp);
          (%o2)                     [05, 10, 24]
          (%i3) map(parse_string,%);
          (%o3)                      [5, 10, 24]

79 to_poly_solve


Functions and Variables for to_poly_solve

79.1 Functions and Variables for to_poly_solve

The packages `topoly' and `to_poly_solve' are experimental; the
specifications of the functions in these packages might change or the
some of the functions in these packages might be merged into other
Maxima functions.

   Barton Willis (Professor of Mathematics, University of Nebraska at
Kearney) wrote the `topoly' and `to_poly_solve' packages and the
English language user documentation for these packages.
Operator: %and

     The operator `%and' is a simplifying nonshort-circuited logical
     conjunction.  Maxima simplifies an `%and' expression to either
     true, false, or a logically equivalent, but simplified,
     expression.  The operator `%and' is associative, commutative, and
     idempotent.  Thus when `%and' returns a noun form, the arguments
     of `%and' form a non-redundant sorted list; for example

          (%i1) a %and (a %and b);
          (%o1)                       a %and b

     If one argument to a conjunction is the explicit the negation of
     another argument, `%and' returns false:

          (%i2) a %and (not a);
          (%o2)                         false

     If any member of the conjunction is false, the conjunction
     simplifies to false even if other members are manifestly
     non-boolean; for example

          (%i3) 42 %and false;
          (%o3)                         false

     Any argument of an `%and' expression that is an inequation (that
     is, an inequality or equation), is simplified using the Fourier
     elimination package.  The Fourier elimination simplifier has a
     pre-processor that converts some, but not all, nonlinear
     inequations into linear inequations; for example the Fourier
     elimination code simplifies `abs(x) + 1 > 0' to true, so

          (%i4) (x < 1) %and (abs(x) + 1 > 0);
          (%o4)                         x < 1

     Notes
        * The option variable `prederror' does not alter the
          simplification `%and' expressions.

        * To avoid operator precedence errors, compound expressions
          involving the operators `%and, %or', and `not' should be
          fully parenthesized.

        * The Maxima operators `and' and `or' are both short-circuited.
          Thus `and' isn't associative or commutative.

Operator: %if (<bool>, <a>, <b>)

     The operator `%if' is a simplifying conditional.  The conditional
     <bool> should be boolean-valued.  When the conditional is true,
     return the second argument; when the conditional is false, return
     the third; in all other cases, return a noun form.

     Maxima inequations (either an inequality or an equality) are not
     boolean-valued; for example, Maxima does not simplify 5 < 6 to
     true, and it does not simplify 5 = 6 to false; however, in the
     context of a conditional to an `%if' statement, Maxima
     automatically attempts to determine the truth value of an
     inequation.  Examples:

          (%i1) f : %if(x # 1, 2, 8);
          (%o1)                 %if(x - 1 # 0, 2, 8)
          (%i2) [subst(x = -1,f), subst(x=1,f)];
          (%o2)                        [2, 8]

     If the conditional involves an inequation, Maxima simplifies it
     using the Fourier elimination package.

     Notes

        * If the conditional is manifestly non-boolean, Maxima returns
          a noun form:

          (%i3) %if(42,1,2);
          (%o3)                     %if(42, 1, 2)

        * The Maxima operator `if' is nary, the operator `%if' isn't
          nary.

     Limitations The Fourier elimination code only simplifies nonlinear
     inequations that are readily convertible to an equivalent linear
     inequation.

     To use: `load(to_poly_solver)'

     Status: The operator `%if' is experimental; its specifications
     might change and its functionality might be merged into other
     Maxima functions.

Operator: %or

     The operator `%or' is a simplifying nonshort-circuited logical
     disjunction.  Maxima simplifies an `%or' expression to either
     true, false, or a logically equivalent, but simplified,
     expression.  The operator `%or' is associative, commutative, and
     idempotent.  Thus when `%or' returns a noun form, the arguments of
     `%or' form a non-redundant sorted list; for example

          (%i1) a %or (a %or b);
          (%o1)                        a %or b

     If one member of the disjunction is the explicit the negation of
     another member, `%or' returns true:

          (%i2) a %or (not a);
          (%o2)                         true

     If any member of the disjunction is true, the disjunction
     simplifies to true even if other members of the disjunction are
     manifestly non-boolean; for example

          (%i3) 42 %or true;
          (%o3)                         true

     Any argument of an `%or' expression that is an inequation (that
     is, an inequality or equation), is simplified using the Fourier
     elimination package.  The Fourier elimination code simplifies
     `abs(x) + 1 > 0' to true, so we have

          (%i4) (x < 1) %or (abs(x) + 1 > 0);
          (%o4)                         true

     Notes
        * The option variable `prederror' does not alter the
          simplification of `%or' expressions.

        * You should parenthesize compound expressions involving the
          operators `%and, %or', and `not'; the binding powers of these
          operators might not match your expectations.

        * The Maxima operators `and' and `or' are both short-circuited.
          Thus `or' isn't associative or commutative.

Function: complex_number_p (<x>)

     The predicate `complex_number_p' returns true if its argument is
     either `a + %i * b', `a', `%i b', or `%i', where `a' and `b' are
     either rational or floating point numbers (including big floating
     point); for all other inputs, `complex_number_p' returns false;
     for example

          (%i1) map('complex_number_p,[2/3, 2 + 1.5 * %i, %i]);
          (%o1)                  [true, true, true]
          (%i2) complex_number_p((2+%i)/(5-%i));
          (%o2)                         false
          (%i3) complex_number_p(cos(5 - 2 * %i));
          (%o3)                         false

     Related functions `isreal_p'

     To use `load(to_poly_solver)'

     Status The operator `complex_number_p' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

Function: compose_functions (<l>)

     The function call `compose_functions(l)' returns a lambda form
     that is the composition of the functions in the list <l>.  The
     functions are applied from right to left; for example

          (%i1) compose_functions([cos, exp]);
                                                  %g151
          (%o1)             lambda([%g151], cos(%e     ))
          (%i2) %(x);
                                            x
          (%o2)                       cos(%e )

     When the function list is empty, return the identity function:

          (%i3) compose_functions([]);
          (%o3)                lambda([%g152], %g152)
          (%i4)  %(x);
          (%o4)                           x

     Notes
        * When Maxima determines that a list member isn't a symbol or a
          lambda form, `funmake' (not `compose_functions') signals an
          error:

          (%i5) compose_functions([a < b]);

          funmake: first argument must be a symbol, subscripted symbol,
          string, or lambda expression; found: a < b
          #0: compose_functions(l=[a < b])(to_poly_solver.mac line 40)
           -- an error. To debug this try: debugmode(true);

        * To avoid name conflicts, the independent variable is
          determined by the function `new_variable'.

          (%i6) compose_functions([%g0]);
          (%o6)              lambda([%g154], %g0(%g154))
          (%i7) compose_functions([%g0]);
          (%o7)              lambda([%g155], %g0(%g155))

          Although the independent variables are different, Maxima is
          able to to deduce that these lambda forms are semantically
          equal:

          (%i8) is(equal(%o6,%o7));
          (%o8)                         true

     To use `load(to_poly_solver)'

     Status  The function `compose_functions' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

Function: dfloat (<x>)

     The function `dfloat' is a similar to `float', but the function
     `dfloat' applies `rectform' when `float' fails to evaluate to an
     IEEE double floating point number; thus

          (%i1) float(4.5^(1 + %i));
                                         %i + 1
          (%o1)                       4.5
          (%i2) dfloat(4.5^(1 + %i));
          (%o2)        4.48998802962884 %i + .3000124893895671

     Notes

        * The rectangular form of an expression might be poorly suited
          for numerical evaluation-for example, the rectangular form
          might needlessly involve the difference of floating point
          numbers (subtractive cancellation).

        * The identifier `float' is both an option variable (default
          value false) and a function name.

Function: elim (<l>, <x>)

     The function `elim' eliminates the variables in the set or list
     `x' from the equations in the set or list `l'.  Each member of `x'
     must be a symbol; the members of `l' can either be equations, or
     expressions that are assumed to equal zero.

     The function `elim' returns a list of two lists; the first is the
     list of expressions with the variables eliminated; the second is
     the list of pivots; thus, the second list is a list of expressions
     that `elim' used to eliminate the variables.

     Here is a example of eliminating between linear equations:

          (%i1) elim(set(x + y + z = 1, x - y  - z = 8, x - z = 1),
                     set(x,y));
          (%o1)            [[2 z - 7], [y + 7, z - x + 1]]

     Eliminating `x' and `y' yields the single equation `2 z - 7 = 0';
     the equations `y + 7 = 0' and `z - z + 1 = 1' were used as pivots.
     Eliminating all three variables from these equations,
     triangularizes the linear system:

          (%i2) elim(set(x + y + z = 1, x - y  - z = 8, x - z = 1),
                     set(x,y,z));
          (%o2)           [[], [2 z - 7, y + 7, z - x + 1]]

     Of course, the equations needn't be linear:

          (%i3) elim(set(x^2 - 2 * y^3 = 1,  x - y = 5), [x,y]);
                               3    2
          (%o3)       [[], [2 y  - y  - 10 y - 24, y - x + 5]]

     The user doesn't control the order the variables are eliminated.
     Instead, the algorithm uses a heuristic to attempt to choose the
     best pivot and the best elimination order.

     Notes

        * Unlike the related function `eliminate', the function `elim'
          does not invoke `solve' when the number of equations equals
          the number of variables.

        * The function `elim' works by applying resultants; the option
          variable `resultant' determines which algorithm Maxima uses.
          Using `sqfr', Maxima factors each resultant and suppresses
          multiple zeros.

        * The `elim' will triangularize a nonlinear set of polynomial
          equations; the solution set of the triangularized set can be
          larger than that solution set of the untriangularized set.
          Thus, the triangularized equations can have spurious
          solutions.

     Related functions elim_allbut, eliminate_using, eliminate

     Option variables resultant

     To use `load(to_poly)'

     Status The function `elim' is experimental; its specifications
     might change and its functionality might be merged into other
     Maxima functions.

Function: elim_allbut (<l>, <x>)

     This function is similar to `elim', except that it eliminates all
     the variables in the list of equations `l' except for those
     variables that in in the list `x'

          (%i1) elim_allbut([x+y = 1, x - 5*y = 1],[]);
          (%o1)                 [[], [y, y + x - 1]]
          (%i2) elim_allbut([x+y = 1, x - 5*y = 1],[x]);
          (%o2)                [[x - 1], [y + x - 1]]

     To use `load(to_poly)'

     Option variables resultant

     Related functions elim, eliminate_using, eliminate

     Status The function `elim_allbut' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

Function: eliminate_using (<l>, <e>, <x>)

     Using `e' as the pivot, eliminate the symbol `x' from the list or
     set of equations in `l'.  The function `eliminate_using' returns a
     set.

          (%i1) eq : [x^2 - y^2 - z^3 , x*y - z^2 - 5, x - y + z];
                         3    2    2     2
          (%o1)      [- z  - y  + x , - z  + x y - 5, z - y + x]
          (%i2) eliminate_using(eq,first(eq),z);
                  3              2      2      3    2
          (%o2) {y  + (1 - 3 x) y  + 3 x  y - x  - x ,
                                  4    3  3       2  2             4
                                 y  - x  y  + 13 x  y  - 75 x y + x  + 125}
          (%i3) eliminate_using(eq,second(eq),z);
                  2            2       4    3  3       2  2             4
          (%o3) {y  - 3 x y + x  + 5, y  - x  y  + 13 x  y  - 75 x y + x
                                                                     + 125}
          (%i4) eliminate_using(eq, third(eq),z);
                  2            2       3              2      2      3    2
          (%o4) {y  - 3 x y + x  + 5, y  + (1 - 3 x) y  + 3 x  y - x  - x }

     Option variables resultant

     Related functions elim, eliminate, elim_allbut

     To use `load(topoly)'

     Status The function `eliminate_using' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

Function: fourier_elim ([<eq1>, <eq2>, ...], [<var1>, <var>, ...])

     Fourier elimination is the analog of Gauss elimination for linear
     inequations (equations or inequalities).  The function call
     `fourier_elim([eq1, eq2, ...], [var1, var2, ...]' does Fourier
     elimination on a list of linear inequations `[eq1, eq2, ...]' with
     respect to the variables `[var1, var2, ...]'; for example

          (%i1) fourier_elim([y-x < 5, x - y < 7, 10 < y],[x,y]);
          (%o1)            [y - 5 < x, x < y + 7, 10 < y]
          (%i2) fourier_elim([y-x < 5, x - y < 7, 10 < y],[y,x]);
          (%o2)        [max(10, x - 7) < y, y < x + 5, 5 < x]

     Eliminating first with respect to x and second with respect to y
     yields lower and upper bounds for x that depend on y, and lower
     and upper bounds for y that are numbers.  Eliminating in the other
     order gives x dependent lower and upper bounds for y, and
     numerical lower and upper bounds for x.

     When necessary, `fourier_elim' returns a _disjunction_ of lists of
     inequations:

          (%i3) fourier_elim([x # 6],[x]);
          (%o3)                  [x < 6] or [6 < x]

     When the solution set is empty,  `fourier_elim' returns `emptyset',
     and when the solution set is all reals, `fourier_elim' returns
     `universalset'; for example

          (%i4) fourier_elim([x < 1, x > 1],[x]);
          (%o4)                       emptyset
          (%i5) fourier_elim([minf < x, x < inf],[x]);
          (%o5)                     universalset

     For nonlinear inequations, `fourier_elim' returns a (somewhat)
     simplified list of inequations:

          (%i6) fourier_elim([x^3 - 1 > 0],[x]);
                         2                             2
          (%o6) [1 < x, x  + x + 1 > 0] or [x < 1, - (x  + x + 1) > 0]
          (%i7) fourier_elim([cos(x) < 1/2],[x]);
          (%o7)                  [1 - 2 cos(x) > 0]

     Instead of a list of inequations, the first argument to
     `fourier_elim' may be a logical disjunction or conjunction:

          (%i8) fourier_elim((x + y < 5) and (x - y >8),[x,y]);
                                                        3
          (%o8)            [y + 8 < x, x < 5 - y, y < - -]
                                                        2
          (%i9) fourier_elim(((x + y < 5) and x < 1) or  (x - y >8),[x,y]);
          (%o9)          [y + 8 < x] or [x < min(1, 5 - y)]

     The function `fourier_elim' supports the inequation operators `<,
     <=, >, >=, #', and `='.

     The Fourier elimination code has a preprocessor that converts some
     nonlinear inequations that involve the absolute value, minimum, and
     maximum functions into linear in equations.  Additionally, the
     preprocessor handles some expressions that are the product or
     quotient of linear terms:

          (%i10) fourier_elim([max(x,y) > 6, x # 8, abs(y-1) > 12],[x,y]);
          (%o10) [6 < x, x < 8, y < - 11] or [8 < x, y < - 11]
           or [x < 8, 13 < y] or [x = y, 13 < y] or [8 < x, x < y, 13 < y]
           or [y < x, 13 < y]
          (%i11) fourier_elim([(x+6)/(x-9) <= 6],[x]);
          (%o11)           [x = 12] or [12 < x] or [x < 9]
          (%i12) fourier_elim([x^2 - 1 # 0],[x]);
          (%o12)      [- 1 < x, x < 1] or [1 < x] or [x < - 1]

     To use `load(fourier_elim)'

Function: isreal_p (<e>)

     The predicate `isreal_p' returns true when Maxima is able to
     determine that `e' is real-valued on the entire real line; it
     returns false when Maxima is able to determine that `e' isn't
     real-valued on some nonempty subset of the real line; and it
     returns a noun form for all other cases.

          (%i1) map('isreal_p, [-1, 0, %i, %pi]);
          (%o1)               [true, true, false, true]

     Maxima variables are assumed to be real; thus

          (%i2) isreal_p(x);
          (%o2)                         true

     The function `isreal_p' examines the fact database:

          (%i3) declare(z,complex)$

          (%i4) isreal_p(z);
          (%o4)                      isreal_p(z)

     Limitations Too often, `isreal_p' returns a noun form when it
     should be able to return false; a simple example: the logarithm
     function isn't real-valued on the entire real line, so
     `isreal_p(log(x))' should return false; however

          (%i5) isreal_p(log(x));
          (%o5)                   isreal_p(log(x))

     To use `load(to_poly_solver)'

     Related functions complex_number_p

     Status The function `real_p' is experimental; its specifications
     might change and its functionality might be merged into other
     Maxima functions.

Function: new_variable (type)

     Return a unique symbol of the form `%[z,n,r,c,g]k', where `k' is
     an integer.  The allowed values for type are integer,
     natural_number, real, natural_number, and general.  (By natural
     number, we mean the nonnegative integers; thus zero is a natural
     number.  Some, but not all,definitions of natural number exclude
     zero.)

     When type isn't one of the allowed values, type defaults to
     general.  For integers, natural numbers, and complex numbers,
     Maxima automatically appends this information to the fact database.

          (%i1) map('new_variable,
                    ['integer, 'natural_number, 'real, 'complex, 'general]);
          (%o1)          [%z144, %n145, %r146, %c147, %g148]
          (%i2) nicedummies(%);
          (%o2)               [%z0, %n0, %r0, %c0, %g0]
          (%i3) featurep(%z0, 'integer);
          (%o3)                         true
          (%i4) featurep(%n0, 'integer);
          (%o4)                         true
          (%i5) is(%n0 >= 0);
          (%o5)                         true
          (%i6) featurep(%c0, 'complex);
          (%o6)                         true

     Note Generally, the argument to `new_variable' should be quoted.
     The quote will protect against errors similar to

          (%i7) integer : 12$

          (%i8) new_variable(integer);
          (%o8)                         %g149
          (%i9) new_variable('integer);
          (%o9)                         %z150

     Related functions nicedummies

     To use `load(to_poly_solver)'

     Status The function `new_variable' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

Function: nicedummies

     Starting with zero, the function `nicedummies' re-indexes the
     variables in an expression that were introduced by `new_variable';

          (%i1) new_variable('integer) + 52 * new_variable('integer);
          (%o1)                   52 %z136 + %z135
          (%i2) new_variable('integer) - new_variable('integer);
          (%o2)                     %z137 - %z138
          (%i3) nicedummies(%);
          (%o3)                       %z0 - %z1

     Related functions new_variable

     To use `load(to_poly_solver)'

     Status The function `nicedummies' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

Function: parg (<x>)

     The function `parg' is a simplifying version of the complex
     argument function `carg'; thus

          (%i1) map('parg,[1,1+%i,%i, -1 + %i, -1]);
                                  %pi  %pi  3 %pi
          (%o1)               [0, ---, ---, -----, %pi]
                                   4    2     4

     Generally, for a non-constant input, `parg' returns a noun form;
     thus

          (%i2) parg(x + %i * sqrt(x));
          (%o2)                 parg(x + %i sqrt(x))

     When `sign' can determine that the input is a positive or negative
     real number, `parg' will return a non-noun form for a non-constant
     input.  Here are two examples:

          (%i3) parg(abs(x));
          (%o3) 0
          (%i4) parg(-x^2-1);
          (%o4)                          %pi

     Note The `sign' function mostly ignores the variables that are
     declared to be complex (`declare(x,complex)'); for variables that
     are declared to be complex, the `parg' can return incorrect
     values; for example

          (%i1) declare(x,complex)$

          (%i2) parg(x^2 + 1);
          (%o2) 0

     Related function carg, isreal_p

     To use `load(to_poly_solver)'

     Status The function `parg' is experimental; its specifications
     might change and its functionality might be merged into other
     Maxima functions.

Function: real_imagpart_to_conjugate (<e>)

     The function `real_imagpart_to_conjugate' replaces all occurrences
     of `realpart' and `imagpart' to algebraically equivalent
     expressions involving the `conjugate'.

          (%i1) declare(x, complex)$

          (%i2) real_imagpart_to_conjugate(realpart(x) +  imagpart(x) = 3);
                    conjugate(x) + x   %i (x - conjugate(x))
          (%o2)     ---------------- - --------------------- = 3
                           2                     2

     To use `load(to_poly_solver)'

     Status The function `real_imagpart_to_conjugate' is experimental;
     its specifications might change and its functionality might be
     merged into other Maxima functions.

Function: rectform_log_if_constant (<e>)

     The function `rectform_if_constant' converts all terms of the form
     ` log(c)' to  `rectform(log(c))', where `c' is either a declared
     constant expression or explicitly declared constant

          (%i1) rectform_log_if_constant(log(1-%i) - log(x - %i));
                                           log(2)   %i %pi
          (%o1)            - log(x - %i) + ------ - ------
                                             2        4
          (%i2) declare(a,constant, b,constant)$

          (%i3) rectform_log_if_constant(log(a + %i*b));
                                 2    2
                            log(b  + a )
          (%o3)             ------------ + %i atan2(b, a)
                                 2

     To use `load(to_poly_solver)'

     Status The function `rectform_log_if_constant' is experimental;
     the specifications of this function might change might change and
     its functionality might be merged into other Maxima functions.

Function: simp_inequality (<e>)

     The function `simp_inequality' applies some simplifications to
     conjunctions and disjunctions of inequations.

     Limitations The function `simp_inequality' is limited in at least
     two ways; first, the simplifications are local; thus

          (%i1) simp_inequality((x > minf) %and (x < 0));
          (%o1) (x>1) %and (x<1)

     And second, `simp_inequality' doesn't consult the fact database:

          (%i2) assume(x > 0)$

          (%i3) simp_inequality(x > 0);
          (%o3)                         x > 0

     To use `load(fourier_elim)'

     Status The function `simp_inequality' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

Function: standardize_inverse_trig (<e>)

     This function applies the identities `cot(x) = atan(1/x), acsc(x)
     = asin(1/x),' and similarly for `asec, acoth, acsch' and `asech'
     to an expression.  See Abramowitz and Stegun, Eqs. 4.4.6 through
     4.4.8 and 4.6.4 through 4.6.6.

     To use `load(to_poly_solver)'

     Status The function `standardize_inverse_trig' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

Function: subst_parallel (<l>, <e>)

     When `l' is a single equation or a list of equations, substitute
     the right hand side of each equation for the left hand side.  The
     substitutions are made in parallel; for example

          (%i1) load(to_poly_solver)$

          (%i2) subst_parallel([x=y,y=x], [x,y]);
          (%o2)                        [y, x]

     Compare this to substitutions made serially:

          (%i3) subst([x=y,y=x],[x,y]);
          (%o3)                        [x, x]

     The function `subst_parallel' is similar to `sublis' except that
     `subst_parallel' allows for substitution of nonatoms; for example

          (%i4) subst_parallel([x^2 = a, y = b], x^2 * y);
          (%o4)                          a b
          (%i5) sublis([x^2 = a, y = b], x^2 * y);

                                                                       2
          sublis: left-hand side of equation must be a symbol; found: x
           -- an error. To debug this try: debugmode(true);

     The substitutions made by `subst_parallel' are literal, not
     semantic; thus `subst_parallel' does not recognize that x * y is a
     subexpression of x^2 * y

          (%i6) subst_parallel([x * y = a], x^2 * y);
                                         2
          (%o6)                         x  y

     The function `subst_parallel' completes all substitutions before
     simplifications.  This allows for substitutions into conditional
     expressions where errors might occur if the simplifications were
     made earlier:

          (%i7) subst_parallel([x = 0], %if(x < 1, 5, log(x)));
          (%o7)                           5
          (%i8) subst([x = 0], %if(x < 1, 5, log(x)));

          log: encountered log(0).
           -- an error. To debug this try: debugmode(true);

     Related functions subst, sublis, ratsubst

     To use `load(to_poly_solve_extra.lisp)'

     Status The function `subst_parallel' is experimental; the
     specifications of this function might change might change and its
     functionality might be merged into other Maxima functions.

Function: to_poly (<e>, <l>)

     The function `to_poly' attempts to convert the equation `e' into a
     polynomial system along with inequality constraints; the solutions
     to the polynomial system that satisfy the constraints are
     solutions to the equation `e'.  Informally, `to_poly' attempts to
     polynomialize the equation <e>; an example might clarify:

          (%i1) load(to_poly_solver)$

          (%i2) to_poly(sqrt(x) = 3, [x]);
                                      2
          (%o2) [[%g130 - 3, x = %g130 ],
                                %pi                               %pi
                             [- --- < parg(%g130), parg(%g130) <= ---], []]
                                 2                                 2

     The conditions `-%pi/2<parg(%g6),parg(%g6)<=%pi/2' tell us that
     `%g6' is in the range of the square root function.  When this is
     true, the solution set to `sqrt(x) = 3' is the same as the
     solution set to `%g6-3,x=%g6^2'.

     To polynomialize trigonometric expressions, it is necessary to
     introduce a non algebraic substitution; these non algebraic
     substitutions are returned in the third list returned by
     `to_poly'; for example

          (%i3) to_poly(cos(x),[x]);
                          2                                 %i x
          (%o3)    [[%g131  + 1], [2 %g131 # 0], [%g131 = %e    ]]

     Constant terms aren't polynomializied unless the number one is a
     member of the variable list; for example

          (%i4) to_poly(x = sqrt(5),[x]);
          (%o4)                [[x - sqrt(5)], [], []]
          (%i5) to_poly(x = sqrt(5),[1,x]);
                                      2
          (%o5) [[x - %g132, 5 = %g132 ],
                                %pi                               %pi
                             [- --- < parg(%g132), parg(%g132) <= ---], []]
                                 2                                 2

     To generate a polynomial with sqrt(5) + sqrt(7) as one of its
     roots, use the commands

          (%i6) first(elim_allbut(first(to_poly(x = sqrt(5) + sqrt(7),
                                                [1,x])), [x]));
                                    4       2
          (%o6)                   [x  - 24 x  + 4]

     Related functions to_poly_solve

     To use `load(to_poly)'

     Status: The function `to_poly' is experimental; its specifications
     might change and its functionality might be merged into other
     Maxima functions.

Function: to_poly_solve (<e>, <l>, [options])

     The function `to_poly_solve' tries to solve the equations e for
     the variables l.  The equation(s) e can either be a single
     expression or a set or list of expressions; similarly, l can
     either be a single symbol or a list of set of symbols.  When a
     member of e isn't explicitly an equation, for example x^2 -1, the
     solver asummes that the expression vanishes.

     The basic strategy of `to_poly_solve' is use `to_poly' to convert
     the input into a polynomial form and call `algsys' on the
     polynomial system.  Thus user options that affect `algsys',
     especially `algexact', also affect `to_poly_solve'.  The default
     for `algexact' is false, but for `to_poly_solve', generally
     `algexact' should be true.  The function `to_poly_solve' does not
     locally set `algexact' to true because this would make it
     impossible to find approximate solutions when the `algsys' is
     unable to determine an exact solution.

     When `to_poly_solve' is able to determine the solution set, each
     member of the solution set is a list in a `%union' object:

          (%i1) load(to_poly_solver)$

          (%i2) to_poly_solve(x*(x-1) = 0, x);
          (%o2)               %union([x = 0], [x = 1])

     When  `to_poly_solve' is unable to determine the solution set, a
     `%solve' nounform is returned (in this case, a warning is printed)

          (%i3) to_poly_solve(x^k + 2* x + 1 = 0, x);

          Nonalgebraic argument given to 'topoly'
          unable to solve
                                    k
          (%o3)            %solve([x  + 2 x + 1 = 0], [x])

     Subsitution into a `%solve' nounform can sometimes result in the
     solution

          (%i4) subst(k = 2, %);
          (%o4)                   %union([x = - 1])

     Especially for trigonometric equations, the solver sometimes needs
     to introduce an arbitary integer.  These arbitary integers have the
     form `%zXXX', where `XXX' is an integer; for example

          (%i5) to_poly_solve(sin(x) = 0, x);
          (%o5)   %union([x = 2 %pi %z33 + %pi], [x = 2 %pi %z35])

     To re-index these variables to zero, use `nicedummies':

          (%i6) nicedummies(%);
          (%o6)    %union([x = 2 %pi %z0 + %pi], [x = 2 %pi %z1])

     Occasionally, the solver introduces an arbitary complex number of
     the form `%cXXX' or an  arbitary real number of the form `%rXXX'.
     The function `nicedummies' will re-index these identifiers to zero.

     The solution set sometimes involves simplifing versions of various
     of logical operators including `%and', `%or', or `%if' for
     conjunction, disjuntion, and implication, respectively; for example

          (%i7) sol : to_poly_solve(abs(x) = a, x);
          (%o7) %union(%if(isnonnegative_p(a), [x = - a], %union()),
                                %if(isnonnegative_p(a), [x = a], %union()))
          (%i8) subst(a = 42, sol);
          (%o8)             %union([x = - 42], [x = 42])
          (%i9) subst(a = -42, sol);
          (%o9)                       %union()

     The empty set is represented by `%union'.

     The function `to_poly_solve' is able to solve some, but not all,
     equations involving rational powers, some nonrational powers,
     absolute values, trigonometric functions, and minimum and maximum.
     Also, some it can solve some equations that are solvable in in
     terms of the Lambert W function; some examples:

          (%i1) load(to_poly_solver)$

          (%i2) to_poly_solve(set(max(x,y) = 5, x+y = 2), set(x,y));
          (%o2)      %union([x = - 3, y = 5], [x = 5, y = - 3])
          (%i3) to_poly_solve(abs(1-abs(1-x)) = 10,x);
          (%o3)             %union([x = - 10], [x = 12])
          (%i4) to_poly_solve(set(sqrt(x) + sqrt(y) = 5, x + y = 10),
                              set(x,y));
                               3/2               3/2
                              5    %i - 10      5    %i + 10
          (%o4) %union([x = - ------------, y = ------------],
                                   2                 2
                                          3/2                 3/2
                                         5    %i + 10        5    %i - 10
                                    [x = ------------, y = - ------------])
                                              2                   2
          (%i5) to_poly_solve(cos(x) * sin(x) = 1/2,x,
                              'simpfuncs = ['expand, 'nicedummies]);
                                                   %pi
          (%o5)              %union([x = %pi %z0 + ---])
                                                    4
          (%i6) to_poly_solve(x^(2*a) + x^a + 1,x);
                                                  2 %i %pi %z81
                                                  -------------
                                            1/a         a
                            (sqrt(3) %i - 1)    %e
          (%o6) %union([x = -----------------------------------],
                                            1/a
                                           2
                                                            2 %i %pi %z83
                                                            -------------
                                                      1/a         a
                                    (- sqrt(3) %i - 1)    %e
                               [x = -------------------------------------])
                                                     1/a
                                                    2
          (%i7) to_poly_solve(x * exp(x) = a, x);
          (%o7)              %union([x = lambert_w(a)])

     For linear inequalities, `to_poly_solve' automatically does Fourier
     elimination:

          (%i8) to_poly_solve([x + y < 1, x - y >= 8], [x,y]);
                                         7
          (%o8) %union([x = y + 8, y < - -],
                                         2
                                                                        7
                                           [y + 8 < x, x < 1 - y, y < - -])
                                                                        2

     Each optional argument to `to_poly_solve' must be an equation;
     generally, the order of these options does not matter.

        * `simpfuncs = l', where `l' is a list of functions.  Apply the
          composition of the members of l to each solution.

          (%i1) to_poly_solve(x^2=%i,x);
                                         1/4             1/4
          (%o1)       %union([x = - (- 1)   ], [x = (- 1)   ])
          (%i2) to_poly_solve(x^2= %i,x, 'simpfuncs = ['rectform]);
                                %i         1             %i         1
          (%o2) %union([x = - ------- - -------], [x = ------- + -------])
                              sqrt(2)   sqrt(2)        sqrt(2)   sqrt(2)

          Sometimes additional simplification can revert a
          simplification; for example

          (%i3) to_poly_solve(x^2=1,x);
          (%o3)              %union([x = - 1], [x = 1])
          (%i4) to_poly_solve(x^2= 1,x, 'simpfuncs = [polarform]);
                                                  %i %pi
          (%o4)            %union([x = 1], [x = %e      ]

          Maxima doesn't try to check that each member of the function
          list `l' is purely a simplification; thus

          (%i5) to_poly_solve(x^2 = %i,x, 'simpfuncs = [lambda([s],s^2)]);
          (%o5)                   %union([x = %i])

          To convert each solution to a double float, use `simpfunc =
          ['dfloat]':

          (%i6) to_poly_solve(x^3 +x + 1 = 0,x,
                              'simpfuncs = ['dfloat]), algexact : true;
          (%o6) %union([x = - .6823278038280178],
          [x = .3411639019140089 - 1.161541399997251 %i],
          [x = 1.161541399997251 %i + .3411639019140089])

        * `use_grobner = true' With this option, the function
          `poly_reduced_grobner' is applied to the equations before
          attempting their solution.  Primarily, this option provides a
          workaround for weakness in the function `algsys'.  Here is an
          example of such a workaround:

          (%i7) to_poly_solve([x^2+y^2=2^2,(x-1)^2+(y-1)^2=2^2],[x,y],
                              'use_grobner = true);
                              sqrt(7) - 1      sqrt(7) + 1
          (%o7) %union([x = - -----------, y = -----------],
                                   2                2
                                           sqrt(7) + 1        sqrt(7) - 1
                                      [x = -----------, y = - -----------])
                                                2                  2
          (%i8) to_poly_solve([x^2+y^2=2^2,(x-1)^2+(y-1)^2=2^2],[x,y]);
          (%o8)                       %union()

        * `maxdepth = k', where `k' is a positive integer.  This
          function controls the maximum recursion depth for the solver.
          The default value for `maxdepth' is five.  When the
          recursions depth is exceeded, the solver signals an error:

          (%i9) to_poly_solve(cos(x) = x,x, 'maxdepth = 2);

          Unable to solve
          Unable to solve
          (%o9)        %solve([cos(x) = x], [x], maxdepth = 2)

        * `parameters = l', where `l' is a list of symbols.  The solver
          attempts to return a solution that is valid for all members
          of the list `l'; for example:

          (%i10) to_poly_solve(a * x = x, x);
          (%o10)                   %union([x = 0])
          (%i11) to_poly_solve(a * x = x, x, 'parameters = [a]);
          (%o11) %union(%if(a - 1 = 0, [x = %c111], %union()),
                                         %if(a - 1 # 0, [x = 0], %union()))

          In `(%o2)', the solver introduced a dummy variable; to
          re-index the these dummy variables, use the function
          `nicedummies':

          (%i12) nicedummies(%);
          (%o12) %union(%if(a - 1 = 0, [x = %c0], %union()),
                                         %if(a - 1 # 0, [x = 0], %union()))

     The `to_poly_solver' uses data stored in the hashed array
     `one_to_one_reduce' to solve equations of the form f(a) = f(b).
     The assignment `one_to_one_reduce['f,'f] : lambda([a,b], a=b)'
     tells `to_poly_solver' that the solution set of f(a) = f(b) equals
     the solution set of a=b; for example

          (%i13) one_to_one_reduce['f,'f] : lambda([a,b], a=b)$

          (%i14) to_poly_solve(f(x^2-1) = f(0),x);
          (%o14)             %union([x = - 1], [x = 1])

     More generally, the assignment `one_to_one_reduce['f,'g] :
     lambda([a,b], w(a, b) = 0' tells `to_poly_solver' that the
     solution set of f(a) = f(b) equals the solution set of w(a,b) = 0;
     for example

          (%i15) one_to_one_reduce['f,'g] : lambda([a,b], a = 1 + b/2)$

          (%i16) to_poly_solve(f(x) - g(x),x);
          (%o16)                   %union([x = 2])

     Additionally, the function `to_poly_solver' uses data stored in
     the hashed array `function_inverse' to solve equations of the form
     f(a) = b.  The assignment `function_inverse['f] : lambda([s],
     g(s))' informs  `to_poly_solver' that the solution set to `f(x) =
     b' equals the solution set to `x = g(b)'; two examples:

          (%i17) function_inverse['Q] : lambda([s], P(s))$

          (%i18) to_poly_solve(Q(x-1) = 2009,x);
          (%o18)              %union([x = P(2009) + 1])
          (%i19) function_inverse['G] : lambda([s], s+new_variable(integer));
          (%o19)       lambda([s], s + new_variable(integer))
          (%i20) to_poly_solve(G(x - a) = b,x);
          (%o20)             %union([x = b + a + %z125])

     Notes

        * The solve variables needn't be symbols; when `fullratsubst' is
          able to appropriately make substitutions, the solve variables
          can be nonsymbols:

          (%i1) to_poly_solve([x^2 + y^2 + x * y = 5, x * y = 8],
                              [x^2 + y^2, x * y]);
                                            2    2
          (%o1)           %union([x y = 8, y  + x  = - 3])

        * For equations that involve complex conjugates, the solver
          automatically appends the conjugate equations; for example

          (%i1) declare(x,complex)$

          (%i2) to_poly_solve(x + (5 + %i) * conjugate(x) = 1, x);
                                             %i + 21
          (%o2)              %union([x = - -----------])
                                           25 %i - 125
          (%i3) declare(y,complex)$

          (%i4) to_poly_solve(set(conjugate(x) - y = 42 + %i,
                                  x + conjugate(y) = 0), set(x,y));
                                     %i - 42        %i + 42
          (%o4)        %union([x = - -------, y = - -------])
                                        2              2

        * For an equation that involves the absolute value function, the
          `to_poly_solver' consults the fact database to decide if the
          argument to the absolute value is complex valued.  When

          (%i1) to_poly_solve(abs(x) = 6, x);
          (%o1)              %union([x = - 6], [x = 6])
          (%i2) declare(z,complex)$

          (%i3) to_poly_solve(abs(z) = 6, z);
          (%o3) %union(%if((%c11 # 0) %and (%c11 conjugate(%c11) - 36 =
                                                 0), [z = %c11], %union()))

          This is the only situation that the solver consults the fact
          database.  If a solve variable is declared to be an integer,
          for example, `to_poly_solve' ignores this declaration.

     Relevant option variables algexact, resultant, algebraic

     Related functions to_poly

     To use `load(to_poly_solver)'

     Status: The function `to_poly_solve' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

80 unit


Introduction to Units
Functions and Variables for Units

80.1 Introduction to Units

The _unit_ package enables the user to convert between arbitrary units
and work with dimensions in equations. The functioning of this package
is radically different from the original Maxima units package - whereas
the original was a basic list of definitions, this package uses
rulesets to allow the user to chose, on a per dimension basis, what
unit final answers should be rendered in.  It will separate units
instead of intermixing them in the display, allowing the user to
readily identify the units associated with a particular answer.  It
will allow a user to simplify an expression to its fundamental Base
Units, as well as providing fine control over simplifying to derived
units.  Dimensional analysis is possible, and a variety of tools are
available to manage conversion and simplification options. In addition
to customizable automatic conversion, _units_ also provides a
traditional manual conversion option.

   Note - when unit conversions are inexact Maxima will make
approximations resulting in fractions. This is a consequence of the
techniques used to simplify units.  The messages warning of this type
of substitution are disabled by default in the case of units (normally
they are on) since this situation occurs frequently and the warnings
clutter the output.  (The existing state of ratprint is restored after
unit conversions, so user changes to that setting will be preserved
otherwise.)  If  the user needs this information for units, they can set
_unitverbose:on_ to reactivate the printing of warnings from the unit
conversion process.

   _unit_ is included in Maxima in the share/contrib/unit directory. It
obeys normal Maxima package loading conventions:

     (%i1) load("unit")$
     *******************************************************************
     *                       Units version 0.50                        *
     *          Definitions based on the NIST Reference on             *
     *              Constants, Units, and Uncertainty                  *
     *       Conversion factors from various sources including         *
     *                   NIST and the GNU units package                *
     *******************************************************************

     Redefining necessary functions...
     WARNING: DEFUN/DEFMACRO: redefining function TOPLEVEL-MACSYMA-EVAL ...
     WARNING: DEFUN/DEFMACRO: redefining function MSETCHK ...
     WARNING: DEFUN/DEFMACRO: redefining function KILL1 ...
     WARNING: DEFUN/DEFMACRO: redefining function NFORMAT ...
     Initializing unit arrays...
     Done.

   The WARNING messages are expected and not a cause for concern - they
indicate the _unit_ package is redefining functions already defined in
Maxima proper.  This is necessary in order to properly handle units.
The user should be aware that if other changes have been made to these
functions by other packages those changes will be overwritten by this
loading process.

   The _unit.mac_ file also loads a lisp file _unit-functions.lisp_
which contains the lisp functions needed for the package.

   Clifford Yapp is the primary author.  He has received valuable
assistance from Barton Willis of the University of Nebraska at Kearney
(UNK), Robert Dodier, and other intrepid folk of the Maxima mailing
list.

   There are probably lots of bugs.  Let me know.  `float' and `numer'
don't do what is expected.

   TODO : dimension functionality, handling of temperature, showabbr
and friends.  Show examples with addition of quantities containing
units.

80.2 Functions and Variables for Units

Function: setunits (<list>)

     By default, the _unit_ package does not use any derived
     dimensions, but will convert all units to the seven fundamental
     dimensions using MKS units.
          (%i2) N;
                                               kg m
          (%o2)                                ----
                                                 2
                                                s
          (%i3) dyn;
                                             1      kg m
          (%o3)                           (------) (----)
                                           100000     2
                                                     s
          (%i4) g;
                                              1
          (%o4)                             (----) (kg)
                                             1000
          (%i5) centigram*inch/minutes^2;
                                            127        kg m
          (%o5)                       (-------------) (----)
                                       1800000000000     2
                                                        s

     In some cases this is the desired behavior.  If the user wishes to
     use other units, this is achieved with the `setunits' command:
          (%i6) setunits([centigram,inch,minute]);
          (%o6)                                done
          (%i7) N;
                                      1800000000000   %in cg
          (%o7)                      (-------------) (------)
                                           127            2
                                                      %min
          (%i8) dyn;
                                         18000000   %in cg
          (%o8)                         (--------) (------)
                                           127          2
                                                    %min
          (%i9) g;
          (%o9)                             (100) (cg)
          (%i10) centigram*inch/minutes^2;
                                              %in cg
          (%o10)                              ------
                                                  2
                                              %min

     The setting of units is quite flexible.  For example, if we want to
     get back to kilograms, meters, and seconds as defaults for those
     dimensions we can do:
          (%i11) setunits([kg,m,s]);
          (%o11)                               done
          (%i12) centigram*inch/minutes^2;
                                            127        kg m
          (%o12)                      (-------------) (----)
                                       1800000000000     2
                                                        s

     Derived units are also handled by this command:
          (%i17) setunits(N);
          (%o17)                               done
          (%i18) N;
          (%o18)                                 N
          (%i19) dyn;
                                              1
          (%o19)                           (------) (N)
                                            100000
          (%i20) kg*m/s^2;
          (%o20)                                 N
          (%i21) centigram*inch/minutes^2;
                                              127
          (%o21)                        (-------------) (N)
                                         1800000000000

     Notice that the _unit_ package recognized the non MKS combination
     of mass, length, and inverse time squared as a force, and
     converted it to Newtons.  This is how Maxima works in general.
     If, for example, we prefer dyne to Newtons, we simply do the
     following:
          (%i22) setunits(dyn);
          (%o22)                               done
          (%i23) kg*m/s^2;
          (%o23)                          (100000) (dyn)
          (%i24) centigram*inch/minutes^2;
                                            127
          (%o24)                         (--------) (dyn)
                                          18000000

     To discontinue simplifying to any force, we use the uforget
     command:
          (%i26) uforget(dyn);
          (%o26)                               false
          (%i27) kg*m/s^2;
                                               kg m
          (%o27)                               ----
                                                 2
                                                s
          (%i28) centigram*inch/minutes^2;
                                            127        kg m
          (%o28)                      (-------------) (----)
                                       1800000000000     2
                                                        s
     This would have worked equally well with `uforget(N)' or
     `uforget(%force)'.

     See also `uforget'. To use this function write first
     `load("unit")'.

Function: uforget (<list>)

     By default, the _unit_ package converts all units to the seven
     fundamental dimensions using MKS units. This behavior can be
     changed with the `setunits' command. After that, the user can
     restore the default behavior for a particular dimension by means
     of the `uforget' command:
          (%i13) setunits([centigram,inch,minute]);
          (%o13)                               done
          (%i14) centigram*inch/minutes^2;
                                              %in cg
          (%o14)                              ------
                                                  2
                                              %min
          (%i15) uforget([cg,%in,%min]);
          (%o15)                      [false, false, false]
          (%i16) centigram*inch/minutes^2;
                                            127        kg m
          (%o16)                      (-------------) (----)
                                       1800000000000     2
                                                        s

     `uforget' operates on dimensions, not units, so any unit of a
     particular dimension will work.  The dimension itself is also a
     legal argument.

     See also `setunits'. To use this function write first
     `load("unit")'.

Function: convert (<expr>, <list>)

     When resetting the global environment is overkill, there is the
     `convert' command, which allows one time conversions.  It can
     accept either a single argument or a list of units to use in
     conversion.  When a convert operation is done, the normal global
     evaluation system is bypassed, in order to avoid the desired
     result being converted again.  As a consequence, for inexact
     calculations "rat" warnings will be visible if the global
     environment controlling this behavior (`ratprint') is true.  This
     is also useful for spot-checking the accuracy of a global
     conversion.  Another feature is `convert' will allow a user to do
     Base Dimension conversions even if the global environment is set to
     simplify to a Derived Dimension.

          (%i2) kg*m/s^2;
                                               kg m
          (%o2)                                ----
                                                 2
                                                s
          (%i3) convert(kg*m/s^2,[g,km,s]);
                                               g km
          (%o3)                                ----
                                                 2
                                                s
          (%i4) convert(kg*m/s^2,[g,inch,minute]);

          `rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
                                        18000000000   %in g
          (%o4)                        (-----------) (-----)
                                            127           2
                                                      %min
          (%i5) convert(kg*m/s^2,[N]);
          (%o5)                                  N
          (%i6) convert(kg*m^2/s^2,[N]);
          (%o6)                                 m N
          (%i7) setunits([N,J]);
          (%o7)                                done
          (%i8) convert(kg*m^2/s^2,[N]);
          (%o8)                                 m N
          (%i9) convert(kg*m^2/s^2,[N,inch]);

          `rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
                                           5000
          (%o9)                           (----) (%in N)
                                           127
          (%i10) convert(kg*m^2/s^2,[J]);
          (%o10)                                 J
          (%i11) kg*m^2/s^2;
          (%o11)                                 J
          (%i12) setunits([g,inch,s]);
          (%o12)                               done
          (%i13) kg*m/s^2;
          (%o13)                                 N
          (%i14) uforget(N);
          (%o14)                               false
          (%i15) kg*m/s^2;
                                          5000000   %in g
          (%o15)                         (-------) (-----)
                                            127       2
                                                     s
          (%i16) convert(kg*m/s^2,[g,inch,s]);

          `rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
                                          5000000   %in g
          (%o16)                         (-------) (-----)
                                            127       2
                                                     s

     See also `setunits' and `uforget'. To use this function write
     first `load("unit")'.

Optional variable: usersetunits

     Default value: none

     If a user wishes to have a default unit behavior other than that
     described, they can make use of _maxima-init.mac_ and the
     _usersetunits_ variable.  The _unit_ package will check on startup
     to see if this variable has been assigned a list.  If it has, it
     will use setunits on that list and take the units from that list
     to be defaults.  `uforget' will revert to the behavior defined by
     usersetunits over its own defaults.  For example, if we have a
     _maxima-init.mac_ file containing:
          usersetunits : [N,J];
     we would see the following behavior:
          (%i1) load("unit")$
          *******************************************************************
          *                       Units version 0.50                        *
          *          Definitions based on the NIST Reference on             *
          *              Constants, Units, and Uncertainty                  *
          *       Conversion factors from various sources including         *
          *                   NIST and the GNU units package                *
          *******************************************************************

          Redefining necessary functions...
          WARNING: DEFUN/DEFMACRO: redefining function
           TOPLEVEL-MACSYMA-EVAL ...
          WARNING: DEFUN/DEFMACRO: redefining function MSETCHK ...
          WARNING: DEFUN/DEFMACRO: redefining function KILL1 ...
          WARNING: DEFUN/DEFMACRO: redefining function NFORMAT ...
          Initializing unit arrays...
          Done.
          User defaults found...
          User defaults initialized.
          (%i2) kg*m/s^2;
          (%o2)                                  N
          (%i3) kg*m^2/s^2;
          (%o3)                                  J
          (%i4) kg*m^3/s^2;
          (%o4)                                 J m
          (%i5) kg*m*km/s^2;
          (%o5)                             (1000) (J)
          (%i6) setunits([dyn,eV]);
          (%o6)                                done
          (%i7) kg*m/s^2;
          (%o7)                           (100000) (dyn)
          (%i8) kg*m^2/s^2;
          (%o8)                     (6241509596477042688) (eV)
          (%i9) kg*m^3/s^2;
          (%o9)                    (6241509596477042688) (eV m)
          (%i10) kg*m*km/s^2;
          (%o10)                   (6241509596477042688000) (eV)
          (%i11) uforget([dyn,eV]);
          (%o11)                           [false, false]
          (%i12) kg*m/s^2;
          (%o12)                                 N
          (%i13) kg*m^2/s^2;
          (%o13)                                 J
          (%i14) kg*m^3/s^2;
          (%o14)                                J m
          (%i15) kg*m*km/s^2;
          (%o15)                            (1000) (J)
     Without `usersetunits', the initial inputs would have been
     converted to MKS, and uforget would have resulted in a return to
     MKS rules.  Instead, the user preferences are respected in both
     cases.  Notice these can still be overridden if desired.  To
     completely eliminate this simplification - i.e.  to have the user
     defaults reset to factory defaults - the `dontusedimension'
     command can be used.  `uforget' can restore user settings again,
     but only if `usedimension' frees it for use.  Alternately,
     `kill(usersetunits)' will completely remove all knowledge of the
     user defaults from the session.  Here are some examples of how
     these various options work.
          (%i2) kg*m/s^2;
          (%o2)                                  N
          (%i3) kg*m^2/s^2;
          (%o3)                                  J
          (%i4) setunits([dyn,eV]);
          (%o4)                                done
          (%i5) kg*m/s^2;
          (%o5)                           (100000) (dyn)
          (%i6) kg*m^2/s^2;
          (%o6)                     (6241509596477042688) (eV)
          (%i7) uforget([dyn,eV]);
          (%o7)                          [false, false]
          (%i8) kg*m/s^2;
          (%o8)                                  N
          (%i9) kg*m^2/s^2;
          (%o9)                                  J
          (%i10) dontusedimension(N);
          (%o10)                             [%force]
          (%i11) dontusedimension(J);
          (%o11)                         [%energy, %force]
          (%i12) kg*m/s^2;
                                               kg m
          (%o12)                               ----
                                                 2
                                                s
          (%i13) kg*m^2/s^2;
                                                   2
                                               kg m
          (%o13)                               -----
                                                 2
                                                s
          (%i14) setunits([dyn,eV]);
          (%o14)                               done
          (%i15) kg*m/s^2;
                                               kg m
          (%o15)                               ----
                                                 2
                                                s
          (%i16) kg*m^2/s^2;
                                                   2
                                               kg m
          (%o16)                               -----
                                                 2
                                                s
          (%i17) uforget([dyn,eV]);
          (%o17)                         [false, false]
          (%i18) kg*m/s^2;
                                               kg m
          (%o18)                               ----
                                                 2
                                                s
          (%i19) kg*m^2/s^2;
                                                   2
                                               kg m
          (%o19)                               -----
                                                 2
                                                s
          (%i20) usedimension(N);
          Done.  To have Maxima simplify to this dimension, use
          setunits([unit]) to select a unit.
          (%o20)                               true
          (%i21) usedimension(J);
          Done.  To have Maxima simplify to this dimension, use
          setunits([unit]) to select a unit.
          (%o21)                               true
          (%i22) kg*m/s^2;
                                               kg m
          (%o22)                               ----
                                                 2
                                                s
          (%i23) kg*m^2/s^2;
                                                   2
                                               kg m
          (%o23)                               -----
                                                 2
                                                s
          (%i24) setunits([dyn,eV]);
          (%o24)                               done
          (%i25) kg*m/s^2;
          (%o25)                          (100000) (dyn)
          (%i26) kg*m^2/s^2;
          (%o26)                    (6241509596477042688) (eV)
          (%i27) uforget([dyn,eV]);
          (%o27)                           [false, false]
          (%i28) kg*m/s^2;
          (%o28)                                 N
          (%i29) kg*m^2/s^2;
          (%o29)                                 J
          (%i30) kill(usersetunits);
          (%o30)                               done
          (%i31) uforget([dyn,eV]);
          (%o31)                          [false, false]
          (%i32) kg*m/s^2;
                                               kg m
          (%o32)                               ----
                                                 2
                                                s
          (%i33) kg*m^2/s^2;
                                                   2
                                               kg m
          (%o33)                               -----
                                                 2
                                                s
     Unfortunately this wide variety of options is a little confusing
     at first, but once the user grows used to them they should find
     they have very full control over their working environment.

Function: metricexpandall (<x>)

     Rebuilds global unit lists automatically creating all desired
     metric units.  <x> is a numerical argument which is used to
     specify how many metric prefixes the user wishes defined.  The
     arguments are as follows, with each higher number defining all
     lower numbers' units:
                     0 - none. Only base units
                     1 - kilo, centi, milli
          (default)  2 - giga, mega, kilo, hecto, deka, deci, centi, milli,
                         micro, nano
                     3 - peta, tera, giga, mega, kilo, hecto, deka, deci,
                         centi, milli, micro, nano, pico, femto
                     4 - all
     Normally, Maxima will not define the full expansion since this
     results in a very large number of units, but `metricexpandall' can
     be used to rebuild the list in a more or less complete fashion.
     The relevant variable in the _unit.mac_ file is <%unitexpand>.

Variable: %unitexpand

     Default value: `2'

     This is the value supplied to `metricexpandall' during the initial
     loading of _unit_.

81 zeilberger


Introduction to zeilberger
Functions and Variables for zeilberger

81.1 Introduction to zeilberger

`zeilberger' is a implementation of Zeilberger's algorithm for definite
hypergeometric summation, and also Gosper's algorithm for indefinite
hypergeometric summation.

   `zeilberger' makes use of the "filtering" optimization method
developed by Axel Riese.

   `zeilberger' was developed by Fabrizio Caruso.

   `load (zeilberger)' loads this package.

81.2 Functions and Variables for zeilberger

Function: AntiDifference (F_k, <k>)

     Returns the hypergeometric anti-difference of F_k, if it exists.
     Otherwise `AntiDifference' returns `no_hyp_antidifference'.

Function: Gosper (F_k, <k>)

     Returns the rational certificate R(k) for F_k, that is, a rational
     function such that F_k = R(k+1) F_(k+1) - R(k) F_k, if it exists.
     Otherwise, `Gosper' returns `no_hyp_sol'.

Function: GosperSum (F_k, <k>, <a>, <b>)

     Returns the summmation of F_k from <k> = <a> to <k> = <b> if F_k
     has a hypergeometric anti-difference.  Otherwise, `GosperSum'
     returns `nongosper_summable'.

     Examples:

          (%i1) load (zeilberger)$
          (%i2) GosperSum ((-1)^k*k / (4*k^2 - 1), k, 1, n);
          Dependent equations eliminated:  (1)
                                     3       n + 1
                                (n + -) (- 1)
                                     2               1
          (%o2)               - ------------------ - -
                                            2        4
                                2 (4 (n + 1)  - 1)
          (%i3) GosperSum (1 / (4*k^2 - 1), k, 1, n);
                                          3
                                    - n - -
                                          2       1
          (%o3)                  -------------- + -
                                          2       2
                                 4 (n + 1)  - 1
          (%i4) GosperSum (x^k, k, 1, n);
                                    n + 1
                                   x          x
          (%o4)                    ------ - -----
                                   x - 1    x - 1
          (%i5) GosperSum ((-1)^k*a! / (k!*(a - k)!), k, 1, n);
                                          n + 1
                          a! (n + 1) (- 1)              a!
          (%o5)       - ------------------------- - ----------
                        a (- n + a - 1)! (n + 1)!   a (a - 1)!
          (%i6) GosperSum (k*k!, k, 1, n);
          Dependent equations eliminated:  (1)
          (%o6)                     (n + 1)! - 1
          (%i7) GosperSum ((k + 1)*k! / (k + 1)!, k, 1, n);
                            (n + 1) (n + 2) (n + 1)!
          (%o7)             ------------------------ - 1
                                    (n + 2)!
          (%i8) GosperSum (1 / ((a - k)!*k!), k, 1, n);
          (%o8)                  NON_GOSPER_SUMMABLE

Function: parGosper (F_(n,k), <k>, <n>, <d>)

     Attempts to find a <d>-th order recurrence for F_(n,k).

     The algorithm yields a sequence [s_1, s_2, ..., s_m] of solutions.
     Each solution has the form

     [R(n, k), [a_0, a_1, ..., a_d]].

     `parGosper' returns `[]' if it fails to find a recurrence.

Function: Zeilberger (F_(n,k), <k>, <n>)

     Attempts to compute the indefinite hypergeometric summation of
     F_(n,k).

     `Zeilberger' first invokes `Gosper', and if that fails to find a
     solution, then invokes `parGosper' with order 1, 2, 3, ..., up to
     `MAX_ORD'.  If Zeilberger finds a solution before reaching
     `MAX_ORD', it stops and returns the solution.

     The algorithms yields a sequence [s_1, s_2, ..., s_m] of solutions.
     Each solution has the form

     [R(n,k), [a_0, a_1, ..., a_d]].

     `Zeilberger' returns `[]' if it fails to find a solution.

     `Zeilberger' invokes `Gosper' only if `Gosper_in_Zeilberger' is
     `true'.

81.3 General global variables

Global variable: MAX_ORD

     Default value: 5

     `MAX_ORD' is the maximum recurrence order attempted by
     `Zeilberger'.

Global variable: simplified_output

     Default value: `false'

     When `simplified_output' is `true', functions in the `zeilberger'
     package attempt further simplification of the solution.

Global variable: linear_solver

     Default value: `linsolve'

     `linear_solver' names the solver which is used to solve the system
     of equations in Zeilberger's algorithm.

Global variable: warnings

     Default value: `true'

     When `warnings' is `true', functions in the `zeilberger' package
     print warning messages during execution.

Global variable: Gosper_in_Zeilberger

     Default value: `true'

     When `Gosper_in_Zeilberger' is `true', the `Zeilberger' function
     calls `Gosper' before calling `parGosper'.  Otherwise,
     `Zeilberger' goes immediately to `parGosper'.

Global variable: trivial_solutions

     Default value: `true'

     When `trivial_solutions' is `true', `Zeilberger' returns solutions
     which have certificate equal to zero, or all coefficients equal to
     zero.

81.4 Variables related to the modular test

Global variable: mod_test

     Default value: `false'

     When `mod_test' is `true', `parGosper' executes a modular test for
     discarding systems with no solutions.

Global variable: modular_linear_solver

     Default value: `linsolve'

     `modular_linear_solver' names the linear solver used by the
     modular test in `parGosper'.

Global variable: ev_point

     Default value: `big_primes[10]'

     `ev_point' is the value at which the variable <n> is evaluated
     when executing the modular test in `parGosper'.

Global variable: mod_big_prime

     Default value: `big_primes[1]'

     `mod_big_prime' is the modulus used by the modular test in
     `parGosper'.

Global variable: mod_threshold

     Default value: 4

     `mod_threshold' is the greatest order for which the modular test
     in `parGosper' is attempted.

82 Indices

Appendix A Function and Variable Index
**************************************

[index]

* !:                                     Combinatorial Functions.
                                                             (line   58)
* !!:                                    Combinatorial Functions.
                                                             (line    7)
* #:                                     Operators for Equations.
                                                             (line    7)
* %:                                     Functions and Variables for Command Line.
                                                             (line   92)
* %%:                                    Functions and Variables for Command Line.
                                                             (line  106)
* %and:                                  Functions and Variables for to_poly_solve.
                                                             (line   16)
* %c:                                    Functions and Variables for contrib_ode.
                                                             (line   76)
* %e:                                    Functions and Variables for Constants.
                                                             (line    7)
* %e_to_numlog:                          Root Exponential and Logarithmic Functions.
                                                             (line    7)
* %edispflag:                            Functions and Variables for Display.
                                                             (line    7)
* %emode:                                Root Exponential and Logarithmic Functions.
                                                             (line   17)
* %enumer:                               Root Exponential and Logarithmic Functions.
                                                             (line   34)
* %f:                                    Hypergeometric Functions.
                                                             (line   22)
* %gamma:                                Functions and Variables for Constants.
                                                             (line   22)
* %i:                                    Functions and Variables for Constants.
                                                             (line   13)
* %iargs:                                Functions and Variables for Trigonometric.
                                                             (line   54)
* %if:                                   Functions and Variables for to_poly_solve.
                                                             (line   91)
* %k1:                                   Functions and Variables for contrib_ode.
                                                             (line   80)
* %k2:                                   Functions and Variables for contrib_ode.
                                                             (line   84)
* %m:                                    Hypergeometric Functions.
                                                             (line   13)
* %or:                                   Functions and Variables for to_poly_solve.
                                                             (line  134)
* %phi:                                  Functions and Variables for Constants.
                                                             (line   49)
* %pi:                                   Functions and Variables for Constants.
                                                             (line  105)
* %piargs:                               Functions and Variables for Trigonometric.
                                                             (line    7)
* %rnum_list:                            Functions and Variables for Equations.
                                                             (line    7)
* %s:                                    Bessel Functions.   (line  174)
* %th:                                   Functions and Variables for Command Line.
                                                             (line  153)
* %unitexpand:                           Functions and Variables for Units.
                                                             (line  414)
* %w:                                    Hypergeometric Functions.
                                                             (line   18)
* ':                                     Functions and Variables for Evaluation.
                                                             (line    7)
* '':                                    Functions and Variables for Evaluation.
                                                             (line  102)
* *:                                     Arithmetic operators.
                                                             (line    9)
* **:                                    Arithmetic operators.
                                                             (line  167)
* +:                                     Arithmetic operators.
                                                             (line    7)
* -:                                     Arithmetic operators.
                                                             (line    8)
* .:                                     Arithmetic operators.
                                                             (line  209)
* /:                                     Arithmetic operators.
                                                             (line   10)
* ::                                     Assignment operators.
                                                             (line    7)
* :::                                    Assignment operators.
                                                             (line   90)
* ::=:                                   Assignment operators.
                                                             (line  116)
* :=:                                    Assignment operators.
                                                             (line  190)
* <:                                     Relational operators.
                                                             (line    7)
* <=:                                    Relational operators.
                                                             (line    8)
* =:                                     Operators for Equations.
                                                             (line   31)
* >:                                     Relational operators.
                                                             (line   10)
* >=:                                    Relational operators.
                                                             (line    9)
* ?:                                     Functions and Variables for Command Line.
                                                             (line  183)
* ??:                                    Functions and Variables for Command Line.
                                                             (line  196)
* @:                                     Functions and Variables for Structures.
                                                             (line   71)
* [:                                     Functions and Variables for Lists.
                                                             (line    7)
* \\:                                    Introduction to Strings.
                                                             (line    6)
* ]:                                     Functions and Variables for Lists.
                                                             (line    8)
* ^:                                     Arithmetic operators.
                                                             (line   11)
* ^^:                                    Arithmetic operators.
                                                             (line  190)
* _:                                     Functions and Variables for Command Line.
                                                             (line   47)
* __:                                    Functions and Variables for Command Line.
                                                             (line    7)
* `:                                     Functions and Variables for ezunits.
                                                             (line    7)
* abasep:                                Functions and Variables for atensor.
                                                             (line  136)
* abs:                                   Functions for Numbers.
                                                             (line    7)
* absboxchar:                            Functions and Variables for Display.
                                                             (line   27)
* absint:                                Functions and Variables for Fourier series.
                                                             (line   32)
* absolute_real_time:                    Functions and Variables for Runtime Environment.
                                                             (line  146)
* acos:                                  Functions and Variables for Trigonometric.
                                                             (line   84)
* acosh:                                 Functions and Variables for Trigonometric.
                                                             (line   88)
* acot:                                  Functions and Variables for Trigonometric.
                                                             (line   92)
* acoth:                                 Functions and Variables for Trigonometric.
                                                             (line   96)
* acsc:                                  Functions and Variables for Trigonometric.
                                                             (line  100)
* acsch:                                 Functions and Variables for Trigonometric.
                                                             (line  104)
* activate:                              Functions and Variables for Facts.
                                                             (line    7)
* activecontexts:                        Functions and Variables for Facts.
                                                             (line   17)
* adapt_depth <1>:                       Functions and Variables for draw.
                                                             (line  195)
* adapt_depth:                           Plotting Options.   (line   15)
* add_edge:                              Functions and Variables for graphs.
                                                             (line 1235)
* add_edges:                             Functions and Variables for graphs.
                                                             (line 1249)
* add_vertex:                            Functions and Variables for graphs.
                                                             (line 1264)
* add_vertices:                          Functions and Variables for graphs.
                                                             (line 1279)
* addcol:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line    7)
* Addition:                              Arithmetic operators.
                                                             (line    7)
* additive:                              Functions and Variables for Simplification.
                                                             (line    7)
* addmatrices:                           Functions and Variables for linearalgebra.
                                                             (line    7)
* addrow:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line   12)
* adim:                                  Functions and Variables for atensor.
                                                             (line   76)
* adjacency_matrix:                      Functions and Variables for graphs.
                                                             (line  361)
* adjoin:                                Functions and Variables for Sets.
                                                             (line    7)
* adjoint:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line   17)
* af:                                    Functions and Variables for atensor.
                                                             (line  104)
* aform:                                 Functions and Variables for atensor.
                                                             (line   84)
* agd:                                   Package functs.     (line  167)
* airy_ai:                               Airy Functions.     (line   16)
* airy_bi:                               Airy Functions.     (line   30)
* airy_dai:                              Airy Functions.     (line   24)
* airy_dbi:                              Airy Functions.     (line   38)
* alg_type:                              Functions and Variables for atensor.
                                                             (line   71)
* algebraic:                             Functions and Variables for Polynomials.
                                                             (line    7)
* algepsilon:                            Functions and Variables for Equations.
                                                             (line   31)
* algexact:                              Functions and Variables for Equations.
                                                             (line   37)
* algsys:                                Functions and Variables for Equations.
                                                             (line   53)
* alias:                                 Functions and Variables for Expressions.
                                                             (line    8)
* aliases:                               Functions and Variables for Expressions.
                                                             (line   13)
* all_dotsimp_denoms:                    Functions and Variables for Affine.
                                                             (line  143)
* allbut:                                Functions and Variables for Expressions.
                                                             (line   21)
* allroots:                              Functions and Variables for Equations.
                                                             (line  156)
* allsym:                                Functions and Variables for itensor.
                                                             (line  525)
* alphabetic:                            Functions and Variables for Properties.
                                                             (line    7)
* alphacharp:                            Functions and Variables for characters.
                                                             (line    7)
* alphanumericp:                         Functions and Variables for characters.
                                                             (line   11)
* amortization:                          Functions and Variables for finance.
                                                             (line  106)
* and:                                   Logical operators.  (line    7)
* annuity_fv:                            Functions and Variables for finance.
                                                             (line   67)
* annuity_pv:                            Functions and Variables for finance.
                                                             (line   54)
* antid:                                 Functions and Variables for Differentiation.
                                                             (line    7)
* antidiff:                              Functions and Variables for Differentiation.
                                                             (line   61)
* AntiDifference:                        Functions and Variables for zeilberger.
                                                             (line    7)
* antisymmetric:                         Functions and Variables for Simplification.
                                                             (line   31)
* append:                                Functions and Variables for Lists.
                                                             (line   41)
* appendfile:                            Functions and Variables for File Input and Output.
                                                             (line    7)
* apply:                                 Functions and Variables for Function Definition.
                                                             (line    7)
* apply1:                                Functions and Variables for Rules and Patterns.
                                                             (line    7)
* apply2:                                Functions and Variables for Rules and Patterns.
                                                             (line   22)
* applyb1:                               Functions and Variables for Rules and Patterns.
                                                             (line   35)
* apropos:                               Functions and Variables for Help.
                                                             (line    7)
* args:                                  Functions and Variables for Expressions.
                                                             (line   52)
* arit_amortization:                     Functions and Variables for finance.
                                                             (line  131)
* arithmetic:                            Package functs.     (line  119)
* arithsum:                              Package functs.     (line  140)
* array:                                 Functions and Variables for Arrays.
                                                             (line    9)
* arrayapply:                            Functions and Variables for Arrays.
                                                             (line   36)
* arrayinfo:                             Functions and Variables for Arrays.
                                                             (line   44)
* arraymake:                             Functions and Variables for Arrays.
                                                             (line  142)
* arrays:                                Functions and Variables for Arrays.
                                                             (line  187)
* ascii:                                 Functions and Variables for characters.
                                                             (line   15)
* asec:                                  Functions and Variables for Trigonometric.
                                                             (line  108)
* asech:                                 Functions and Variables for Trigonometric.
                                                             (line  112)
* asin:                                  Functions and Variables for Trigonometric.
                                                             (line  116)
* asinh:                                 Functions and Variables for Trigonometric.
                                                             (line  120)
* askexp:                                Functions and Variables for Miscellaneous Options.
                                                             (line    7)
* askinteger:                            Functions and Variables for Facts.
                                                             (line   28)
* asksign:                               Functions and Variables for Facts.
                                                             (line   40)
* Assignment operator:                   Assignment operators.
                                                             (line    7)
* Assignment operator (evaluates left-hand side): Assignment operators.
                                                             (line   90)
* assoc:                                 Functions and Variables for Lists.
                                                             (line   50)
* assoc_legendre_p:                      Functions and Variables for orthogonal polynomials.
                                                             (line    7)
* assoc_legendre_q:                      Functions and Variables for orthogonal polynomials.
                                                             (line   15)
* assume:                                Functions and Variables for Facts.
                                                             (line   49)
* assume_external_byte_order:            Functions and Variables for binary input and output.
                                                             (line    7)
* assume_pos:                            Functions and Variables for Facts.
                                                             (line  134)
* assume_pos_pred:                       Functions and Variables for Facts.
                                                             (line  159)
* assumescalar:                          Functions and Variables for Facts.
                                                             (line  110)
* asymbol:                               Functions and Variables for atensor.
                                                             (line   91)
* asympa:                                Introduction to asympa.
                                                             (line    7)
* at:                                    Functions and Variables for Differentiation.
                                                             (line  112)
* atan:                                  Functions and Variables for Trigonometric.
                                                             (line  124)
* atan2:                                 Functions and Variables for Trigonometric.
                                                             (line  128)
* atanh:                                 Functions and Variables for Trigonometric.
                                                             (line  133)
* atensimp:                              Functions and Variables for atensor.
                                                             (line   62)
* atom:                                  Functions and Variables for Expressions.
                                                             (line   65)
* atomgrad:                              Functions and Variables for Differentiation.
                                                             (line  157)
* atrig1:                                Functions and Variables for Trigonometric.
                                                             (line  137)
* atvalue:                               Functions and Variables for Differentiation.
                                                             (line  162)
* augcoefmatrix:                         Functions and Variables for Matrices and Linear Algebra.
                                                             (line   23)
* augmented_lagrangian_method:           Functions and Variables for augmented_lagrangian.
                                                             (line   13)
* av:                                    Functions and Variables for atensor.
                                                             (line  111)
* average_degree:                        Functions and Variables for graphs.
                                                             (line  377)
* axes:                                  Plotting Options.   (line   22)
* axis_3d:                               Functions and Variables for draw.
                                                             (line  204)
* axis_bottom:                           Functions and Variables for draw.
                                                             (line  223)
* axis_left:                             Functions and Variables for draw.
                                                             (line  240)
* axis_right:                            Functions and Variables for draw.
                                                             (line  257)
* axis_top:                              Functions and Variables for draw.
                                                             (line  274)
* azimuth:                               Plotting Options.   (line   32)
* background_color:                      Functions and Variables for draw.
                                                             (line  291)
* backslash:                             Introduction to Strings.
                                                             (line    6)
* backsubst:                             Functions and Variables for Equations.
                                                             (line  224)
* backtrace:                             Functions and Variables for Program Flow.
                                                             (line    8)
* bars:                                  Functions and Variables for draw.
                                                             (line 2590)
* barsplot:                              Functions and Variables for statistical graphs.
                                                             (line    8)
* barsplot_description:                  Functions and Variables for statistical graphs.
                                                             (line    9)
* Base of natural logarithm:             Functions and Variables for Constants.
                                                             (line    7)
* bashindices:                           Sums and Products.  (line    7)
* batch:                                 Functions and Variables for File Input and Output.
                                                             (line   16)
* batchload:                             Functions and Variables for File Input and Output.
                                                             (line   68)
* bc2:                                   Functions and Variables for Differential Equations.
                                                             (line    7)
* bdvac:                                 Functions and Variables for ctensor.
                                                             (line  801)
* belln:                                 Functions and Variables for Sets.
                                                             (line   25)
* benefit_cost:                          Functions and Variables for finance.
                                                             (line  241)
* berlefact:                             Functions and Variables for Polynomials.
                                                             (line   14)
* bern:                                  Functions and Variables for Number Theory.
                                                             (line    7)
* bernpoly:                              Functions and Variables for Number Theory.
                                                             (line   25)
* bernstein_approx:                      Functions and Variables for Bernstein.
                                                             (line   76)
* bernstein_expand:                      Functions and Variables for Bernstein.
                                                             (line  103)
* bernstein_explicit:                    Functions and Variables for Bernstein.
                                                             (line   50)
* bernstein_poly:                        Functions and Variables for Bernstein.
                                                             (line    7)
* bessel_i:                              Bessel Functions.   (line   35)
* bessel_j:                              Bessel Functions.   (line    7)
* bessel_k:                              Bessel Functions.   (line   51)
* bessel_y:                              Bessel Functions.   (line   23)
* besselexpand:                          Bessel Functions.   (line  133)
* beta:                                  Gamma and factorial Functions.
                                                             (line  173)
* beta_args_sum_to_integer:              Gamma and factorial Functions.
                                                             (line  665)
* beta_expand:                           Gamma and factorial Functions.
                                                             (line  658)
* beta_incomplete:                       Gamma and factorial Functions.
                                                             (line  288)
* beta_incomplete_generalized:           Gamma and factorial Functions.
                                                             (line  536)
* beta_incomplete_regularized:           Gamma and factorial Functions.
                                                             (line  437)
* bezout:                                Functions and Variables for Polynomials.
                                                             (line   22)
* bf_find_root:                          Functions and Variables for fast Fourier transform.
                                                             (line  236)
* bf_fmin_cobyla:                        Functions and Variables for cobyla.
                                                             (line   72)
* bfallroots:                            Functions and Variables for Equations.
                                                             (line  214)
* bffac:                                 Gamma and factorial Functions.
                                                             (line   11)
* bfhzeta:                               Functions and Variables for Number Theory.
                                                             (line   35)
* bfloat:                                Functions and Variables for Numbers.
                                                             (line    7)
* bfloatp:                               Functions and Variables for Numbers.
                                                             (line   17)
* bfpsi:                                 Gamma and factorial Functions.
                                                             (line   17)
* bfpsi0:                                Gamma and factorial Functions.
                                                             (line   18)
* bftorat:                               Functions and Variables for Numbers.
                                                             (line   21)
* bftrunc:                               Functions and Variables for Numbers.
                                                             (line   49)
* bfzeta:                                Functions and Variables for Number Theory.
                                                             (line   29)
* biconnected_components:                Functions and Variables for graphs.
                                                             (line  388)
* bimetric:                              Functions and Variables for ctensor.
                                                             (line  825)
* bindtest:                              Functions and Variables for Properties.
                                                             (line   27)
* binomial:                              Combinatorial Functions.
                                                             (line   24)
* bipartition:                           Functions and Variables for graphs.
                                                             (line  404)
* block:                                 Functions and Variables for Function Definition.
                                                             (line   67)
* blockmatrixp:                          Functions and Variables for linearalgebra.
                                                             (line   22)
* bode_gain:                             Functions and Variables for bode.
                                                             (line    7)
* bode_phase:                            Functions and Variables for bode.
                                                             (line   53)
* border:                                Functions and Variables for draw.
                                                             (line  300)
* bothcoef:                              Functions and Variables for Polynomials.
                                                             (line   40)
* boundaries_array:                      Functions and Variables for worldmap.
                                                             (line   12)
* box <1>:                               Functions and Variables for Expressions.
                                                             (line   72)
* box:                                   Plotting Options.   (line   45)
* boxchar:                               Functions and Variables for Expressions.
                                                             (line  121)
* boxplot:                               Functions and Variables for statistical graphs.
                                                             (line  145)
* boxplot_description:                   Functions and Variables for statistical graphs.
                                                             (line  146)
* break:                                 Functions and Variables for Function Definition.
                                                             (line  119)
* breakup:                               Functions and Variables for Equations.
                                                             (line  243)
* bug_report:                            Functions and Variables for Bug Detection and Reporting.
                                                             (line   65)
* build_info:                            Functions and Variables for Bug Detection and Reporting.
                                                             (line   76)
* buildq:                                Macros.             (line    7)
* burn:                                  Functions and Variables for Number Theory.
                                                             (line   53)
* cabs:                                  Functions for Complex Numbers.
                                                             (line    7)
* canform:                               Functions and Variables for itensor.
                                                             (line  588)
* canten:                                Functions and Variables for itensor.
                                                             (line  505)
* cardinality:                           Functions and Variables for Sets.
                                                             (line   52)
* carg:                                  Functions for Complex Numbers.
                                                             (line   73)
* cartan:                                Functions and Variables for Differentiation.
                                                             (line  219)
* cartesian_product:                     Functions and Variables for Sets.
                                                             (line   70)
* catch:                                 Functions and Variables for Function Definition.
                                                             (line  125)
* cauchy_matrix:                         Functions and Variables for Matrices and Linear Algebra.
                                                             (line   39)
* cauchysum:                             Functions and Variables for Series.
                                                             (line    7)
* cbffac:                                Gamma and factorial Functions.
                                                             (line   27)
* cbrange:                               Functions and Variables for draw.
                                                             (line  320)
* cbtics:                                Functions and Variables for draw.
                                                             (line  348)
* cdf_bernoulli:                         Functions and Variables for discrete distributions.
                                                             (line  257)
* cdf_beta:                              Functions and Variables for continuous distributions.
                                                             (line  902)
* cdf_binomial:                          Functions and Variables for discrete distributions.
                                                             (line  112)
* cdf_cauchy:                            Functions and Variables for continuous distributions.
                                                             (line 1448)
* cdf_chi2:                              Functions and Variables for continuous distributions.
                                                             (line  303)
* cdf_continuous_uniform:                Functions and Variables for continuous distributions.
                                                             (line  969)
* cdf_discrete_uniform:                  Functions and Variables for discrete distributions.
                                                             (line  428)
* cdf_exp:                               Functions and Variables for continuous distributions.
                                                             (line  602)
* cdf_f:                                 Functions and Variables for continuous distributions.
                                                             (line  508)
* cdf_gamma:                             Functions and Variables for continuous distributions.
                                                             (line  829)
* cdf_general_finite_discrete:           Functions and Variables for discrete distributions.
                                                             (line   26)
* cdf_geometric:                         Functions and Variables for discrete distributions.
                                                             (line  370)
* cdf_gumbel:                            Functions and Variables for continuous distributions.
                                                             (line 1478)
* cdf_hypergeometric:                    Functions and Variables for discrete distributions.
                                                             (line  496)
* cdf_laplace:                           Functions and Variables for continuous distributions.
                                                             (line 1390)
* cdf_logistic:                          Functions and Variables for continuous distributions.
                                                             (line 1031)
* cdf_lognormal:                         Functions and Variables for continuous distributions.
                                                             (line  753)
* cdf_negative_binomial:                 Functions and Variables for discrete distributions.
                                                             (line  568)
* cdf_noncentral_chi2:                   Functions and Variables for continuous distributions.
                                                             (line  446)
* cdf_noncentral_student_t:              Functions and Variables for continuous distributions.
                                                             (line  192)
* cdf_normal:                            Functions and Variables for continuous distributions.
                                                             (line   13)
* cdf_pareto:                            Functions and Variables for continuous distributions.
                                                             (line 1089)
* cdf_poisson:                           Functions and Variables for discrete distributions.
                                                             (line  181)
* cdf_rank_sum:                          Functions and Variables for special distributions.
                                                             (line   31)
* cdf_rayleigh:                          Functions and Variables for continuous distributions.
                                                             (line 1219)
* cdf_signed_rank:                       Functions and Variables for special distributions.
                                                             (line   15)
* cdf_student_t:                         Functions and Variables for continuous distributions.
                                                             (line   93)
* cdf_weibull:                           Functions and Variables for continuous distributions.
                                                             (line 1147)
* cdisplay:                              Functions and Variables for ctensor.
                                                             (line  851)
* ceiling:                               Functions for Numbers.
                                                             (line   95)
* central_moment:                        Functions and Variables for descriptive statistics.
                                                             (line  143)
* cequal:                                Functions and Variables for characters.
                                                             (line   26)
* cequalignore:                          Functions and Variables for characters.
                                                             (line   30)
* cf:                                    Functions and Variables for Number Theory.
                                                             (line   74)
* cfdisrep:                              Functions and Variables for Number Theory.
                                                             (line  140)
* cfexpand:                              Functions and Variables for Number Theory.
                                                             (line  157)
* cflength:                              Functions and Variables for Number Theory.
                                                             (line  174)
* cframe_flag:                           Functions and Variables for ctensor.
                                                             (line 1100)
* cgeodesic:                             Functions and Variables for ctensor.
                                                             (line  794)
* cgreaterp:                             Functions and Variables for characters.
                                                             (line   34)
* cgreaterpignore:                       Functions and Variables for characters.
                                                             (line   39)
* changename:                            Functions and Variables for itensor.
                                                             (line   24)
* changevar:                             Functions and Variables for Integration.
                                                             (line    7)
* chaosgame:                             Functions and Variables for dynamics.
                                                             (line    8)
* charat:                                Functions and Variables for strings.
                                                             (line   11)
* charfun:                               Functions and Variables for Predicates.
                                                             (line    7)
* charfun2:                              Functions and Variables for interpol.
                                                             (line  105)
* charlist:                              Functions and Variables for strings.
                                                             (line   19)
* charp:                                 Functions and Variables for characters.
                                                             (line   43)
* charpoly:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line   69)
* chebyshev_t:                           Functions and Variables for orthogonal polynomials.
                                                             (line   22)
* chebyshev_u:                           Functions and Variables for orthogonal polynomials.
                                                             (line   28)
* check_overlaps:                        Functions and Variables for Affine.
                                                             (line   97)
* checkdiv:                              Functions and Variables for ctensor.
                                                             (line  785)
* cholesky:                              Functions and Variables for linearalgebra.
                                                             (line   72)
* christof:                              Functions and Variables for ctensor.
                                                             (line  196)
* chromatic_index:                       Functions and Variables for graphs.
                                                             (line  417)
* chromatic_number:                      Functions and Variables for graphs.
                                                             (line  427)
* cint:                                  Functions and Variables for characters.
                                                             (line   48)
* circulant_graph:                       Functions and Variables for graphs.
                                                             (line   71)
* clear_edge_weight:                     Functions and Variables for graphs.
                                                             (line  438)
* clear_rules:                           Functions and Variables for Rules and Patterns.
                                                             (line  856)
* clear_vertex_label:                    Functions and Variables for graphs.
                                                             (line  452)
* clebsch_graph:                         Functions and Variables for graphs.
                                                             (line   92)
* clessp:                                Functions and Variables for characters.
                                                             (line   52)
* clesspignore:                          Functions and Variables for characters.
                                                             (line   57)
* close:                                 Functions and Variables for input and output.
                                                             (line   26)
* closefile:                             Functions and Variables for File Input and Output.
                                                             (line   84)
* cmetric:                               Functions and Variables for ctensor.
                                                             (line   17)
* cnonmet_flag:                          Functions and Variables for ctensor.
                                                             (line 1115)
* coeff:                                 Functions and Variables for Polynomials.
                                                             (line   55)
* coefmatrix:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  106)
* cograd:                                Functions and Variables for ctensor.
                                                             (line  731)
* col:                                   Functions and Variables for Matrices and Linear Algebra.
                                                             (line  116)
* collapse:                              Functions and Variables for Expressions.
                                                             (line  133)
* collectterms:                          Package facexp.     (line  106)
* color <1>:                             Functions and Variables for draw.
                                                             (line  372)
* color:                                 Plotting Options.   (line   52)
* colorbox <1>:                          Plotting Options.   (line   70)
* colorbox:                              Functions and Variables for draw.
                                                             (line  421)
* columnop:                              Functions and Variables for linearalgebra.
                                                             (line   27)
* columns:                               Functions and Variables for draw.
                                                             (line  453)
* columnspace:                           Functions and Variables for linearalgebra.
                                                             (line   38)
* columnswap:                            Functions and Variables for linearalgebra.
                                                             (line   33)
* columnvector:                          Functions and Variables for Matrices and Linear Algebra.
                                                             (line  121)
* combination:                           Package functs.     (line  198)
* combine:                               Functions and Variables for Simplification.
                                                             (line   58)
* commutative:                           Functions and Variables for Simplification.
                                                             (line   63)
* comp2pui:                              Functions and Variables for Symmetries.
                                                             (line   10)
* compare:                               Functions and Variables for Predicates.
                                                             (line   24)
* compfile:                              Functions and Variables for Function Definition.
                                                             (line  151)
* compile:                               Functions and Variables for Function Definition.
                                                             (line  167)
* compile_file:                          Functions and Variables for Function Definition.
                                                             (line 1336)
* complement_graph:                      Functions and Variables for graphs.
                                                             (line   96)
* complete_bipartite_graph:              Functions and Variables for graphs.
                                                             (line  100)
* complete_graph:                        Functions and Variables for graphs.
                                                             (line  104)
* complex:                               Functions and Variables for Properties.
                                                             (line  548)
* Complex infinity:                      Functions and Variables for Constants.
                                                             (line   41)
* complex_number_p:                      Functions and Variables for to_poly_solve.
                                                             (line  209)
* components:                            Functions and Variables for itensor.
                                                             (line  198)
* compose_functions:                     Functions and Variables for to_poly_solve.
                                                             (line  232)
* concan:                                Functions and Variables for itensor.
                                                             (line  518)
* concat:                                Functions and Variables for Strings.
                                                             (line    7)
* conditional evaluation:                Functions and Variables for to_poly_solve.
                                                             (line   91)
* conjugate:                             Functions for Complex Numbers.
                                                             (line  104)
* conmetderiv:                           Functions and Variables for itensor.
                                                             (line  788)
* connect_vertices:                      Functions and Variables for graphs.
                                                             (line 1283)
* connected_components:                  Functions and Variables for graphs.
                                                             (line  466)
* cons:                                  Functions and Variables for Lists.
                                                             (line   61)
* constant:                              Functions and Variables for Properties.
                                                             (line   43)
* constantp:                             Functions and Variables for Properties.
                                                             (line   62)
* constituent:                           Functions and Variables for characters.
                                                             (line   61)
* constvalue:                            Functions and Variables for ezunits.
                                                             (line  257)
* cont2part:                             Functions and Variables for Symmetries.
                                                             (line  223)
* content:                               Functions and Variables for Polynomials.
                                                             (line  145)
* context:                               Functions and Variables for Facts.
                                                             (line  229)
* contexts:                              Functions and Variables for Facts.
                                                             (line  245)
* continuous_freq:                       Functions and Variables for data manipulation.
                                                             (line    7)
* contortion:                            Functions and Variables for ctensor.
                                                             (line  664)
* contour:                               Functions and Variables for draw.
                                                             (line  473)
* contour_levels:                        Functions and Variables for draw.
                                                             (line  504)
* contour_plot:                          Functions and Variables for Plotting.
                                                             (line    8)
* contract <1>:                          Functions and Variables for Symmetries.
                                                             (line  235)
* contract:                              Functions and Variables for itensor.
                                                             (line  182)
* contract_edge:                         Functions and Variables for graphs.
                                                             (line 1302)
* Contraction with a vector:             Functions and Variables for itensor.
                                                             (line 1428)
* contragrad:                            Functions and Variables for ctensor.
                                                             (line  737)
* contrib_ode:                           Functions and Variables for contrib_ode.
                                                             (line    7)
* convert:                               Functions and Variables for Units.
                                                             (line  144)
* coord:                                 Functions and Variables for itensor.
                                                             (line  766)
* copy:                                  Functions and Variables for linearalgebra.
                                                             (line   45)
* copy_graph:                            Functions and Variables for graphs.
                                                             (line   67)
* copylist:                              Functions and Variables for Lists.
                                                             (line   67)
* copymatrix:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  148)
* cor:                                   Functions and Variables for descriptive statistics.
                                                             (line  605)
* cos:                                   Functions and Variables for Trigonometric.
                                                             (line  146)
* cosh:                                  Functions and Variables for Trigonometric.
                                                             (line  150)
* cosnpiflag:                            Functions and Variables for Fourier series.
                                                             (line   63)
* cot:                                   Functions and Variables for Trigonometric.
                                                             (line  154)
* coth:                                  Functions and Variables for Trigonometric.
                                                             (line  158)
* cov:                                   Functions and Variables for descriptive statistics.
                                                             (line  499)
* cov1:                                  Functions and Variables for descriptive statistics.
                                                             (line  529)
* covdiff:                               Functions and Variables for itensor.
                                                             (line  938)
* covect:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line  122)
* covers:                                Package functs.     (line  180)
* create_graph:                          Functions and Variables for graphs.
                                                             (line   11)
* create_list:                           Functions and Variables for Lists.
                                                             (line   72)
* csc:                                   Functions and Variables for Trigonometric.
                                                             (line  162)
* csch:                                  Functions and Variables for Trigonometric.
                                                             (line  166)
* csetup:                                Functions and Variables for ctensor.
                                                             (line   10)
* cspline:                               Functions and Variables for interpol.
                                                             (line  178)
* ct_coords:                             Functions and Variables for ctensor.
                                                             (line 1276)
* ct_coordsys:                           Functions and Variables for ctensor.
                                                             (line   37)
* ctaylor:                               Functions and Variables for ctensor.
                                                             (line  341)
* ctaypov:                               Functions and Variables for ctensor.
                                                             (line 1133)
* ctaypt:                                Functions and Variables for ctensor.
                                                             (line 1138)
* ctayswitch:                            Functions and Variables for ctensor.
                                                             (line 1122)
* ctayvar:                               Functions and Variables for ctensor.
                                                             (line 1128)
* ctorsion_flag:                         Functions and Variables for ctensor.
                                                             (line 1109)
* ctransform:                            Functions and Variables for ctensor.
                                                             (line  677)
* ctranspose:                            Functions and Variables for linearalgebra.
                                                             (line   80)
* ctrgsimp:                              Functions and Variables for ctensor.
                                                             (line 1094)
* cube_graph:                            Functions and Variables for graphs.
                                                             (line  120)
* cuboctahedron_graph:                   Functions and Variables for graphs.
                                                             (line  116)
* cunlisp:                               Functions and Variables for characters.
                                                             (line   74)
* Current input expression:              Functions and Variables for Command Line.
                                                             (line    7)
* current_let_rule_package:              Functions and Variables for Rules and Patterns.
                                                             (line   52)
* cv:                                    Functions and Variables for descriptive statistics.
                                                             (line  168)
* cycle_digraph:                         Functions and Variables for graphs.
                                                             (line  108)
* cycle_graph:                           Functions and Variables for graphs.
                                                             (line  112)
* cylindrical:                           Functions and Variables for draw.
                                                             (line 2619)
* data_file_name:                        Functions and Variables for draw.
                                                             (line  556)
* days360:                               Functions and Variables for finance.
                                                             (line    7)
* dblint:                                Functions and Variables for Integration.
                                                             (line   60)
* deactivate:                            Functions and Variables for Facts.
                                                             (line  279)
* debugmode:                             Functions and Variables for Debugging.
                                                             (line    7)
* declare:                               Functions and Variables for Properties.
                                                             (line   98)
* declare_constvalue:                    Functions and Variables for ezunits.
                                                             (line  258)
* declare_dimensions:                    Functions and Variables for ezunits.
                                                             (line  491)
* declare_fundamental_dimensions:        Functions and Variables for ezunits.
                                                             (line  521)
* declare_fundamental_units:             Functions and Variables for ezunits.
                                                             (line  553)
* declare_qty:                           Functions and Variables for ezunits.
                                                             (line  363)
* declare_translated:                    Functions and Variables for Function Definition.
                                                             (line 1363)
* declare_unit_conversion:               Functions and Variables for ezunits.
                                                             (line  448)
* declare_units:                         Functions and Variables for ezunits.
                                                             (line  302)
* declare_weights:                       Functions and Variables for Affine.
                                                             (line   62)
* decreasing:                            Functions and Variables for Properties.
                                                             (line  241)
* decsym:                                Functions and Variables for itensor.
                                                             (line  534)
* default_let_rule_package:              Functions and Variables for Rules and Patterns.
                                                             (line   65)
* defcon:                                Functions and Variables for itensor.
                                                             (line  157)
* define:                                Functions and Variables for Function Definition.
                                                             (line  185)
* define_variable:                       Functions and Variables for Function Definition.
                                                             (line  274)
* defint:                                Functions and Variables for Integration.
                                                             (line  106)
* defmatch:                              Functions and Variables for Rules and Patterns.
                                                             (line   74)
* defrule:                               Functions and Variables for Rules and Patterns.
                                                             (line  175)
* defstruct:                             Functions and Variables for Structures.
                                                             (line   13)
* deftaylor:                             Functions and Variables for Series.
                                                             (line   41)
* degree_sequence:                       Functions and Variables for graphs.
                                                             (line  506)
* del:                                   Functions and Variables for Differentiation.
                                                             (line  231)
* delay:                                 Functions and Variables for draw.
                                                             (line  569)
* delete:                                Functions and Variables for Lists.
                                                             (line  101)
* deleten:                               Functions and Variables for ctensor.
                                                             (line 1067)
* delta:                                 Functions and Variables for Differentiation.
                                                             (line  251)
* demo:                                  Functions and Variables for Help.
                                                             (line   33)
* demoivre:                              Functions and Variables for Simplification.
                                                             (line   69)
* denom:                                 Functions and Variables for Polynomials.
                                                             (line  158)
* dependencies:                          Functions and Variables for Differentiation.
                                                             (line  266)
* depends:                               Functions and Variables for Differentiation.
                                                             (line  277)
* derivabbrev:                           Functions and Variables for Differentiation.
                                                             (line  333)
* derivdegree:                           Functions and Variables for Differentiation.
                                                             (line  341)
* derivlist:                             Functions and Variables for Differentiation.
                                                             (line  358)
* derivsubst:                            Functions and Variables for Differentiation.
                                                             (line  363)
* describe:                              Functions and Variables for Help.
                                                             (line   70)
* desolve:                               Functions and Variables for Differential Equations.
                                                             (line   20)
* determinant:                           Functions and Variables for Matrices and Linear Algebra.
                                                             (line  159)
* detout:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line  170)
* dfloat:                                Functions and Variables for to_poly_solve.
                                                             (line  284)
* dgauss_a:                              Functions and Variables for contrib_ode.
                                                             (line  105)
* dgauss_b:                              Functions and Variables for contrib_ode.
                                                             (line  110)
* dgeev:                                 Functions and Variables for lapack.
                                                             (line    7)
* dgemm:                                 Functions and Variables for lapack.
                                                             (line  299)
* dgeqrf:                                Functions and Variables for lapack.
                                                             (line   73)
* dgesv:                                 Functions and Variables for lapack.
                                                             (line  104)
* dgesvd:                                Functions and Variables for lapack.
                                                             (line  176)
* diag:                                  Functions and Variables for diag.
                                                             (line    7)
* diag_matrix:                           Functions and Variables for linearalgebra.
                                                             (line   86)
* diagmatrix:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  198)
* diagmatrixp:                           Functions and Variables for ctensor.
                                                             (line  835)
* diagmetric:                            Functions and Variables for ctensor.
                                                             (line 1083)
* diameter:                              Functions and Variables for graphs.
                                                             (line  477)
* diff <1>:                              Functions and Variables for Differentiation.
                                                             (line  372)
* diff <2>:                              Functions and Variables for itensor.
                                                             (line  610)
* diff:                                  Functions and Variables for Differentiation.
                                                             (line  373)
* digitcharp:                            Functions and Variables for characters.
                                                             (line   79)
* dim:                                   Functions and Variables for ctensor.
                                                             (line 1075)
* dimacs_export:                         Functions and Variables for graphs.
                                                             (line 1355)
* dimacs_import:                         Functions and Variables for graphs.
                                                             (line 1361)
* dimension:                             Functions and Variables for Equations.
                                                             (line  314)
* dimensionless:                         Functions and Variables for ezunits.
                                                             (line  672)
* dimensions <1>:                        Functions and Variables for ezunits.
                                                             (line  593)
* dimensions:                            Functions and Variables for draw.
                                                             (line  597)
* dimensions_as_list:                    Functions and Variables for ezunits.
                                                             (line  594)
* direct:                                Functions and Variables for Symmetries.
                                                             (line  295)
* discrete_freq:                         Functions and Variables for data manipulation.
                                                             (line   50)
* disjoin:                               Functions and Variables for Sets.
                                                             (line   89)
* disjointp:                             Functions and Variables for Sets.
                                                             (line  109)
* disolate:                              Functions and Variables for Expressions.
                                                             (line  144)
* disp:                                  Functions and Variables for Display.
                                                             (line   40)
* dispcon:                               Functions and Variables for itensor.
                                                             (line   10)
* dispflag:                              Functions and Variables for Equations.
                                                             (line  321)
* dispform:                              Functions and Variables for Expressions.
                                                             (line  155)
* dispfun:                               Functions and Variables for Function Definition.
                                                             (line  368)
* dispJordan:                            Functions and Variables for diag.
                                                             (line   98)
* display:                               Functions and Variables for Display.
                                                             (line   63)
* display2d:                             Functions and Variables for Display.
                                                             (line   89)
* display_format_internal:               Functions and Variables for Display.
                                                             (line  110)
* disprule:                              Functions and Variables for Rules and Patterns.
                                                             (line  191)
* dispterms:                             Functions and Variables for Display.
                                                             (line  134)
* distrib:                               Functions and Variables for Simplification.
                                                             (line   86)
* distribute_over:                       Functions and Variables for Simplification.
                                                             (line  108)
* divide:                                Functions and Variables for Polynomials.
                                                             (line  162)
* Division:                              Arithmetic operators.
                                                             (line   10)
* divisors:                              Functions and Variables for Sets.
                                                             (line  122)
* divsum:                                Functions and Variables for Number Theory.
                                                             (line  193)
* dkummer_m:                             Functions and Variables for contrib_ode.
                                                             (line  133)
* dkummer_u:                             Functions and Variables for contrib_ode.
                                                             (line  137)
* dlange:                                Functions and Variables for lapack.
                                                             (line  277)
* do:                                    Functions and Variables for Program Flow.
                                                             (line   54)
* doallmxops:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  211)
* dodecahedron_graph:                    Functions and Variables for graphs.
                                                             (line  124)
* domain:                                Functions and Variables for Simplification.
                                                             (line  156)
* domxexpt:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line  219)
* domxmxops:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line  252)
* domxnctimes:                           Functions and Variables for Matrices and Linear Algebra.
                                                             (line  260)
* dontfactor:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  267)
* doscmxops:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line  278)
* doscmxplus:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  285)
* dot0nscsimp:                           Functions and Variables for Matrices and Linear Algebra.
                                                             (line  292)
* dot0simp:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line  299)
* dot1simp:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line  306)
* dotassoc:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line  313)
* dotconstrules:                         Functions and Variables for Matrices and Linear Algebra.
                                                             (line  320)
* dotdistrib:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  330)
* dotexptsimp:                           Functions and Variables for Matrices and Linear Algebra.
                                                             (line  337)
* dotident:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line  344)
* dotproduct:                            Functions and Variables for linearalgebra.
                                                             (line  108)
* dotscrules:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  350)
* dotsimp:                               Functions and Variables for Affine.
                                                             (line   76)
* Double factorial:                      Combinatorial Functions.
                                                             (line    7)
* dpart:                                 Functions and Variables for Expressions.
                                                             (line  207)
* draw:                                  Functions and Variables for draw.
                                                             (line   55)
* draw2d:                                Functions and Variables for draw.
                                                             (line  103)
* draw3d:                                Functions and Variables for draw.
                                                             (line  114)
* draw_file:                             Functions and Variables for draw.
                                                             (line  125)
* draw_graph:                            Functions and Variables for graphs.
                                                             (line 1404)
* draw_graph_program:                    Functions and Variables for graphs.
                                                             (line 1509)
* draw_realpart:                         Functions and Variables for draw.
                                                             (line  642)
* drawdf:                                Functions and Variables for drawdf.
                                                             (line   18)
* dscalar <1>:                           Functions and Variables for Differentiation.
                                                             (line  453)
* dscalar:                               Functions and Variables for ctensor.
                                                             (line  764)
* e:                                     Functions and Variables for Constants.
                                                             (line    7)
* echelon:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line  357)
* edge_color:                            Functions and Variables for graphs.
                                                             (line 1589)
* edge_coloring:                         Functions and Variables for graphs.
                                                             (line  486)
* edge_connectivity:                     Functions and Variables for graphs.
                                                             (line  515)
* edge_partition:                        Functions and Variables for graphs.
                                                             (line 1620)
* edge_type:                             Functions and Variables for graphs.
                                                             (line 1597)
* edge_width:                            Functions and Variables for graphs.
                                                             (line 1593)
* edges:                                 Functions and Variables for graphs.
                                                             (line  521)
* eigens_by_jacobi:                      Functions and Variables for linearalgebra.
                                                             (line  114)
* eigenvalues:                           Functions and Variables for Matrices and Linear Algebra.
                                                             (line  388)
* eigenvectors:                          Functions and Variables for Matrices and Linear Algebra.
                                                             (line  417)
* eighth:                                Functions and Variables for Lists.
                                                             (line  165)
* einstein:                              Functions and Variables for ctensor.
                                                             (line  235)
* eivals:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line  389)
* eivects:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line  418)
* elapsed_real_time:                     Functions and Variables for Runtime Environment.
                                                             (line  160)
* elapsed_run_time:                      Functions and Variables for Runtime Environment.
                                                             (line  176)
* ele2comp:                              Functions and Variables for Symmetries.
                                                             (line   30)
* ele2polynome:                          Functions and Variables for Symmetries.
                                                             (line  524)
* ele2pui:                               Functions and Variables for Symmetries.
                                                             (line   23)
* elem:                                  Functions and Variables for Symmetries.
                                                             (line   37)
* elementp:                              Functions and Variables for Sets.
                                                             (line  165)
* elevation:                             Plotting Options.   (line   80)
* elevation_grid:                        Functions and Variables for draw.
                                                             (line 2640)
* elim:                                  Functions and Variables for to_poly_solve.
                                                             (line  315)
* elim_allbut:                           Functions and Variables for to_poly_solve.
                                                             (line  379)
* eliminate:                             Functions and Variables for Polynomials.
                                                             (line  179)
* eliminate_using:                       Functions and Variables for to_poly_solve.
                                                             (line  400)
* ellipse:                               Functions and Variables for draw.
                                                             (line 2668)
* elliptic_e:                            Functions and Variables for Elliptic Integrals.
                                                             (line   15)
* elliptic_ec:                           Functions and Variables for Elliptic Integrals.
                                                             (line   55)
* elliptic_eu:                           Functions and Variables for Elliptic Integrals.
                                                             (line   23)
* elliptic_f:                            Functions and Variables for Elliptic Integrals.
                                                             (line    7)
* elliptic_kc:                           Functions and Variables for Elliptic Integrals.
                                                             (line   46)
* elliptic_pi:                           Functions and Variables for Elliptic Integrals.
                                                             (line   38)
* ematrix:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line  528)
* empty_graph:                           Functions and Variables for graphs.
                                                             (line  128)
* emptyp:                                Functions and Variables for Sets.
                                                             (line  178)
* endcons:                               Functions and Variables for Lists.
                                                             (line  170)
* enhanced3d:                            Functions and Variables for draw.
                                                             (line  666)
* entermatrix:                           Functions and Variables for Matrices and Linear Algebra.
                                                             (line  533)
* entertensor:                           Functions and Variables for itensor.
                                                             (line   17)
* entier:                                Functions for Numbers.
                                                             (line  140)
* epsilon_lp:                            Functions and Variables for simplex.
                                                             (line    7)
* equal:                                 Functions and Variables for Predicates.
                                                             (line   55)
* Equal (syntactic equality):            Operators for Equations.
                                                             (line   31)
* equalp:                                Functions and Variables for Fourier series.
                                                             (line    7)
* Equation operator:                     Operators for Equations.
                                                             (line   31)
* equiv_classes:                         Functions and Variables for Sets.
                                                             (line  190)
* erf:                                   Error Function.     (line   10)
* erf_generalized:                       Error Function.     (line   28)
* erf_representation:                    Error Function.     (line   58)
* erfc:                                  Error Function.     (line   16)
* erfflag:                               Functions and Variables for Integration.
                                                             (line  119)
* erfi:                                  Error Function.     (line   22)
* errcatch:                              Functions and Variables for Program Flow.
                                                             (line  244)
* error:                                 Functions and Variables for Program Flow.
                                                             (line  256)
* error_size:                            Functions and Variables for Program Flow.
                                                             (line  271)
* error_syms:                            Functions and Variables for Program Flow.
                                                             (line  316)
* error_type:                            Functions and Variables for draw.
                                                             (line  805)
* errormsg:                              Functions and Variables for Program Flow.
                                                             (line  337)
* errors:                                Functions and Variables for draw.
                                                             (line 2700)
* euler:                                 Functions and Variables for Number Theory.
                                                             (line  209)
* Euler's number:                        Functions and Variables for Constants.
                                                             (line    7)
* Euler-Mascheroni constant:             Functions and Variables for Constants.
                                                             (line   22)
* ev:                                    Functions and Variables for Evaluation.
                                                             (line  238)
* ev_point:                              Functions and Variables for zeilberger.
                                                             (line  157)
* eval:                                  Functions and Variables for Evaluation.
                                                             (line  428)
* eval_string:                           Functions and Variables for strings.
                                                             (line   28)
* even:                                  Functions and Variables for Properties.
                                                             (line  261)
* evenfun:                               Functions and Variables for Simplification.
                                                             (line  163)
* evenp:                                 Functions and Variables for Numbers.
                                                             (line   58)
* every:                                 Functions and Variables for Sets.
                                                             (line  226)
* evflag:                                Functions and Variables for Evaluation.
                                                             (line  446)
* evfun:                                 Functions and Variables for Evaluation.
                                                             (line  510)
* evolution:                             Functions and Variables for dynamics.
                                                             (line   18)
* evolution2d:                           Functions and Variables for dynamics.
                                                             (line   32)
* evundiff:                              Functions and Variables for itensor.
                                                             (line  683)
* example:                               Functions and Variables for Help.
                                                             (line  134)
* exp:                                   Root Exponential and Logarithmic Functions.
                                                             (line   46)
* expand:                                Functions and Variables for Simplification.
                                                             (line  184)
* expandwrt:                             Functions and Variables for Simplification.
                                                             (line  275)
* expandwrt_denom:                       Functions and Variables for Simplification.
                                                             (line  288)
* expandwrt_factored:                    Functions and Variables for Simplification.
                                                             (line  298)
* expintegral_chi:                       Exponential Integrals.
                                                             (line   38)
* expintegral_ci:                        Exponential Integrals.
                                                             (line   30)
* expintegral_e:                         Exponential Integrals.
                                                             (line   22)
* expintegral_e1:                        Exponential Integrals.
                                                             (line   10)
* expintegral_ei:                        Exponential Integrals.
                                                             (line   14)
* expintegral_li:                        Exponential Integrals.
                                                             (line   18)
* expintegral_shi:                       Exponential Integrals.
                                                             (line   34)
* expintegral_si:                        Exponential Integrals.
                                                             (line   26)
* expintexpand:                          Exponential Integrals.
                                                             (line   50)
* expintrep:                             Exponential Integrals.
                                                             (line   42)
* explicit:                              Functions and Variables for draw.
                                                             (line 2742)
* explose:                               Functions and Variables for Symmetries.
                                                             (line  253)
* expon:                                 Functions and Variables for Simplification.
                                                             (line  306)
* exponentialize:                        Functions and Variables for Simplification.
                                                             (line  315)
* Exponentiation:                        Arithmetic operators.
                                                             (line   11)
* expop:                                 Functions and Variables for Simplification.
                                                             (line  330)
* express:                               Functions and Variables for Differentiation.
                                                             (line  459)
* expt:                                  Functions and Variables for Display.
                                                             (line  161)
* exptdispflag:                          Functions and Variables for Display.
                                                             (line  169)
* exptisolate:                           Functions and Variables for Expressions.
                                                             (line  222)
* exptsubst:                             Functions and Variables for Expressions.
                                                             (line  229)
* exsec:                                 Package functs.     (line  186)
* extdiff:                               Functions and Variables for itensor.
                                                             (line 1459)
* extract_linear_equations:              Functions and Variables for Affine.
                                                             (line  126)
* extremal_subset:                       Functions and Variables for Sets.
                                                             (line  293)
* ezgcd:                                 Functions and Variables for Polynomials.
                                                             (line  207)
* f90:                                   Functions and Variables for f90.
                                                             (line    7)
* facexpand:                             Functions and Variables for Polynomials.
                                                             (line  239)
* facsum:                                Package facexp.     (line   28)
* facsum_combine:                        Package facexp.     (line   83)
* factcomb:                              Combinatorial Functions.
                                                             (line   48)
* factlim:                               Combinatorial Functions.
                                                             (line  145)
* factor:                                Functions and Variables for Polynomials.
                                                             (line  247)
* factorfacsum:                          Package facexp.     (line   99)
* factorflag:                            Functions and Variables for Polynomials.
                                                             (line  379)
* factorial:                             Combinatorial Functions.
                                                             (line   57)
* factorial_expand:                      Combinatorial Functions.
                                                             (line  152)
* factorout:                             Functions and Variables for Polynomials.
                                                             (line  386)
* factorsum:                             Functions and Variables for Polynomials.
                                                             (line  405)
* facts:                                 Functions and Variables for Facts.
                                                             (line  283)
* false:                                 Functions and Variables for Constants.
                                                             (line   17)
* fast_central_elements:                 Functions and Variables for Affine.
                                                             (line   84)
* fast_linsolve:                         Functions and Variables for Affine.
                                                             (line    8)
* fasttimes:                             Functions and Variables for Polynomials.
                                                             (line  425)
* fb:                                    Functions and Variables for ctensor.
                                                             (line 1226)
* feature:                               Functions and Variables for Properties.
                                                             (line  282)
* featurep:                              Functions and Variables for Properties.
                                                             (line  291)
* features:                              Functions and Variables for Properties.
                                                             (line  308)
* fernfale:                              Definitions for IFS fractals.
                                                             (line   43)
* Fetch documentation:                   Functions and Variables for Command Line.
                                                             (line  183)
* Fetch documentation (inexact search):  Functions and Variables for Command Line.
                                                             (line  196)
* fft:                                   Functions and Variables for fast Fourier transform.
                                                             (line  102)
* fib:                                   Functions and Variables for Number Theory.
                                                             (line  218)
* fibtophi:                              Functions and Variables for Number Theory.
                                                             (line  230)
* fifth:                                 Functions and Variables for Lists.
                                                             (line  176)
* file_name <1>:                         Functions and Variables for draw.
                                                             (line  815)
* file_name:                             Functions and Variables for graphs.
                                                             (line 1662)
* file_output_append:                    Functions and Variables for File Input and Output.
                                                             (line   88)
* file_search:                           Functions and Variables for File Input and Output.
                                                             (line  113)
* file_search_demo:                      Functions and Variables for File Input and Output.
                                                             (line  154)
* file_search_lisp:                      Functions and Variables for File Input and Output.
                                                             (line  153)
* file_search_maxima:                    Functions and Variables for File Input and Output.
                                                             (line  152)
* file_search_tests:                     Functions and Variables for File Input and Output.
                                                             (line  156)
* file_search_usage:                     Functions and Variables for File Input and Output.
                                                             (line  155)
* file_type:                             Functions and Variables for File Input and Output.
                                                             (line  190)
* file_type_lisp:                        Functions and Variables for File Input and Output.
                                                             (line  214)
* file_type_maxima:                      Functions and Variables for File Input and Output.
                                                             (line  223)
* filename_merge:                        Functions and Variables for File Input and Output.
                                                             (line  103)
* fill_color:                            Functions and Variables for draw.
                                                             (line  836)
* fill_density:                          Functions and Variables for draw.
                                                             (line  845)
* fillarray:                             Functions and Variables for Arrays.
                                                             (line  217)
* filled_func:                           Functions and Variables for draw.
                                                             (line  853)
* find_root:                             Functions and Variables for fast Fourier transform.
                                                             (line  233)
* find_root_abs:                         Functions and Variables for fast Fourier transform.
                                                             (line  238)
* find_root_error:                       Functions and Variables for fast Fourier transform.
                                                             (line  237)
* find_root_rel:                         Functions and Variables for fast Fourier transform.
                                                             (line  239)
* findde:                                Functions and Variables for ctensor.
                                                             (line  684)
* first:                                 Functions and Variables for Lists.
                                                             (line  181)
* fix:                                   Functions for Numbers.
                                                             (line  191)
* fixed_vertices:                        Functions and Variables for graphs.
                                                             (line 1675)
* flatten:                               Functions and Variables for Sets.
                                                             (line  314)
* flength:                               Functions and Variables for input and output.
                                                             (line   30)
* flipflag:                              Functions and Variables for itensor.
                                                             (line  144)
* float:                                 Functions and Variables for Numbers.
                                                             (line   63)
* float2bf:                              Functions and Variables for Numbers.
                                                             (line   70)
* floatnump:                             Functions and Variables for Numbers.
                                                             (line   78)
* floor:                                 Functions for Numbers.
                                                             (line  146)
* flower_snark:                          Functions and Variables for graphs.
                                                             (line  132)
* flush:                                 Functions and Variables for itensor.
                                                             (line  739)
* flush1deriv:                           Functions and Variables for itensor.
                                                             (line  886)
* flushd:                                Functions and Variables for itensor.
                                                             (line  744)
* flushnd:                               Functions and Variables for itensor.
                                                             (line  749)
* fmin_cobyla:                           Functions and Variables for cobyla.
                                                             (line    7)
* font:                                  Functions and Variables for draw.
                                                             (line  886)
* font_size:                             Functions and Variables for draw.
                                                             (line  973)
* for:                                   Functions and Variables for Program Flow.
                                                             (line  373)
* forget:                                Functions and Variables for Facts.
                                                             (line  296)
* fortindent:                            Functions and Variables for Fortran Output.
                                                             (line    7)
* fortran:                               Functions and Variables for Fortran Output.
                                                             (line   16)
* fortspaces:                            Functions and Variables for Fortran Output.
                                                             (line   72)
* fourcos:                               Functions and Variables for Fourier series.
                                                             (line   75)
* fourexpand:                            Functions and Variables for Fourier series.
                                                             (line   69)
* fourier:                               Functions and Variables for Fourier series.
                                                             (line   47)
* fourier_elim:                          Functions and Variables for to_poly_solve.
                                                             (line  432)
* fourint:                               Functions and Variables for Fourier series.
                                                             (line   90)
* fourintcos:                            Functions and Variables for Fourier series.
                                                             (line   95)
* fourintsin:                            Functions and Variables for Fourier series.
                                                             (line  100)
* foursimp:                              Functions and Variables for Fourier series.
                                                             (line   52)
* foursin:                               Functions and Variables for Fourier series.
                                                             (line   80)
* fourth:                                Functions and Variables for Lists.
                                                             (line  195)
* fposition:                             Functions and Variables for input and output.
                                                             (line   35)
* fpprec:                                Functions and Variables for Numbers.
                                                             (line   83)
* fpprintprec:                           Functions and Variables for Numbers.
                                                             (line   93)
* frame_bracket:                         Functions and Variables for ctensor.
                                                             (line  472)
* freeof:                                Functions and Variables for Expressions.
                                                             (line  236)
* freshline:                             Functions and Variables for input and output.
                                                             (line   42)
* fresnel_c:                             Error Function.     (line   32)
* fresnel_s:                             Error Function.     (line   45)
* from_adjacency_matrix:                 Functions and Variables for graphs.
                                                             (line  142)
* frucht_graph:                          Functions and Variables for graphs.
                                                             (line  146)
* full_listify:                          Functions and Variables for Sets.
                                                             (line  381)
* fullmap:                               Functions and Variables for Function Definition.
                                                             (line  451)
* fullmapl:                              Functions and Variables for Function Definition.
                                                             (line  470)
* fullratsimp:                           Functions and Variables for Polynomials.
                                                             (line  434)
* fullratsubst:                          Functions and Variables for Polynomials.
                                                             (line  473)
* fullsetify:                            Functions and Variables for Sets.
                                                             (line  396)
* funcsolve:                             Functions and Variables for Equations.
                                                             (line  330)
* Function definition operator:          Assignment operators.
                                                             (line  190)
* functions:                             Functions and Variables for Function Definition.
                                                             (line  480)
* fundamental_dimensions:                Functions and Variables for ezunits.
                                                             (line  523)
* fundamental_units:                     Functions and Variables for ezunits.
                                                             (line  637)
* fundef:                                Functions and Variables for Function Definition.
                                                             (line  526)
* funmake:                               Functions and Variables for Function Definition.
                                                             (line  546)
* funp:                                  Functions and Variables for Fourier series.
                                                             (line   22)
* fv:                                    Functions and Variables for finance.
                                                             (line   18)
* gamma:                                 Gamma and factorial Functions.
                                                             (line   33)
* gamma_incomplete:                      Gamma and factorial Functions.
                                                             (line  121)
* gamma_incomplete_generalized:          Gamma and factorial Functions.
                                                             (line  142)
* gamma_incomplete_regularized:          Gamma and factorial Functions.
                                                             (line  133)
* gammalim:                              Gamma and factorial Functions.
                                                             (line  155)
* gauss_a:                               Functions and Variables for contrib_ode.
                                                             (line   88)
* gauss_b:                               Functions and Variables for contrib_ode.
                                                             (line  101)
* gaussprob:                             Package functs.     (line  154)
* gcd:                                   Functions and Variables for Polynomials.
                                                             (line  530)
* gcdex:                                 Functions and Variables for Polynomials.
                                                             (line  578)
* gcdivide:                              Package functs.     (line   76)
* gcfac:                                 Package scifac.     (line    7)
* gcfactor:                              Functions and Variables for Polynomials.
                                                             (line  619)
* gd:                                    Package functs.     (line  161)
* gdet:                                  Functions and Variables for ctensor.
                                                             (line 1143)
* gen_laguerre:                          Functions and Variables for orthogonal polynomials.
                                                             (line   34)
* genfact:                               Combinatorial Functions.
                                                             (line  160)
* genindex:                              Functions and Variables for Miscellaneous Options.
                                                             (line   15)
* genmatrix:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line  572)
* gensumnum:                             Functions and Variables for Miscellaneous Options.
                                                             (line   22)
* gensym:                                Functions and Variables for Miscellaneous Options.
                                                             (line   30)
* geo_amortization:                      Functions and Variables for finance.
                                                             (line  160)
* geo_annuity_fv:                        Functions and Variables for finance.
                                                             (line   93)
* geo_annuity_pv:                        Functions and Variables for finance.
                                                             (line   80)
* geomap:                                Functions and Variables for worldmap.
                                                             (line  154)
* geometric:                             Package functs.     (line  126)
* geometric_mean:                        Functions and Variables for descriptive statistics.
                                                             (line  377)
* geosum:                                Package functs.     (line  146)
* get:                                   Functions and Variables for Properties.
                                                             (line  347)
* get_edge_weight:                       Functions and Variables for graphs.
                                                             (line  531)
* get_lu_factors:                        Functions and Variables for linearalgebra.
                                                             (line  173)
* get_pixel:                             Functions and Variables for pictures.
                                                             (line    7)
* get_plot_option:                       Functions and Variables for Plotting.
                                                             (line   38)
* get_tex_environment:                   Functions and Variables for TeX Output.
                                                             (line  209)
* get_tex_environment_default:           Functions and Variables for TeX Output.
                                                             (line  249)
* get_vertex_label:                      Functions and Variables for graphs.
                                                             (line  549)
* gfactor:                               Functions and Variables for Polynomials.
                                                             (line  626)
* gfactorsum:                            Functions and Variables for Polynomials.
                                                             (line  637)
* ggf:                                   Functions and Variables for ggf.
                                                             (line   38)
* GGFCFMAX:                              Functions and Variables for ggf.
                                                             (line   23)
* GGFINFINITY:                           Functions and Variables for ggf.
                                                             (line    7)
* girth:                                 Functions and Variables for graphs.
                                                             (line  622)
* global_variances:                      Functions and Variables for descriptive statistics.
                                                             (line  560)
* globalsolve:                           Functions and Variables for Equations.
                                                             (line  352)
* gnuplot_close:                         Gnuplot_pipes Format Functions.
                                                             (line   13)
* gnuplot_curve_styles:                  Gnuplot Options.    (line   75)
* gnuplot_curve_titles:                  Gnuplot Options.    (line   70)
* gnuplot_default_term_command:          Gnuplot Options.    (line   79)
* gnuplot_dumb_term_command:             Gnuplot Options.    (line   84)
* gnuplot_file_name:                     Functions and Variables for draw.
                                                             (line  987)
* gnuplot_out_file:                      Gnuplot Options.    (line   47)
* gnuplot_pm3d:                          Gnuplot Options.    (line   57)
* gnuplot_preamble:                      Gnuplot Options.    (line   62)
* gnuplot_ps_term_command:               Gnuplot Options.    (line   90)
* gnuplot_replot:                        Gnuplot_pipes Format Functions.
                                                             (line   23)
* gnuplot_reset:                         Gnuplot_pipes Format Functions.
                                                             (line   30)
* gnuplot_restart:                       Gnuplot_pipes Format Functions.
                                                             (line   18)
* gnuplot_start:                         Gnuplot_pipes Format Functions.
                                                             (line    7)
* gnuplot_term:                          Gnuplot Options.    (line   14)
* go:                                    Functions and Variables for Program Flow.
                                                             (line  378)
* Golden mean:                           Functions and Variables for Constants.
                                                             (line   49)
* Gosper:                                Functions and Variables for zeilberger.
                                                             (line   12)
* Gosper_in_Zeilberger:                  Functions and Variables for zeilberger.
                                                             (line  124)
* GosperSum:                             Functions and Variables for zeilberger.
                                                             (line   18)
* gr2d:                                  Functions and Variables for draw.
                                                             (line   11)
* gr3d:                                  Functions and Variables for draw.
                                                             (line   31)
* gradef:                                Functions and Variables for Differentiation.
                                                             (line  521)
* gradefs:                               Functions and Variables for Differentiation.
                                                             (line  566)
* gramschmidt:                           Functions and Variables for Matrices and Linear Algebra.
                                                             (line  632)
* graph6_decode:                         Functions and Variables for graphs.
                                                             (line 1365)
* graph6_encode:                         Functions and Variables for graphs.
                                                             (line 1369)
* graph6_export:                         Functions and Variables for graphs.
                                                             (line 1373)
* graph6_import:                         Functions and Variables for graphs.
                                                             (line 1378)
* graph_center:                          Functions and Variables for graphs.
                                                             (line  571)
* graph_charpoly:                        Functions and Variables for graphs.
                                                             (line  559)
* graph_eigenvalues:                     Functions and Variables for graphs.
                                                             (line  581)
* graph_flow:                            Functions and Variables for finance.
                                                             (line   42)
* graph_order:                           Functions and Variables for graphs.
                                                             (line  612)
* graph_periphery:                       Functions and Variables for graphs.
                                                             (line  592)
* graph_product:                         Functions and Variables for graphs.
                                                             (line  150)
* graph_size:                            Functions and Variables for graphs.
                                                             (line  602)
* graph_union:                           Functions and Variables for graphs.
                                                             (line  159)
* great_rhombicosidodecahedron_graph:    Functions and Variables for graphs.
                                                             (line  167)
* great_rhombicuboctahedron_graph:       Functions and Variables for graphs.
                                                             (line  171)
* Greater than:                          Relational operators.
                                                             (line   10)
* Greater than or equal:                 Relational operators.
                                                             (line    9)
* grid <1>:                              Plotting Options.   (line   94)
* grid:                                  Functions and Variables for draw.
                                                             (line 1010)
* grid_graph:                            Functions and Variables for graphs.
                                                             (line  163)
* grind:                                 Functions and Variables for Display.
                                                             (line  191)
* grobner_basis:                         Functions and Variables for Affine.
                                                             (line   25)
* grotzch_graph:                         Functions and Variables for graphs.
                                                             (line  175)
* halfangles:                            Functions and Variables for Trigonometric.
                                                             (line  170)
* hamilton_cycle:                        Functions and Variables for graphs.
                                                             (line  632)
* hamilton_path:                         Functions and Variables for graphs.
                                                             (line  644)
* hankel:                                Functions and Variables for linearalgebra.
                                                             (line  180)
* hankel_1:                              Bessel Functions.   (line   64)
* hankel_2:                              Bessel Functions.   (line  114)
* harmonic:                              Package functs.     (line  133)
* harmonic_mean:                         Functions and Variables for descriptive statistics.
                                                             (line  351)
* hav:                                   Package functs.     (line  192)
* head_angle <1>:                        Functions and Variables for draw.
                                                             (line 1025)
* head_angle:                            Functions and Variables for graphs.
                                                             (line 1638)
* head_both:                             Functions and Variables for draw.
                                                             (line 1061)
* head_length <1>:                       Functions and Variables for graphs.
                                                             (line 1644)
* head_length:                           Functions and Variables for draw.
                                                             (line 1082)
* head_type:                             Functions and Variables for draw.
                                                             (line 1106)
* heawood_graph:                         Functions and Variables for graphs.
                                                             (line  179)
* Help:                                  Functions and Variables for Help.
                                                             (line   72)
* hermite:                               Functions and Variables for orthogonal polynomials.
                                                             (line   40)
* hessian:                               Functions and Variables for linearalgebra.
                                                             (line  187)
* hgfred:                                Functions and Variables for Special Functions.
                                                             (line  100)
* hilbert_matrix:                        Functions and Variables for linearalgebra.
                                                             (line  214)
* hilbertmap:                            Definitions for Peano maps.
                                                             (line   10)
* hipow:                                 Functions and Variables for Polynomials.
                                                             (line  642)
* histogram:                             Functions and Variables for statistical graphs.
                                                             (line  208)
* histogram_description:                 Functions and Variables for statistical graphs.
                                                             (line  209)
* hodge:                                 Functions and Variables for itensor.
                                                             (line 1490)
* horner:                                Functions and Variables for fast Fourier transform.
                                                             (line  207)
* hypergeometric:                        Hypergeometric Functions.
                                                             (line   27)
* hypergeometric_representation:         Error Function.     (line   64)
* i:                                     Functions and Variables for Constants.
                                                             (line   13)
* ibase:                                 Functions and Variables for Display.
                                                             (line  291)
* ic1:                                   Functions and Variables for Differential Equations.
                                                             (line   70)
* ic2:                                   Functions and Variables for Differential Equations.
                                                             (line   80)
* ic_convert:                            Functions and Variables for itensor.
                                                             (line 1584)
* icc1:                                  Functions and Variables for itensor.
                                                             (line 1133)
* icc2:                                  Functions and Variables for itensor.
                                                             (line 1148)
* ichr1:                                 Functions and Variables for itensor.
                                                             (line  909)
* ichr2:                                 Functions and Variables for itensor.
                                                             (line  918)
* icosahedron_graph:                     Functions and Variables for graphs.
                                                             (line  183)
* icosidodecahedron_graph:               Functions and Variables for graphs.
                                                             (line  187)
* icounter:                              Functions and Variables for itensor.
                                                             (line  351)
* icurvature:                            Functions and Variables for itensor.
                                                             (line  926)
* ident:                                 Functions and Variables for Matrices and Linear Algebra.
                                                             (line  687)
* identfor:                              Functions and Variables for linearalgebra.
                                                             (line  219)
* identity:                              Functions and Variables for Sets.
                                                             (line  414)
* idiff:                                 Functions and Variables for itensor.
                                                             (line  634)
* idim:                                  Functions and Variables for itensor.
                                                             (line  903)
* idummy:                                Functions and Variables for itensor.
                                                             (line  338)
* idummyx:                               Functions and Variables for itensor.
                                                             (line  345)
* ieqn:                                  Functions and Variables for Equations.
                                                             (line  408)
* ieqnprint:                             Functions and Variables for Equations.
                                                             (line  439)
* if:                                    Functions and Variables for Program Flow.
                                                             (line  391)
* ifactors:                              Functions and Variables for Number Theory.
                                                             (line  256)
* ifb:                                   Functions and Variables for itensor.
                                                             (line 1105)
* ifc1:                                  Functions and Variables for itensor.
                                                             (line 1165)
* ifc2:                                  Functions and Variables for itensor.
                                                             (line 1178)
* ifg:                                   Functions and Variables for itensor.
                                                             (line 1202)
* ifgi:                                  Functions and Variables for itensor.
                                                             (line 1207)
* ifr:                                   Functions and Variables for itensor.
                                                             (line 1191)
* iframe_bracket_form:                   Functions and Variables for itensor.
                                                             (line 1212)
* iframes:                               Functions and Variables for itensor.
                                                             (line 1099)
* ifri:                                  Functions and Variables for itensor.
                                                             (line 1196)
* ifs:                                   Functions and Variables for dynamics.
                                                             (line   45)
* igcdex:                                Functions and Variables for Number Theory.
                                                             (line  270)
* igeodesic_coords:                      Functions and Variables for itensor.
                                                             (line  990)
* igeowedge_flag:                        Functions and Variables for itensor.
                                                             (line 1528)
* ikt1:                                  Functions and Variables for itensor.
                                                             (line 1265)
* ikt2:                                  Functions and Variables for itensor.
                                                             (line 1279)
* ilt:                                   Functions and Variables for Integration.
                                                             (line  127)
* image:                                 Functions and Variables for draw.
                                                             (line 2790)
* imaginary:                             Functions and Variables for Properties.
                                                             (line  547)
* Imaginary unit:                        Functions and Variables for Constants.
                                                             (line   13)
* imagpart:                              Functions for Complex Numbers.
                                                             (line  124)
* imetric:                               Functions and Variables for itensor.
                                                             (line  895)
* implicit:                              Functions and Variables for draw.
                                                             (line 2881)
* implicit_derivative:                   Functions and Variables for impdiff.
                                                             (line    8)
* implicit_plot:                         Functions and Variables for Plotting.
                                                             (line   49)
* in_neighbors:                          Functions and Variables for graphs.
                                                             (line  668)
* inchar:                                Functions and Variables for Command Line.
                                                             (line  204)
* increasing:                            Functions and Variables for Properties.
                                                             (line  242)
* ind:                                   Functions and Variables for Constants.
                                                             (line   26)
* Indeterminate:                         Functions and Variables for Constants.
                                                             (line   26)
* indexed_tensor:                        Functions and Variables for itensor.
                                                             (line  192)
* indices:                               Functions and Variables for itensor.
                                                             (line   55)
* induced_subgraph:                      Functions and Variables for graphs.
                                                             (line  191)
* inf:                                   Functions and Variables for Constants.
                                                             (line   37)
* inference_result:                      Functions and Variables for inference_result.
                                                             (line    7)
* inferencep:                            Functions and Variables for inference_result.
                                                             (line   49)
* infeval:                               Functions and Variables for Evaluation.
                                                             (line  579)
* infinity:                              Functions and Variables for Constants.
                                                             (line   41)
* infix:                                 User defined operators.
                                                             (line    7)
* inflag:                                Functions and Variables for Expressions.
                                                             (line  327)
* infolists:                             Functions and Variables for Command Line.
                                                             (line  230)
* init_atensor:                          Functions and Variables for atensor.
                                                             (line    7)
* init_ctensor:                          Functions and Variables for ctensor.
                                                             (line  153)
* inm:                                   Functions and Variables for itensor.
                                                             (line 1226)
* inmc1:                                 Functions and Variables for itensor.
                                                             (line 1237)
* inmc2:                                 Functions and Variables for itensor.
                                                             (line 1250)
* innerproduct:                          Functions and Variables for Matrices and Linear Algebra.
                                                             (line  691)
* inpart:                                Functions and Variables for Expressions.
                                                             (line  348)
* inprod:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line  692)
* inrt:                                  Functions and Variables for Number Theory.
                                                             (line  291)
* intanalysis:                           Functions and Variables for Integration.
                                                             (line  171)
* integer:                               Functions and Variables for Properties.
                                                             (line  374)
* integer_partitions:                    Functions and Variables for Sets.
                                                             (line  425)
* integerp:                              Functions and Variables for Numbers.
                                                             (line  112)
* integervalued:                         Functions and Variables for Properties.
                                                             (line  391)
* integrate:                             Functions and Variables for Integration.
                                                             (line  211)
* integrate_use_rootsof:                 Functions and Variables for Integration.
                                                             (line  447)
* integration_constant:                  Functions and Variables for Integration.
                                                             (line  384)
* integration_constant_counter:          Functions and Variables for Integration.
                                                             (line  410)
* intersect:                             Functions and Variables for Sets.
                                                             (line  470)
* intersection:                          Functions and Variables for Sets.
                                                             (line  474)
* intervalp:                             Functions and Variables for orthogonal polynomials.
                                                             (line   46)
* intfaclim:                             Functions and Variables for Polynomials.
                                                             (line  666)
* intopois:                              Poisson series.     (line    7)
* intosum:                               Sums and Products.  (line   36)
* inv_mod:                               Functions and Variables for Number Theory.
                                                             (line  299)
* invariant1:                            Functions and Variables for ctensor.
                                                             (line  811)
* invariant2:                            Functions and Variables for ctensor.
                                                             (line  817)
* inverse_fft:                           Functions and Variables for fast Fourier transform.
                                                             (line   45)
* inverse_jacobi_cd:                     Functions and Variables for Elliptic Functions.
                                                             (line   87)
* inverse_jacobi_cn:                     Functions and Variables for Elliptic Functions.
                                                             (line   59)
* inverse_jacobi_cs:                     Functions and Variables for Elliptic Functions.
                                                             (line   83)
* inverse_jacobi_dc:                     Functions and Variables for Elliptic Functions.
                                                             (line   99)
* inverse_jacobi_dn:                     Functions and Variables for Elliptic Functions.
                                                             (line   63)
* inverse_jacobi_ds:                     Functions and Variables for Elliptic Functions.
                                                             (line   95)
* inverse_jacobi_nc:                     Functions and Variables for Elliptic Functions.
                                                             (line   79)
* inverse_jacobi_nd:                     Functions and Variables for Elliptic Functions.
                                                             (line   91)
* inverse_jacobi_ns:                     Functions and Variables for Elliptic Functions.
                                                             (line   67)
* inverse_jacobi_sc:                     Functions and Variables for Elliptic Functions.
                                                             (line   71)
* inverse_jacobi_sd:                     Functions and Variables for Elliptic Functions.
                                                             (line   75)
* inverse_jacobi_sn:                     Functions and Variables for Elliptic Functions.
                                                             (line   55)
* invert:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line  704)
* invert_by_lu:                          Functions and Variables for linearalgebra.
                                                             (line  234)
* ip_grid:                               Functions and Variables for draw.
                                                             (line 1131)
* ip_grid_in:                            Functions and Variables for draw.
                                                             (line 1139)
* irr:                                   Functions and Variables for finance.
                                                             (line  228)
* irrational:                            Functions and Variables for Properties.
                                                             (line  539)
* is:                                    Functions and Variables for Facts.
                                                             (line  306)
* is_biconnected:                        Functions and Variables for graphs.
                                                             (line  681)
* is_bipartite:                          Functions and Variables for graphs.
                                                             (line  692)
* is_connected:                          Functions and Variables for graphs.
                                                             (line  704)
* is_digraph:                            Functions and Variables for graphs.
                                                             (line  714)
* is_edge_in_graph:                      Functions and Variables for graphs.
                                                             (line  725)
* is_graph:                              Functions and Variables for graphs.
                                                             (line  742)
* is_graph_or_digraph:                   Functions and Variables for graphs.
                                                             (line  753)
* is_isomorphic:                         Functions and Variables for graphs.
                                                             (line  765)
* is_planar:                             Functions and Variables for graphs.
                                                             (line  778)
* is_sconnected:                         Functions and Variables for graphs.
                                                             (line  794)
* is_tree:                               Functions and Variables for graphs.
                                                             (line  819)
* is_vertex_in_graph:                    Functions and Variables for graphs.
                                                             (line  806)
* ishow:                                 Functions and Variables for itensor.
                                                             (line   47)
* isolate:                               Functions and Variables for Expressions.
                                                             (line  376)
* isolate_wrt_times:                     Functions and Variables for Expressions.
                                                             (line  395)
* isomorphism:                           Functions and Variables for graphs.
                                                             (line  656)
* isqrt:                                 Functions and Variables for Number Theory.
                                                             (line  311)
* isreal_p:                              Functions and Variables for to_poly_solve.
                                                             (line  505)
* items_inference:                       Functions and Variables for inference_result.
                                                             (line   54)
* itr:                                   Functions and Variables for itensor.
                                                             (line 1291)
* jacobi:                                Functions and Variables for Number Theory.
                                                             (line  316)
* jacobi_cd:                             Functions and Variables for Elliptic Functions.
                                                             (line   39)
* jacobi_cn:                             Functions and Variables for Elliptic Functions.
                                                             (line   11)
* jacobi_cs:                             Functions and Variables for Elliptic Functions.
                                                             (line   35)
* jacobi_dc:                             Functions and Variables for Elliptic Functions.
                                                             (line   51)
* jacobi_dn:                             Functions and Variables for Elliptic Functions.
                                                             (line   15)
* jacobi_ds:                             Functions and Variables for Elliptic Functions.
                                                             (line   47)
* jacobi_nc:                             Functions and Variables for Elliptic Functions.
                                                             (line   31)
* jacobi_nd:                             Functions and Variables for Elliptic Functions.
                                                             (line   43)
* jacobi_ns:                             Functions and Variables for Elliptic Functions.
                                                             (line   19)
* jacobi_p:                              Functions and Variables for orthogonal polynomials.
                                                             (line   51)
* jacobi_sc:                             Functions and Variables for Elliptic Functions.
                                                             (line   23)
* jacobi_sd:                             Functions and Variables for Elliptic Functions.
                                                             (line   27)
* jacobi_sn:                             Functions and Variables for Elliptic Functions.
                                                             (line    7)
* jacobian:                              Functions and Variables for linearalgebra.
                                                             (line  239)
* JF:                                    Functions and Variables for diag.
                                                             (line   34)
* join:                                  Functions and Variables for Lists.
                                                             (line  200)
* jordan:                                Functions and Variables for diag.
                                                             (line   58)
* julia:                                 Functions and Variables for dynamics.
                                                             (line   60)
* julia_parameter:                       Definitions for complex fractals.
                                                             (line   41)
* julia_set:                             Definitions for complex fractals.
                                                             (line   23)
* julia_sin:                             Definitions for complex fractals.
                                                             (line   49)
* kdels:                                 Functions and Variables for itensor.
                                                             (line  375)
* kdelta:                                Functions and Variables for itensor.
                                                             (line  359)
* keepfloat:                             Functions and Variables for Polynomials.
                                                             (line  683)
* key:                                   Functions and Variables for draw.
                                                             (line 1148)
* kill:                                  Functions and Variables for Command Line.
                                                             (line  291)
* killcontext:                           Functions and Variables for Facts.
                                                             (line  359)
* kinvariant:                            Functions and Variables for ctensor.
                                                             (line 1230)
* kostka:                                Functions and Variables for Symmetries.
                                                             (line  471)
* kron_delta:                            Functions and Variables for Sets.
                                                             (line  500)
* kronecker_product:                     Functions and Variables for linearalgebra.
                                                             (line  262)
* kt:                                    Functions and Variables for ctensor.
                                                             (line 1258)
* kummer_m:                              Functions and Variables for contrib_ode.
                                                             (line  115)
* kummer_u:                              Functions and Variables for contrib_ode.
                                                             (line  126)
* kurtosis:                              Functions and Variables for descriptive statistics.
                                                             (line  402)
* kurtosis_bernoulli:                    Functions and Variables for discrete distributions.
                                                             (line  333)
* kurtosis_beta:                         Functions and Variables for continuous distributions.
                                                             (line  944)
* kurtosis_binomial:                     Functions and Variables for discrete distributions.
                                                             (line  156)
* kurtosis_chi2:                         Functions and Variables for continuous distributions.
                                                             (line  407)
* kurtosis_continuous_uniform:           Functions and Variables for continuous distributions.
                                                             (line 1006)
* kurtosis_discrete_uniform:             Functions and Variables for discrete distributions.
                                                             (line  466)
* kurtosis_exp:                          Functions and Variables for continuous distributions.
                                                             (line  718)
* kurtosis_f:                            Functions and Variables for continuous distributions.
                                                             (line  561)
* kurtosis_gamma:                        Functions and Variables for continuous distributions.
                                                             (line  871)
* kurtosis_general_finite_discrete:      Functions and Variables for discrete distributions.
                                                             (line   82)
* kurtosis_geometric:                    Functions and Variables for discrete distributions.
                                                             (line  405)
* kurtosis_gumbel:                       Functions and Variables for continuous distributions.
                                                             (line 1527)
* kurtosis_hypergeometric:               Functions and Variables for discrete distributions.
                                                             (line  541)
* kurtosis_laplace:                      Functions and Variables for continuous distributions.
                                                             (line 1425)
* kurtosis_logistic:                     Functions and Variables for continuous distributions.
                                                             (line 1066)
* kurtosis_lognormal:                    Functions and Variables for continuous distributions.
                                                             (line  805)
* kurtosis_negative_binomial:            Functions and Variables for discrete distributions.
                                                             (line  614)
* kurtosis_noncentral_chi2:              Functions and Variables for continuous distributions.
                                                             (line  486)
* kurtosis_noncentral_student_t:         Functions and Variables for continuous distributions.
                                                             (line  253)
* kurtosis_normal:                       Functions and Variables for continuous distributions.
                                                             (line   68)
* kurtosis_pareto:                       Functions and Variables for continuous distributions.
                                                             (line 1124)
* kurtosis_poisson:                      Functions and Variables for discrete distributions.
                                                             (line  221)
* kurtosis_rayleigh:                     Functions and Variables for continuous distributions.
                                                             (line 1348)
* kurtosis_student_t:                    Functions and Variables for continuous distributions.
                                                             (line  143)
* kurtosis_weibull:                      Functions and Variables for continuous distributions.
                                                             (line 1182)
* label:                                 Functions and Variables for draw.
                                                             (line 2931)
* label_alignment <1>:                   Functions and Variables for graphs.
                                                             (line 1528)
* label_alignment:                       Functions and Variables for draw.
                                                             (line 1171)
* label_orientation:                     Functions and Variables for draw.
                                                             (line 1197)
* labels:                                Functions and Variables for Command Line.
                                                             (line  355)
* lagrange:                              Functions and Variables for interpol.
                                                             (line    8)
* laguerre:                              Functions and Variables for orthogonal polynomials.
                                                             (line   61)
* lambda:                                Functions and Variables for Function Definition.
                                                             (line  632)
* lambert_w:                             Functions and Variables for Special Functions.
                                                             (line  149)
* laplace:                               Functions and Variables for Differentiation.
                                                             (line  575)
* laplacian_matrix:                      Functions and Variables for graphs.
                                                             (line  830)
* lassociative:                          Functions and Variables for Simplification.
                                                             (line  341)
* last:                                  Functions and Variables for Lists.
                                                             (line  221)
* lbfgs:                                 Functions and Variables for lbfgs.
                                                             (line    8)
* lbfgs_ncorrections:                    Functions and Variables for lbfgs.
                                                             (line  293)
* lbfgs_nfeval_max:                      Functions and Variables for lbfgs.
                                                             (line  284)
* lc2kdt:                                Functions and Variables for itensor.
                                                             (line  403)
* lc_l:                                  Functions and Variables for itensor.
                                                             (line  475)
* lc_u:                                  Functions and Variables for itensor.
                                                             (line  498)
* lcharp:                                Functions and Variables for characters.
                                                             (line   83)
* lcm:                                   Functions and Variables for Number Theory.
                                                             (line  324)
* ldefint:                               Functions and Variables for Integration.
                                                             (line  498)
* ldisp:                                 Functions and Variables for Display.
                                                             (line  349)
* ldisplay:                              Functions and Variables for Display.
                                                             (line  380)
* leftjust:                              Functions and Variables for Display.
                                                             (line  414)
* legend:                                Plotting Options.   (line  101)
* legendre_p:                            Functions and Variables for orthogonal polynomials.
                                                             (line   68)
* legendre_q:                            Functions and Variables for orthogonal polynomials.
                                                             (line   75)
* leinstein:                             Functions and Variables for ctensor.
                                                             (line  247)
* length:                                Functions and Variables for Lists.
                                                             (line  225)
* Less than:                             Relational operators.
                                                             (line    7)
* Less than or equal:                    Relational operators.
                                                             (line    8)
* let:                                   Functions and Variables for Rules and Patterns.
                                                             (line  226)
* let_rule_packages:                     Functions and Variables for Rules and Patterns.
                                                             (line  362)
* letrat:                                Functions and Variables for Rules and Patterns.
                                                             (line  307)
* letrules:                              Functions and Variables for Rules and Patterns.
                                                             (line  333)
* letsimp:                               Functions and Variables for Rules and Patterns.
                                                             (line  347)
* levi_civita:                           Functions and Variables for itensor.
                                                             (line  396)
* lfg:                                   Functions and Variables for ctensor.
                                                             (line 1177)
* lfreeof:                               Functions and Variables for Expressions.
                                                             (line  466)
* lg:                                    Functions and Variables for ctensor.
                                                             (line 1213)
* lgtreillis:                            Functions and Variables for Symmetries.
                                                             (line  479)
* lhospitallim:                          Functions and Variables for Limits.
                                                             (line    7)
* lhs:                                   Functions and Variables for Equations.
                                                             (line  457)
* li:                                    Root Exponential and Logarithmic Functions.
                                                             (line   61)
* liediff:                               Functions and Variables for itensor.
                                                             (line  649)
* limit:                                 Functions and Variables for Limits.
                                                             (line   17)
* limsubst:                              Functions and Variables for Limits.
                                                             (line   59)
* Lindstedt:                             Functions and Variables for lindstedt.
                                                             (line    7)
* line_graph:                            Functions and Variables for graphs.
                                                             (line  210)
* line_type:                             Functions and Variables for draw.
                                                             (line 1223)
* line_width:                            Functions and Variables for draw.
                                                             (line 1247)
* linear <1>:                            Package functs.     (line   68)
* linear:                                Functions and Variables for Simplification.
                                                             (line  347)
* linear_program:                        Functions and Variables for simplex.
                                                             (line   15)
* linear_solver:                         Functions and Variables for zeilberger.
                                                             (line  110)
* linearinterpol:                        Functions and Variables for interpol.
                                                             (line  111)
* linechar:                              Functions and Variables for Command Line.
                                                             (line  392)
* linel:                                 Functions and Variables for Display.
                                                             (line  433)
* linenum:                               Functions and Variables for Command Line.
                                                             (line  411)
* linsolve:                              Functions and Variables for Equations.
                                                             (line  499)
* linsolve_params:                       Functions and Variables for Equations.
                                                             (line  576)
* linsolvewarn:                          Functions and Variables for Equations.
                                                             (line  569)
* lispdisp:                              Functions and Variables for Display.
                                                             (line  444)
* List delimiters:                       Functions and Variables for Lists.
                                                             (line    8)
* list_correlations:                     Functions and Variables for descriptive statistics.
                                                             (line  649)
* list_matrix_entries:                   Functions and Variables for Matrices and Linear Algebra.
                                                             (line  735)
* list_nc_monomials:                     Functions and Variables for Affine.
                                                             (line  136)
* listarith:                             Functions and Variables for Lists.
                                                             (line  237)
* listarray:                             Functions and Variables for Arrays.
                                                             (line  267)
* listconstvars:                         Functions and Variables for Expressions.
                                                             (line  422)
* listdummyvars:                         Functions and Variables for Expressions.
                                                             (line  431)
* listify:                               Functions and Variables for Sets.
                                                             (line  523)
* listoftens:                            Functions and Variables for itensor.
                                                             (line   32)
* listofvars:                            Functions and Variables for Expressions.
                                                             (line  451)
* listp <1>:                             Functions and Variables for linearalgebra.
                                                             (line  267)
* listp:                                 Functions and Variables for Lists.
                                                             (line  247)
* lmax:                                  Functions for Numbers.
                                                             (line  195)
* lmin:                                  Functions for Numbers.
                                                             (line  201)
* lmxchar:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line  744)
* load:                                  Functions and Variables for File Input and Output.
                                                             (line  232)
* load_pathname:                         Functions and Variables for File Input and Output.
                                                             (line  260)
* loadfile:                              Functions and Variables for File Input and Output.
                                                             (line  286)
* loadprint:                             Functions and Variables for File Input and Output.
                                                             (line  297)
* local:                                 Functions and Variables for Function Definition.
                                                             (line  763)
* locate_matrix_entry:                   Functions and Variables for linearalgebra.
                                                             (line  275)
* log:                                   Root Exponential and Logarithmic Functions.
                                                             (line  120)
* log_gamma:                             Gamma and factorial Functions.
                                                             (line  117)
* logabs:                                Root Exponential and Logarithmic Functions.
                                                             (line  156)
* logand:                                Package functs.     (line   43)
* logarc:                                Root Exponential and Logarithmic Functions.
                                                             (line  167)
* logcb:                                 Functions and Variables for draw.
                                                             (line 1271)
* logconcoeffp:                          Root Exponential and Logarithmic Functions.
                                                             (line  177)
* logcontract:                           Root Exponential and Logarithmic Functions.
                                                             (line  188)
* logexpand:                             Root Exponential and Logarithmic Functions.
                                                             (line  210)
* Logical conjunction <1>:               Logical operators.  (line    7)
* Logical conjunction:                   Functions and Variables for to_poly_solve.
                                                             (line   16)
* Logical disjunction <1>:               Functions and Variables for to_poly_solve.
                                                             (line  134)
* Logical disjunction:                   Logical operators.  (line   43)
* Logical negation:                      Logical operators.  (line   30)
* lognegint:                             Root Exponential and Logarithmic Functions.
                                                             (line  221)
* logor:                                 Package functs.     (line   49)
* logsimp:                               Root Exponential and Logarithmic Functions.
                                                             (line  228)
* logx <1>:                              Functions and Variables for draw.
                                                             (line 1297)
* logx:                                  Plotting Options.   (line  112)
* logxor:                                Package functs.     (line   55)
* logy <1>:                              Plotting Options.   (line  117)
* logy:                                  Functions and Variables for draw.
                                                             (line 1315)
* logz:                                  Functions and Variables for draw.
                                                             (line 1333)
* lopow:                                 Functions and Variables for Polynomials.
                                                             (line  714)
* lorentz_gauge:                         Functions and Variables for itensor.
                                                             (line  984)
* lowercasep:                            Functions and Variables for characters.
                                                             (line   87)
* lpart:                                 Functions and Variables for Expressions.
                                                             (line  471)
* lratsubst:                             Functions and Variables for Polynomials.
                                                             (line  722)
* lreduce:                               Functions and Variables for Sets.
                                                             (line  537)
* lriem:                                 Functions and Variables for ctensor.
                                                             (line 1197)
* lriemann:                              Functions and Variables for ctensor.
                                                             (line  281)
* lsquares_estimates:                    Functions and Variables for lsquares.
                                                             (line    9)
* lsquares_estimates_approximate:        Functions and Variables for lsquares.
                                                             (line  149)
* lsquares_estimates_exact:              Functions and Variables for lsquares.
                                                             (line   99)
* lsquares_mse:                          Functions and Variables for lsquares.
                                                             (line  213)
* lsquares_residual_mse:                 Functions and Variables for lsquares.
                                                             (line  335)
* lsquares_residuals:                    Functions and Variables for lsquares.
                                                             (line  285)
* lsum:                                  Sums and Products.  (line   16)
* ltreillis:                             Functions and Variables for Symmetries.
                                                             (line  487)
* lu_backsub:                            Functions and Variables for linearalgebra.
                                                             (line  300)
* lu_factor:                             Functions and Variables for linearalgebra.
                                                             (line  305)
* m1pbranch:                             Functions and Variables for Numbers.
                                                             (line  141)
* Macro function definition operator:    Assignment operators.
                                                             (line  116)
* macroexpand:                           Macros.             (line  102)
* macroexpand1:                          Macros.             (line  136)
* macroexpansion:                        Functions and Variables for Function Definition.
                                                             (line  797)
* macros:                                Macros.             (line  168)
* mainvar:                               Functions and Variables for Expressions.
                                                             (line  477)
* make_array:                            Functions and Variables for Arrays.
                                                             (line  358)
* make_graph:                            Functions and Variables for graphs.
                                                             (line  214)
* make_level_picture:                    Functions and Variables for pictures.
                                                             (line   12)
* make_poly_continent:                   Functions and Variables for worldmap.
                                                             (line   40)
* make_poly_country:                     Functions and Variables for worldmap.
                                                             (line   55)
* make_polygon:                          Functions and Variables for worldmap.
                                                             (line   67)
* make_random_state:                     Random Numbers.     (line    7)
* make_rgb_picture:                      Functions and Variables for pictures.
                                                             (line   46)
* make_transform:                        Functions and Variables for Plotting.
                                                             (line   69)
* makebox:                               Functions and Variables for itensor.
                                                             (line  781)
* makefact:                              Gamma and factorial Functions.
                                                             (line  723)
* makegamma:                             Gamma and factorial Functions.
                                                             (line  166)
* makelist:                              Functions and Variables for Lists.
                                                             (line  251)
* makeOrders:                            Functions and Variables for makeOrders.
                                                             (line    7)
* makeset:                               Functions and Variables for Sets.
                                                             (line  577)
* mandelbrot:                            Functions and Variables for dynamics.
                                                             (line   85)
* mandelbrot_set:                        Definitions for complex fractals.
                                                             (line    7)
* manual_demo:                           Functions and Variables for Help.
                                                             (line  170)
* map:                                   Functions and Variables for Program Flow.
                                                             (line  446)
* mapatom:                               Functions and Variables for Program Flow.
                                                             (line  481)
* maperror:                              Functions and Variables for Program Flow.
                                                             (line  487)
* maplist:                               Functions and Variables for Program Flow.
                                                             (line  515)
* mapprint:                              Functions and Variables for Program Flow.
                                                             (line  504)
* mat_cond:                              Functions and Variables for linearalgebra.
                                                             (line  422)
* mat_fullunblocker:                     Functions and Variables for linearalgebra.
                                                             (line  456)
* mat_function:                          Functions and Variables for diag.
                                                             (line  189)
* mat_norm:                              Functions and Variables for linearalgebra.
                                                             (line  435)
* mat_trace:                             Functions and Variables for linearalgebra.
                                                             (line  461)
* mat_unblocker:                         Functions and Variables for linearalgebra.
                                                             (line  467)
* matchdeclare:                          Functions and Variables for Rules and Patterns.
                                                             (line  369)
* matchfix:                              User defined operators.
                                                             (line  105)
* matrix:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line  761)
* matrix_element_add:                    Functions and Variables for Matrices and Linear Algebra.
                                                             (line  926)
* matrix_element_mult:                   Functions and Variables for Matrices and Linear Algebra.
                                                             (line  958)
* matrix_element_transpose:              Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1000)
* matrix_size:                           Functions and Variables for linearalgebra.
                                                             (line  451)
* matrixmap:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line  916)
* matrixp <1>:                           Functions and Variables for linearalgebra.
                                                             (line  442)
* matrixp:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line  922)
* mattrace:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1051)
* max:                                   Functions for Numbers.
                                                             (line  207)
* max_clique:                            Functions and Variables for graphs.
                                                             (line  847)
* max_degree:                            Functions and Variables for graphs.
                                                             (line  857)
* max_flow:                              Functions and Variables for graphs.
                                                             (line  870)
* max_independent_set:                   Functions and Variables for graphs.
                                                             (line  901)
* max_matching:                          Functions and Variables for graphs.
                                                             (line  912)
* MAX_ORD:                               Functions and Variables for zeilberger.
                                                             (line   96)
* maxapplydepth:                         Functions and Variables for Rules and Patterns.
                                                             (line  527)
* maxapplyheight:                        Functions and Variables for Rules and Patterns.
                                                             (line  534)
* maxima_tempdir:                        Functions and Variables for Runtime Environment.
                                                             (line    7)
* maxima_userdir:                        Functions and Variables for Runtime Environment.
                                                             (line   19)
* maximize_lp:                           Functions and Variables for simplex.
                                                             (line   42)
* maxnegex:                              Functions and Variables for Simplification.
                                                             (line  380)
* maxposex:                              Functions and Variables for Simplification.
                                                             (line  387)
* maxpsifracdenom:                       Gamma and factorial Functions.
                                                             (line  715)
* maxpsifracnum:                         Gamma and factorial Functions.
                                                             (line  707)
* maxpsinegint:                          Gamma and factorial Functions.
                                                             (line  698)
* maxpsiposint:                          Gamma and factorial Functions.
                                                             (line  691)
* maxtayorder:                           Functions and Variables for Series.
                                                             (line   72)
* maybe:                                 Functions and Variables for Facts.
                                                             (line  377)
* mean:                                  Functions and Variables for descriptive statistics.
                                                             (line    7)
* mean_bernoulli:                        Functions and Variables for discrete distributions.
                                                             (line  270)
* mean_beta:                             Functions and Variables for continuous distributions.
                                                             (line  922)
* mean_binomial:                         Functions and Variables for discrete distributions.
                                                             (line  132)
* mean_chi2:                             Functions and Variables for continuous distributions.
                                                             (line  335)
* mean_continuous_uniform:               Functions and Variables for continuous distributions.
                                                             (line  982)
* mean_deviation:                        Functions and Variables for descriptive statistics.
                                                             (line  296)
* mean_discrete_uniform:                 Functions and Variables for discrete distributions.
                                                             (line  442)
* mean_exp:                              Functions and Variables for continuous distributions.
                                                             (line  640)
* mean_f:                                Functions and Variables for continuous distributions.
                                                             (line  539)
* mean_gamma:                            Functions and Variables for continuous distributions.
                                                             (line  849)
* mean_general_finite_discrete:          Functions and Variables for discrete distributions.
                                                             (line   54)
* mean_geometric:                        Functions and Variables for discrete distributions.
                                                             (line  383)
* mean_gumbel:                           Functions and Variables for continuous distributions.
                                                             (line 1491)
* mean_hypergeometric:                   Functions and Variables for discrete distributions.
                                                             (line  513)
* mean_laplace:                          Functions and Variables for continuous distributions.
                                                             (line 1403)
* mean_logistic:                         Functions and Variables for continuous distributions.
                                                             (line 1044)
* mean_lognormal:                        Functions and Variables for continuous distributions.
                                                             (line  783)
* mean_negative_binomial:                Functions and Variables for discrete distributions.
                                                             (line  590)
* mean_noncentral_chi2:                  Functions and Variables for continuous distributions.
                                                             (line  464)
* mean_noncentral_student_t:             Functions and Variables for continuous distributions.
                                                             (line  215)
* mean_normal:                           Functions and Variables for continuous distributions.
                                                             (line   44)
* mean_pareto:                           Functions and Variables for continuous distributions.
                                                             (line 1102)
* mean_poisson:                          Functions and Variables for discrete distributions.
                                                             (line  199)
* mean_rayleigh:                         Functions and Variables for continuous distributions.
                                                             (line 1260)
* mean_student_t:                        Functions and Variables for continuous distributions.
                                                             (line  115)
* mean_weibull:                          Functions and Variables for continuous distributions.
                                                             (line 1160)
* median:                                Functions and Variables for descriptive statistics.
                                                             (line  255)
* median_deviation:                      Functions and Variables for descriptive statistics.
                                                             (line  324)
* member:                                Functions and Variables for Lists.
                                                             (line  293)
* mesh:                                  Functions and Variables for draw.
                                                             (line 2973)
* mesh_lines_color:                      Plotting Options.   (line  122)
* method:                                Functions and Variables for contrib_ode.
                                                             (line   72)
* metricexpandall:                       Functions and Variables for Units.
                                                             (line  395)
* min:                                   Functions for Numbers.
                                                             (line  219)
* min_degree:                            Functions and Variables for graphs.
                                                             (line  924)
* min_edge_cut:                          Functions and Variables for graphs.
                                                             (line  937)
* min_vertex_cover:                      Functions and Variables for graphs.
                                                             (line  943)
* min_vertex_cut:                        Functions and Variables for graphs.
                                                             (line  947)
* minf:                                  Functions and Variables for Constants.
                                                             (line   45)
* minfactorial:                          Combinatorial Functions.
                                                             (line  166)
* minimalPoly:                           Functions and Variables for diag.
                                                             (line  129)
* minimize_lp:                           Functions and Variables for simplex.
                                                             (line   50)
* minimum_spanning_tree:                 Functions and Variables for graphs.
                                                             (line  953)
* minor:                                 Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1061)
* minpack_lsquares:                      Functions and Variables for minpack.
                                                             (line    8)
* minpack_solve:                         Functions and Variables for minpack.
                                                             (line   73)
* Minus infinity:                        Functions and Variables for Constants.
                                                             (line   45)
* mnewton:                               Functions and Variables for mnewton.
                                                             (line   25)
* mod:                                   Functions and Variables for Number Theory.
                                                             (line  331)
* mod_big_prime:                         Functions and Variables for zeilberger.
                                                             (line  164)
* mod_test:                              Functions and Variables for zeilberger.
                                                             (line  143)
* mod_threshold:                         Functions and Variables for zeilberger.
                                                             (line  171)
* mode_check_errorp:                     Functions and Variables for Function Definition.
                                                             (line  942)
* mode_check_warnp:                      Functions and Variables for Function Definition.
                                                             (line  948)
* mode_checkp:                           Functions and Variables for Function Definition.
                                                             (line  935)
* mode_declare:                          Functions and Variables for Function Definition.
                                                             (line  954)
* mode_identity:                         Functions and Variables for Function Definition.
                                                             (line  998)
* ModeMatrix:                            Functions and Variables for diag.
                                                             (line  152)
* modular_linear_solver:                 Functions and Variables for zeilberger.
                                                             (line  150)
* modulus:                               Functions and Variables for Polynomials.
                                                             (line  752)
* moebius:                               Functions and Variables for Sets.
                                                             (line  611)
* mon2schur:                             Functions and Variables for Symmetries.
                                                             (line   67)
* mono:                                  Functions and Variables for Affine.
                                                             (line  110)
* monomial_dimensions:                   Functions and Variables for Affine.
                                                             (line  118)
* multi_elem:                            Functions and Variables for Symmetries.
                                                             (line   95)
* multi_orbit:                           Functions and Variables for Symmetries.
                                                             (line  375)
* multi_pui:                             Functions and Variables for Symmetries.
                                                             (line  112)
* multibernstein_poly:                   Functions and Variables for Bernstein.
                                                             (line   67)
* multinomial:                           Functions and Variables for Symmetries.
                                                             (line  891)
* multinomial_coeff:                     Functions and Variables for Sets.
                                                             (line  644)
* Multiplication:                        Arithmetic operators.
                                                             (line    9)
* multiplicative:                        Functions and Variables for Simplification.
                                                             (line  394)
* multiplicities:                        Functions and Variables for Equations.
                                                             (line  586)
* multiplot_mode:                        Functions and Variables for draw.
                                                             (line  139)
* multsym:                               Functions and Variables for Symmetries.
                                                             (line  389)
* multthru:                              Functions and Variables for Simplification.
                                                             (line  420)
* mycielski_graph:                       Functions and Variables for graphs.
                                                             (line  245)
* myoptions:                             Functions and Variables for Command Line.
                                                             (line  416)
* N'th previous output:                  Functions and Variables for Command Line.
                                                             (line  153)
* nary <1>:                              User defined operators.
                                                             (line  209)
* nary <2>:                              Functions and Variables for Simplification.
                                                             (line  466)
* nary:                                  User defined operators.
                                                             (line  208)
* natural_unit:                          Functions and Variables for ezunits.
                                                             (line  703)
* nc_degree:                             Functions and Variables for Affine.
                                                             (line   69)
* ncexpt:                                Functions and Variables for Display.
                                                             (line  162)
* ncharpoly:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1066)
* Negative infinity:                     Functions and Variables for Constants.
                                                             (line   45)
* negative_picture:                      Functions and Variables for pictures.
                                                             (line   76)
* negdistrib:                            Functions and Variables for Simplification.
                                                             (line  485)
* negsumdispflag:                        Functions and Variables for Display.
                                                             (line  461)
* neighbors:                             Functions and Variables for graphs.
                                                             (line  963)
* new:                                   Functions and Variables for Structures.
                                                             (line   50)
* new_graph:                             Functions and Variables for graphs.
                                                             (line  249)
* new_variable:                          Functions and Variables for to_poly_solve.
                                                             (line  543)
* newcontext:                            Functions and Variables for Facts.
                                                             (line  401)
* newdet:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1083)
* newline <1>:                           Functions and Variables for input and output.
                                                             (line   49)
* newline:                               Functions and Variables for characters.
                                                             (line   91)
* newton:                                Functions and Variables for fast Fourier transform.
                                                             (line  322)
* newtonepsilon:                         Functions and Variables for mnewton.
                                                             (line    7)
* newtonmaxiter:                         Functions and Variables for mnewton.
                                                             (line   16)
* next_prime:                            Functions and Variables for Number Theory.
                                                             (line  359)
* nextlayerfactor:                       Package facexp.     (line   65)
* nicedummies:                           Functions and Variables for to_poly_solve.
                                                             (line  588)
* niceindices:                           Functions and Variables for Series.
                                                             (line   80)
* niceindicespref:                       Functions and Variables for Series.
                                                             (line  115)
* ninth:                                 Functions and Variables for Lists.
                                                             (line  329)
* nm:                                    Functions and Variables for ctensor.
                                                             (line 1262)
* nmc:                                   Functions and Variables for ctensor.
                                                             (line 1266)
* noeval:                                Functions and Variables for Evaluation.
                                                             (line  587)
* nofix:                                 User defined operators.
                                                             (line  222)
* nolabels:                              Functions and Variables for Command Line.
                                                             (line  423)
* nonarray:                              Functions and Variables for Properties.
                                                             (line  409)
* noncentral_moment:                     Functions and Variables for descriptive statistics.
                                                             (line  118)
* Noncommutative exponentiation:         Arithmetic operators.
                                                             (line  190)
* Noncommutative multiplication:         Arithmetic operators.
                                                             (line  209)
* nonegative_lp:                         Functions and Variables for simplex.
                                                             (line   91)
* noninteger:                            Functions and Variables for Properties.
                                                             (line  375)
* nonmetricity:                          Functions and Variables for ctensor.
                                                             (line  669)
* nonnegintegerp:                        Functions and Variables for Numbers.
                                                             (line  160)
* nonscalar:                             Functions and Variables for Properties.
                                                             (line  430)
* nonscalarp:                            Functions and Variables for Properties.
                                                             (line  437)
* nonzeroandfreeof:                      Package functs.     (line   61)
* not:                                   Logical operators.  (line   30)
* Not equal (syntactic inequality):      Operators for Equations.
                                                             (line    7)
* notequal:                              Functions and Variables for Predicates.
                                                             (line  157)
* noun:                                  Functions and Variables for Expressions.
                                                             (line  490)
* noundisp:                              Functions and Variables for Expressions.
                                                             (line  509)
* nounify:                               Functions and Variables for Expressions.
                                                             (line  516)
* nouns:                                 Functions and Variables for Evaluation.
                                                             (line  593)
* np:                                    Functions and Variables for ctensor.
                                                             (line 1234)
* npi:                                   Functions and Variables for ctensor.
                                                             (line 1238)
* nptetrad:                              Functions and Variables for ctensor.
                                                             (line  490)
* npv:                                   Functions and Variables for finance.
                                                             (line  216)
* nroots:                                Functions and Variables for Equations.
                                                             (line  593)
* nterms:                                Functions and Variables for Expressions.
                                                             (line  526)
* ntermst:                               Functions and Variables for ctensor.
                                                             (line  843)
* nthroot:                               Functions and Variables for Equations.
                                                             (line  605)
* nticks <1>:                            Functions and Variables for draw.
                                                             (line 1351)
* nticks:                                Plotting Options.   (line  133)
* ntrig:                                 Functions and Variables for Trigonometric.
                                                             (line  224)
* nullity:                               Functions and Variables for linearalgebra.
                                                             (line  508)
* nullspace:                             Functions and Variables for linearalgebra.
                                                             (line  501)
* num:                                   Functions and Variables for Polynomials.
                                                             (line  796)
* num_distinct_partitions:               Functions and Variables for Sets.
                                                             (line  678)
* num_partitions:                        Functions and Variables for Sets.
                                                             (line  699)
* numbered_boundaries:                   Functions and Variables for worldmap.
                                                             (line   24)
* numberp:                               Functions and Variables for Numbers.
                                                             (line  164)
* numer:                                 Functions and Variables for Numbers.
                                                             (line  192)
* numer_pbranch:                         Functions and Variables for Numbers.
                                                             (line  211)
* numerval:                              Functions and Variables for Numbers.
                                                             (line  240)
* numfactor:                             Gamma and factorial Functions.
                                                             (line  730)
* nusum:                                 Functions and Variables for Series.
                                                             (line  145)
* nzeta:                                 Functions and Variables for Special Functions.
                                                             (line  154)
* nzetai:                                Functions and Variables for Special Functions.
                                                             (line  163)
* nzetar:                                Functions and Variables for Special Functions.
                                                             (line  159)
* obase:                                 Functions and Variables for Display.
                                                             (line  471)
* odd:                                   Functions and Variables for Properties.
                                                             (line  262)
* odd_girth:                             Functions and Variables for graphs.
                                                             (line  973)
* oddfun:                                Functions and Variables for Simplification.
                                                             (line  164)
* oddp:                                  Functions and Variables for Numbers.
                                                             (line  250)
* ode2:                                  Functions and Variables for Differential Equations.
                                                             (line   93)
* ode_check:                             Functions and Variables for contrib_ode.
                                                             (line   42)
* odelin:                                Functions and Variables for contrib_ode.
                                                             (line   12)
* op:                                    Functions and Variables for Expressions.
                                                             (line  534)
* opena:                                 Functions and Variables for input and output.
                                                             (line   55)
* opena_binary:                          Functions and Variables for binary input and output.
                                                             (line   28)
* openr:                                 Functions and Variables for input and output.
                                                             (line   60)
* openr_binary:                          Functions and Variables for binary input and output.
                                                             (line   18)
* openw:                                 Functions and Variables for input and output.
                                                             (line   65)
* openw_binary:                          Functions and Variables for binary input and output.
                                                             (line   23)
* operatorp:                             Functions and Variables for Expressions.
                                                             (line  583)
* opproperties:                          Functions and Variables for Simplification.
                                                             (line  496)
* opsubst <1>:                           Functions and Variables for opsubst.
                                                             (line    9)
* opsubst <2>:                           Functions and Variables for Expressions.
                                                             (line  591)
* opsubst:                               Functions and Variables for opsubst.
                                                             (line    8)
* optimize:                              Functions and Variables for Expressions.
                                                             (line  599)
* optimprefix:                           Functions and Variables for Expressions.
                                                             (line  607)
* optionset:                             Functions and Variables for Command Line.
                                                             (line  442)
* or:                                    Logical operators.  (line   43)
* orbit:                                 Functions and Variables for Symmetries.
                                                             (line  409)
* orbits:                                Functions and Variables for dynamics.
                                                             (line  110)
* ordergreat:                            Functions and Variables for Expressions.
                                                             (line  614)
* ordergreatp:                           Functions and Variables for Expressions.
                                                             (line  632)
* orderless:                             Functions and Variables for Expressions.
                                                             (line  615)
* orderlessp:                            Functions and Variables for Expressions.
                                                             (line  633)
* orthogonal_complement:                 Functions and Variables for linearalgebra.
                                                             (line  512)
* orthopoly_recur:                       Functions and Variables for orthogonal polynomials.
                                                             (line   81)
* orthopoly_returns_intervals:           Functions and Variables for orthogonal polynomials.
                                                             (line  110)
* orthopoly_weight:                      Functions and Variables for orthogonal polynomials.
                                                             (line  120)
* out_neighbors:                         Functions and Variables for graphs.
                                                             (line  985)
* outative:                              Functions and Variables for Simplification.
                                                             (line  504)
* outchar:                               Functions and Variables for Command Line.
                                                             (line  460)
* outermap:                              Functions and Variables for Program Flow.
                                                             (line  597)
* outofpois:                             Poisson series.     (line   11)
* packagefile:                           Functions and Variables for Miscellaneous Options.
                                                             (line   57)
* pade:                                  Functions and Variables for Series.
                                                             (line  196)
* palette <1>:                           Plotting Options.   (line  143)
* palette:                               Functions and Variables for draw.
                                                             (line 1372)
* parabolic_cylinder_d:                  Parabolic Cylinder Functions.
                                                             (line   13)
* parametric:                            Functions and Variables for draw.
                                                             (line 3022)
* parametric_surface:                    Functions and Variables for draw.
                                                             (line 3062)
* parg:                                  Functions and Variables for to_poly_solve.
                                                             (line  608)
* parGosper:                             Functions and Variables for zeilberger.
                                                             (line   62)
* parse_string:                          Functions and Variables for strings.
                                                             (line   46)
* part:                                  Functions and Variables for Expressions.
                                                             (line  717)
* part2cont:                             Functions and Variables for Symmetries.
                                                             (line  261)
* partfrac:                              Functions and Variables for Number Theory.
                                                             (line  366)
* partition:                             Functions and Variables for Expressions.
                                                             (line  755)
* partition_set:                         Functions and Variables for Sets.
                                                             (line  723)
* partpol:                               Functions and Variables for Symmetries.
                                                             (line  271)
* partswitch:                            Functions and Variables for Expressions.
                                                             (line  771)
* path_digraph:                          Functions and Variables for graphs.
                                                             (line  253)
* path_graph:                            Functions and Variables for graphs.
                                                             (line  257)
* pathname_directory:                    Functions and Variables for File Input and Output.
                                                             (line  313)
* pathname_name:                         Functions and Variables for File Input and Output.
                                                             (line  314)
* pathname_type:                         Functions and Variables for File Input and Output.
                                                             (line  315)
* pdf_bernoulli:                         Functions and Variables for discrete distributions.
                                                             (line  241)
* pdf_beta:                              Functions and Variables for continuous distributions.
                                                             (line  896)
* pdf_binomial:                          Functions and Variables for discrete distributions.
                                                             (line  104)
* pdf_cauchy:                            Functions and Variables for continuous distributions.
                                                             (line 1442)
* pdf_chi2:                              Functions and Variables for continuous distributions.
                                                             (line  281)
* pdf_continuous_uniform:                Functions and Variables for continuous distributions.
                                                             (line  963)
* pdf_discrete_uniform:                  Functions and Variables for discrete distributions.
                                                             (line  422)
* pdf_exp:                               Functions and Variables for continuous distributions.
                                                             (line  583)
* pdf_f:                                 Functions and Variables for continuous distributions.
                                                             (line  502)
* pdf_gamma:                             Functions and Variables for continuous distributions.
                                                             (line  823)
* pdf_general_finite_discrete:           Functions and Variables for discrete distributions.
                                                             (line    7)
* pdf_geometric:                         Functions and Variables for discrete distributions.
                                                             (line  364)
* pdf_gumbel:                            Functions and Variables for continuous distributions.
                                                             (line 1472)
* pdf_hypergeometric:                    Functions and Variables for discrete distributions.
                                                             (line  485)
* pdf_laplace:                           Functions and Variables for continuous distributions.
                                                             (line 1384)
* pdf_logistic:                          Functions and Variables for continuous distributions.
                                                             (line 1025)
* pdf_lognormal:                         Functions and Variables for continuous distributions.
                                                             (line  747)
* pdf_negative_binomial:                 Functions and Variables for discrete distributions.
                                                             (line  562)
* pdf_noncentral_chi2:                   Functions and Variables for continuous distributions.
                                                             (line  439)
* pdf_noncentral_student_t:              Functions and Variables for continuous distributions.
                                                             (line  169)
* pdf_normal:                            Functions and Variables for continuous distributions.
                                                             (line    7)
* pdf_pareto:                            Functions and Variables for continuous distributions.
                                                             (line 1083)
* pdf_poisson:                           Functions and Variables for discrete distributions.
                                                             (line  175)
* pdf_rank_sum:                          Functions and Variables for special distributions.
                                                             (line   23)
* pdf_rayleigh:                          Functions and Variables for continuous distributions.
                                                             (line 1199)
* pdf_signed_rank:                       Functions and Variables for special distributions.
                                                             (line    7)
* pdf_student_t:                         Functions and Variables for continuous distributions.
                                                             (line   87)
* pdf_weibull:                           Functions and Variables for continuous distributions.
                                                             (line 1141)
* pearson_skewness:                      Functions and Variables for descriptive statistics.
                                                             (line  452)
* permanent:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1089)
* permut:                                Functions and Variables for Symmetries.
                                                             (line  898)
* permutation:                           Package functs.     (line  205)
* permutations:                          Functions and Variables for Sets.
                                                             (line  744)
* petersen_graph:                        Functions and Variables for graphs.
                                                             (line  261)
* petrov:                                Functions and Variables for ctensor.
                                                             (line  536)
* pfeformat:                             Functions and Variables for Display.
                                                             (line  506)
* phi:                                   Functions and Variables for Constants.
                                                             (line   49)
* pi:                                    Functions and Variables for Constants.
                                                             (line  105)
* pickapart:                             Functions and Variables for Expressions.
                                                             (line  779)
* picture_equalp:                        Functions and Variables for pictures.
                                                             (line   80)
* picturep:                              Functions and Variables for pictures.
                                                             (line   84)
* piece:                                 Functions and Variables for Expressions.
                                                             (line  866)
* piechart:                              Functions and Variables for statistical graphs.
                                                             (line  289)
* piechart_description:                  Functions and Variables for statistical graphs.
                                                             (line  296)
* planar_embedding:                      Functions and Variables for graphs.
                                                             (line  998)
* playback:                              Functions and Variables for Command Line.
                                                             (line  490)
* plog:                                  Root Exponential and Logarithmic Functions.
                                                             (line  235)
* plot2d:                                Functions and Variables for Plotting.
                                                             (line   91)
* plot3d:                                Functions and Variables for Plotting.
                                                             (line  237)
* plot_format:                           Plotting Options.   (line  172)
* plot_options:                          Functions and Variables for Plotting.
                                                             (line  344)
* plot_realpart:                         Plotting Options.   (line  182)
* plotdf:                                Functions and Variables for plotdf.
                                                             (line    7)
* plsquares:                             Functions and Variables for lsquares.
                                                             (line  386)
* pochhammer:                            Functions and Variables for orthogonal polynomials.
                                                             (line  138)
* pochhammer_max_index:                  Functions and Variables for orthogonal polynomials.
                                                             (line  170)
* point_size:                            Functions and Variables for draw.
                                                             (line 1448)
* point_type <1>:                        Plotting Options.   (line  196)
* point_type:                            Functions and Variables for draw.
                                                             (line 1470)
* points:                                Functions and Variables for draw.
                                                             (line 3100)
* points_joined:                         Functions and Variables for draw.
                                                             (line 1513)
* poisdiff:                              Poisson series.     (line   19)
* poisexpt:                              Poisson series.     (line   24)
* poisint:                               Poisson series.     (line   29)
* poislim:                               Poisson series.     (line   35)
* poismap:                               Poisson series.     (line   44)
* poisplus:                              Poisson series.     (line   51)
* poissimp:                              Poisson series.     (line   55)
* poisson:                               Poisson series.     (line   60)
* poissubst:                             Poisson series.     (line   65)
* poistimes:                             Poisson series.     (line   85)
* poistrim:                              Poisson series.     (line   89)
* polar:                                 Functions and Variables for draw.
                                                             (line 3255)
* polar_to_xy:                           Functions and Variables for Plotting.
                                                             (line   83)
* polarform:                             Functions for Complex Numbers.
                                                             (line  134)
* polartorect:                           Functions and Variables for fast Fourier transform.
                                                             (line    7)
* poly_add:                              Functions and Variables for grobner.
                                                             (line   95)
* poly_buchberger:                       Functions and Variables for grobner.
                                                             (line  215)
* poly_buchberger_criterion:             Functions and Variables for grobner.
                                                             (line  208)
* poly_coefficient_ring:                 Functions and Variables for grobner.
                                                             (line   18)
* poly_colon_ideal:                      Functions and Variables for grobner.
                                                             (line  274)
* poly_content:                          Functions and Variables for grobner.
                                                             (line  177)
* poly_depends_p:                        Functions and Variables for grobner.
                                                             (line  263)
* poly_elimination_ideal:                Functions and Variables for grobner.
                                                             (line  268)
* poly_elimination_order:                Functions and Variables for grobner.
                                                             (line   41)
* poly_exact_divide:                     Functions and Variables for grobner.
                                                             (line  197)
* poly_expand:                           Functions and Variables for grobner.
                                                             (line  140)
* poly_expt:                             Functions and Variables for grobner.
                                                             (line  169)
* poly_gcd:                              Functions and Variables for grobner.
                                                             (line  291)
* poly_grobner:                          Functions and Variables for grobner.
                                                             (line  253)
* poly_grobner_algorithm:                Functions and Variables for grobner.
                                                             (line   66)
* poly_grobner_debug:                    Functions and Variables for grobner.
                                                             (line   60)
* poly_grobner_equal:                    Functions and Variables for grobner.
                                                             (line  309)
* poly_grobner_member:                   Functions and Variables for grobner.
                                                             (line  330)
* poly_grobner_subsetp:                  Functions and Variables for grobner.
                                                             (line  323)
* poly_ideal_intersection:               Functions and Variables for grobner.
                                                             (line  283)
* poly_ideal_polysaturation:             Functions and Variables for grobner.
                                                             (line  374)
* poly_ideal_polysaturation1:            Functions and Variables for grobner.
                                                             (line  362)
* poly_ideal_saturation:                 Functions and Variables for grobner.
                                                             (line  351)
* poly_ideal_saturation1:                Functions and Variables for grobner.
                                                             (line  340)
* poly_lcm:                              Functions and Variables for grobner.
                                                             (line  287)
* poly_minimization:                     Functions and Variables for grobner.
                                                             (line  240)
* poly_monomial_order:                   Functions and Variables for grobner.
                                                             (line   10)
* poly_multiply:                         Functions and Variables for grobner.
                                                             (line  111)
* poly_normal_form:                      Functions and Variables for grobner.
                                                             (line  203)
* poly_normalize:                        Functions and Variables for grobner.
                                                             (line  131)
* poly_normalize_list:                   Functions and Variables for grobner.
                                                             (line  247)
* poly_polysaturation_extension:         Functions and Variables for grobner.
                                                             (line  388)
* poly_primary_elimination_order:        Functions and Variables for grobner.
                                                             (line   27)
* poly_primitive_part:                   Functions and Variables for grobner.
                                                             (line  123)
* poly_pseudo_divide:                    Functions and Variables for grobner.
                                                             (line  184)
* poly_reduced_grobner:                  Functions and Variables for grobner.
                                                             (line  258)
* poly_reduction:                        Functions and Variables for grobner.
                                                             (line  234)
* poly_return_term_list:                 Functions and Variables for grobner.
                                                             (line   52)
* poly_s_polynomial:                     Functions and Variables for grobner.
                                                             (line  118)
* poly_saturation_extension:             Functions and Variables for grobner.
                                                             (line  383)
* poly_secondary_elimination_order:      Functions and Variables for grobner.
                                                             (line   34)
* poly_subtract:                         Functions and Variables for grobner.
                                                             (line  103)
* poly_top_reduction_only:               Functions and Variables for grobner.
                                                             (line   79)
* polydecomp:                            Functions and Variables for Polynomials.
                                                             (line  803)
* polyfactor:                            Functions and Variables for Equations.
                                                             (line  613)
* polygon:                               Functions and Variables for draw.
                                                             (line 3280)
* polymod:                               Functions and Variables for Polynomials.
                                                             (line  855)
* polynome2ele:                          Functions and Variables for Symmetries.
                                                             (line  543)
* polynomialp:                           Functions and Variables for linearalgebra.
                                                             (line  519)
* polytocompanion:                       Functions and Variables for linearalgebra.
                                                             (line  557)
* pop:                                   Functions and Variables for Lists.
                                                             (line  334)
* posfun:                                Functions and Variables for Properties.
                                                             (line  444)
* postfix:                               User defined operators.
                                                             (line  234)
* potential:                             Functions and Variables for Integration.
                                                             (line  512)
* power_mod:                             Functions and Variables for Number Theory.
                                                             (line  391)
* powerdisp:                             Functions and Variables for Display.
                                                             (line  531)
* powers:                                Functions and Variables for Polynomials.
                                                             (line  867)
* powerseries:                           Functions and Variables for Series.
                                                             (line  259)
* powerset:                              Functions and Variables for Sets.
                                                             (line  763)
* pred:                                  Functions and Variables for Evaluation.
                                                             (line  601)
* prederror:                             Functions and Variables for Program Flow.
                                                             (line  526)
* prefix:                                User defined operators.
                                                             (line  246)
* prev_prime:                            Functions and Variables for Number Theory.
                                                             (line  429)
* Previous input:                        Functions and Variables for Command Line.
                                                             (line   47)
* Previous output:                       Functions and Variables for Command Line.
                                                             (line   92)
* Previous result in compound expression: Functions and Variables for Command Line.
                                                             (line  106)
* primep:                                Functions and Variables for Number Theory.
                                                             (line  406)
* primep_number_of_tests:                Functions and Variables for Number Theory.
                                                             (line  423)
* print:                                 Functions and Variables for Display.
                                                             (line  556)
* print_graph:                           Functions and Variables for graphs.
                                                             (line 1014)
* printf:                                Functions and Variables for input and output.
                                                             (line   72)
* printfile:                             Functions and Variables for File Input and Output.
                                                             (line  328)
* printpois:                             Poisson series.     (line   98)
* printprops:                            Functions and Variables for Properties.
                                                             (line  452)
* prodrac:                               Functions and Variables for Symmetries.
                                                             (line  556)
* product:                               Sums and Products.  (line   48)
* product_use_gamma:                     Functions and Variables for solve_rec.
                                                             (line  175)
* program:                               Functions and Variables for graphs.
                                                             (line 1666)
* programmode:                           Functions and Variables for Equations.
                                                             (line  624)
* prompt:                                Functions and Variables for Command Line.
                                                             (line  537)
* properties:                            Functions and Variables for Properties.
                                                             (line  463)
* proportional_axes:                     Functions and Variables for draw.
                                                             (line 1545)
* props:                                 Functions and Variables for Properties.
                                                             (line  468)
* propvars:                              Functions and Variables for Properties.
                                                             (line  479)
* psexpand:                              Functions and Variables for Series.
                                                             (line  304)
* psfile:                                Plotting Options.   (line  210)
* psi <1>:                               Functions and Variables for ctensor.
                                                             (line  513)
* psi:                                   Gamma and factorial Functions.
                                                             (line  672)
* psubst:                                Functions and Variables for Expressions.
                                                             (line  873)
* ptriangularize:                        Functions and Variables for linearalgebra.
                                                             (line  565)
* pui:                                   Functions and Variables for Symmetries.
                                                             (line  124)
* pui2comp:                              Functions and Variables for Symmetries.
                                                             (line  151)
* pui2ele:                               Functions and Variables for Symmetries.
                                                             (line  178)
* pui2polynome:                          Functions and Variables for Symmetries.
                                                             (line  564)
* pui_direct:                            Functions and Variables for Symmetries.
                                                             (line  424)
* puireduc:                              Functions and Variables for Symmetries.
                                                             (line  188)
* push:                                  Functions and Variables for Lists.
                                                             (line  348)
* put:                                   Functions and Variables for Properties.
                                                             (line  485)
* pv:                                    Functions and Variables for finance.
                                                             (line   30)
* qput:                                  Functions and Variables for Properties.
                                                             (line  512)
* qrange:                                Functions and Variables for descriptive statistics.
                                                             (line  278)
* qty:                                   Functions and Variables for ezunits.
                                                             (line  362)
* quad_control:                          Functions and Variables for QUADPACK.
                                                             (line  704)
* quad_qag:                              Functions and Variables for QUADPACK.
                                                             (line   10)
* quad_qagi:                             Functions and Variables for QUADPACK.
                                                             (line  171)
* quad_qagp:                             Functions and Variables for QUADPACK.
                                                             (line  621)
* quad_qags:                             Functions and Variables for QUADPACK.
                                                             (line   91)
* quad_qawc:                             Functions and Variables for QUADPACK.
                                                             (line  255)
* quad_qawf:                             Functions and Variables for QUADPACK.
                                                             (line  347)
* quad_qawo:                             Functions and Variables for QUADPACK.
                                                             (line  435)
* quad_qaws:                             Functions and Variables for QUADPACK.
                                                             (line  529)
* quadrilateral:                         Functions and Variables for draw.
                                                             (line 3306)
* quantile:                              Functions and Variables for descriptive statistics.
                                                             (line  234)
* quantile_bernoulli:                    Functions and Variables for discrete distributions.
                                                             (line  263)
* quantile_beta:                         Functions and Variables for continuous distributions.
                                                             (line  915)
* quantile_binomial:                     Functions and Variables for discrete distributions.
                                                             (line  125)
* quantile_cauchy:                       Functions and Variables for continuous distributions.
                                                             (line 1454)
* quantile_chi2:                         Functions and Variables for continuous distributions.
                                                             (line  316)
* quantile_continuous_uniform:           Functions and Variables for continuous distributions.
                                                             (line  975)
* quantile_discrete_uniform:             Functions and Variables for discrete distributions.
                                                             (line  434)
* quantile_exp:                          Functions and Variables for continuous distributions.
                                                             (line  621)
* quantile_f:                            Functions and Variables for continuous distributions.
                                                             (line  521)
* quantile_gamma:                        Functions and Variables for continuous distributions.
                                                             (line  842)
* quantile_general_finite_discrete:      Functions and Variables for discrete distributions.
                                                             (line   47)
* quantile_geometric:                    Functions and Variables for discrete distributions.
                                                             (line  376)
* quantile_gumbel:                       Functions and Variables for continuous distributions.
                                                             (line 1484)
* quantile_hypergeometric:               Functions and Variables for discrete distributions.
                                                             (line  505)
* quantile_laplace:                      Functions and Variables for continuous distributions.
                                                             (line 1396)
* quantile_logistic:                     Functions and Variables for continuous distributions.
                                                             (line 1037)
* quantile_lognormal:                    Functions and Variables for continuous distributions.
                                                             (line  769)
* quantile_negative_binomial:            Functions and Variables for discrete distributions.
                                                             (line  582)
* quantile_noncentral_chi2:              Functions and Variables for continuous distributions.
                                                             (line  453)
* quantile_noncentral_student_t:         Functions and Variables for continuous distributions.
                                                             (line  207)
* quantile_normal:                       Functions and Variables for continuous distributions.
                                                             (line   29)
* quantile_pareto:                       Functions and Variables for continuous distributions.
                                                             (line 1095)
* quantile_poisson:                      Functions and Variables for discrete distributions.
                                                             (line  192)
* quantile_rayleigh:                     Functions and Variables for continuous distributions.
                                                             (line 1239)
* quantile_student_t:                    Functions and Variables for continuous distributions.
                                                             (line  108)
* quantile_weibull:                      Functions and Variables for continuous distributions.
                                                             (line 1153)
* quartile_skewness:                     Functions and Variables for descriptive statistics.
                                                             (line  476)
* quit:                                  Functions and Variables for Command Line.
                                                             (line  544)
* qunit:                                 Functions and Variables for Number Theory.
                                                             (line  436)
* Quote operator:                        Functions and Variables for Evaluation.
                                                             (line    7)
* Quote-quote operator:                  Functions and Variables for Evaluation.
                                                             (line  102)
* quotient:                              Functions and Variables for Polynomials.
                                                             (line  873)
* radcan:                                Functions and Variables for Simplification.
                                                             (line  536)
* radexpand:                             Functions and Variables for Simplification.
                                                             (line  565)
* radius:                                Functions and Variables for graphs.
                                                             (line 1041)
* radsubstflag:                          Functions and Variables for Polynomials.
                                                             (line 1245)
* random:                                Random Numbers.     (line   33)
* random_bernoulli:                      Functions and Variables for discrete distributions.
                                                             (line  352)
* random_beta:                           Functions and Variables for continuous distributions.
                                                             (line  950)
* random_binomial:                       Functions and Variables for discrete distributions.
                                                             (line  162)
* random_bipartite_graph:                Functions and Variables for graphs.
                                                             (line  267)
* random_cauchy:                         Functions and Variables for continuous distributions.
                                                             (line 1461)
* random_chi2:                           Functions and Variables for continuous distributions.
                                                             (line  427)
* random_continuous_uniform:             Functions and Variables for continuous distributions.
                                                             (line 1012)
* random_digraph:                        Functions and Variables for graphs.
                                                             (line  272)
* random_discrete_uniform:               Functions and Variables for discrete distributions.
                                                             (line  472)
* random_exp:                            Functions and Variables for continuous distributions.
                                                             (line  736)
* random_f:                              Functions and Variables for continuous distributions.
                                                             (line  567)
* random_gamma:                          Functions and Variables for continuous distributions.
                                                             (line  877)
* random_general_finite_discrete:        Functions and Variables for discrete distributions.
                                                             (line   89)
* random_geometric:                      Functions and Variables for discrete distributions.
                                                             (line  411)
* random_graph:                          Functions and Variables for graphs.
                                                             (line  283)
* random_graph1:                         Functions and Variables for graphs.
                                                             (line  288)
* random_gumbel:                         Functions and Variables for continuous distributions.
                                                             (line 1533)
* random_hypergeometric:                 Functions and Variables for discrete distributions.
                                                             (line  549)
* random_laplace:                        Functions and Variables for continuous distributions.
                                                             (line 1431)
* random_logistic:                       Functions and Variables for continuous distributions.
                                                             (line 1072)
* random_lognormal:                      Functions and Variables for continuous distributions.
                                                             (line  811)
* random_negative_binomial:              Functions and Variables for discrete distributions.
                                                             (line  620)
* random_network:                        Functions and Variables for graphs.
                                                             (line  292)
* random_noncentral_chi2:                Functions and Variables for continuous distributions.
                                                             (line  493)
* random_noncentral_student_t:           Functions and Variables for continuous distributions.
                                                             (line  260)
* random_normal:                         Functions and Variables for continuous distributions.
                                                             (line   74)
* random_pareto:                         Functions and Variables for continuous distributions.
                                                             (line 1131)
* random_permutation:                    Functions and Variables for Sets.
                                                             (line  792)
* random_poisson:                        Functions and Variables for discrete distributions.
                                                             (line  227)
* random_rayleigh:                       Functions and Variables for continuous distributions.
                                                             (line 1373)
* random_regular_graph:                  Functions and Variables for graphs.
                                                             (line  278)
* random_student_t:                      Functions and Variables for continuous distributions.
                                                             (line  150)
* random_tournament:                     Functions and Variables for graphs.
                                                             (line  306)
* random_tree:                           Functions and Variables for graphs.
                                                             (line  310)
* random_weibull:                        Functions and Variables for continuous distributions.
                                                             (line 1189)
* range:                                 Functions and Variables for descriptive statistics.
                                                             (line  218)
* rank <1>:                              Functions and Variables for linearalgebra.
                                                             (line  586)
* rank:                                  Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1097)
* rassociative:                          Functions and Variables for Simplification.
                                                             (line  587)
* rat:                                   Functions and Variables for Polynomials.
                                                             (line  882)
* ratalgdenom:                           Functions and Variables for Polynomials.
                                                             (line  935)
* ratchristof:                           Functions and Variables for ctensor.
                                                             (line 1148)
* ratcoef:                               Functions and Variables for Polynomials.
                                                             (line  945)
* ratdenom:                              Functions and Variables for Polynomials.
                                                             (line  972)
* ratdenomdivide:                        Functions and Variables for Polynomials.
                                                             (line  986)
* ratdiff:                               Functions and Variables for Polynomials.
                                                             (line 1031)
* ratdisrep:                             Functions and Variables for Polynomials.
                                                             (line 1074)
* rateinstein:                           Functions and Variables for ctensor.
                                                             (line 1152)
* ratepsilon:                            Functions and Variables for Numbers.
                                                             (line  255)
* ratexpand:                             Functions and Variables for Polynomials.
                                                             (line 1087)
* ratfac:                                Functions and Variables for Polynomials.
                                                             (line 1135)
* ratinterpol:                           Functions and Variables for interpol.
                                                             (line  279)
* rational <1>:                          Package functs.     (line   34)
* rational:                              Functions and Variables for Properties.
                                                             (line  538)
* rationalize:                           Functions and Variables for Numbers.
                                                             (line  263)
* ratmx:                                 Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1105)
* ratnumer:                              Functions and Variables for Polynomials.
                                                             (line 1158)
* ratnump:                               Functions and Variables for Numbers.
                                                             (line  329)
* ratp:                                  Functions and Variables for Polynomials.
                                                             (line 1172)
* ratprint:                              Functions and Variables for Polynomials.
                                                             (line 1180)
* ratriemann:                            Functions and Variables for ctensor.
                                                             (line 1160)
* ratsimp:                               Functions and Variables for Polynomials.
                                                             (line 1188)
* ratsimpexpons:                         Functions and Variables for Polynomials.
                                                             (line 1238)
* ratsubst:                              Functions and Variables for Polynomials.
                                                             (line 1252)
* ratvars:                               Functions and Variables for Polynomials.
                                                             (line 1291)
* ratvarswitch:                          Functions and Variables for Polynomials.
                                                             (line 1311)
* ratweight:                             Functions and Variables for Polynomials.
                                                             (line 1364)
* ratweights:                            Functions and Variables for Polynomials.
                                                             (line 1392)
* ratweyl:                               Functions and Variables for ctensor.
                                                             (line 1169)
* ratwtlvl:                              Functions and Variables for Polynomials.
                                                             (line 1402)
* read:                                  Functions and Variables for Command Line.
                                                             (line  554)
* read_array:                            Functions and Variables for plain-text input and output.
                                                             (line   28)
* read_binary_array:                     Functions and Variables for binary input and output.
                                                             (line   44)
* read_binary_list:                      Functions and Variables for binary input and output.
                                                             (line   57)
* read_binary_matrix:                    Functions and Variables for binary input and output.
                                                             (line   33)
* read_hashed_array:                     Functions and Variables for plain-text input and output.
                                                             (line   43)
* read_list:                             Functions and Variables for plain-text input and output.
                                                             (line   72)
* read_matrix:                           Functions and Variables for plain-text input and output.
                                                             (line   10)
* read_nested_list:                      Functions and Variables for plain-text input and output.
                                                             (line   58)
* read_xpm:                              Functions and Variables for pictures.
                                                             (line   89)
* readline:                              Functions and Variables for input and output.
                                                             (line  128)
* readonly:                              Functions and Variables for Command Line.
                                                             (line  572)
* real:                                  Functions and Variables for Properties.
                                                             (line  546)
* Real infinity:                         Functions and Variables for Constants.
                                                             (line   37)
* real_imagpart_to_conjugate:            Functions and Variables for to_poly_solve.
                                                             (line  651)
* realonly:                              Functions and Variables for Equations.
                                                             (line  638)
* realpart:                              Functions for Complex Numbers.
                                                             (line  139)
* realroots:                             Functions and Variables for Equations.
                                                             (line  648)
* rearray:                               Functions and Variables for Arrays.
                                                             (line  403)
* rectangle:                             Functions and Variables for draw.
                                                             (line 3338)
* rectform:                              Functions for Complex Numbers.
                                                             (line  145)
* rectform_log_if_constant:              Functions and Variables for to_poly_solve.
                                                             (line  670)
* recttopolar:                           Functions and Variables for fast Fourier transform.
                                                             (line   25)
* rediff:                                Functions and Variables for itensor.
                                                             (line  668)
* redraw:                                Functions and Variables for graphs.
                                                             (line 1631)
* reduce_consts:                         Package rducon.     (line    7)
* reduce_order:                          Functions and Variables for solve_rec.
                                                             (line    7)
* refcheck:                              Functions and Variables for Debugging.
                                                             (line   18)
* region:                                Functions and Variables for draw.
                                                             (line 3368)
* region_boundaries:                     Functions and Variables for worldmap.
                                                             (line  122)
* region_boundaries_plus:                Functions and Variables for worldmap.
                                                             (line  136)
* rem:                                   Functions and Variables for Properties.
                                                             (line  556)
* remainder:                             Functions and Variables for Polynomials.
                                                             (line 1411)
* remarray:                              Functions and Variables for Arrays.
                                                             (line  411)
* rembox:                                Functions and Variables for Expressions.
                                                             (line  893)
* remcomps:                              Functions and Variables for itensor.
                                                             (line  289)
* remcon:                                Functions and Variables for itensor.
                                                             (line  175)
* remcoord:                              Functions and Variables for itensor.
                                                             (line  774)
* remfun:                                Functions and Variables for Fourier series.
                                                             (line   13)
* remfunction:                           Functions and Variables for Function Definition.
                                                             (line 1023)
* remlet:                                Functions and Variables for Rules and Patterns.
                                                             (line  542)
* remove:                                Functions and Variables for Properties.
                                                             (line  566)
* remove_constvalue:                     Functions and Variables for ezunits.
                                                             (line  259)
* remove_dimensions:                     Functions and Variables for ezunits.
                                                             (line  492)
* remove_edge:                           Functions and Variables for graphs.
                                                             (line 1333)
* remove_fundamental_dimensions:         Functions and Variables for ezunits.
                                                             (line  522)
* remove_fundamental_units:              Functions and Variables for ezunits.
                                                             (line  554)
* remove_vertex:                         Functions and Variables for graphs.
                                                             (line 1348)
* rempart:                               Package functs.     (line    7)
* remrule:                               Functions and Variables for Rules and Patterns.
                                                             (line  565)
* remsym:                                Functions and Variables for itensor.
                                                             (line  583)
* remvalue:                              Functions and Variables for Miscellaneous Options.
                                                             (line   72)
* rename:                                Functions and Variables for itensor.
                                                             (line   78)
* reset:                                 Functions and Variables for Command Line.
                                                             (line  594)
* residue:                               Functions and Variables for Integration.
                                                             (line  524)
* resolvante:                            Functions and Variables for Symmetries.
                                                             (line  593)
* resolvante_alternee1:                  Functions and Variables for Symmetries.
                                                             (line  778)
* resolvante_bipartite:                  Functions and Variables for Symmetries.
                                                             (line  788)
* resolvante_diedrale:                   Functions and Variables for Symmetries.
                                                             (line  802)
* resolvante_klein:                      Functions and Variables for Symmetries.
                                                             (line  821)
* resolvante_klein3:                     Functions and Variables for Symmetries.
                                                             (line  831)
* resolvante_produit_sym:                Functions and Variables for Symmetries.
                                                             (line  841)
* resolvante_unitaire:                   Functions and Variables for Symmetries.
                                                             (line  868)
* resolvante_vierer:                     Functions and Variables for Symmetries.
                                                             (line  878)
* rest:                                  Functions and Variables for Lists.
                                                             (line  389)
* resultant:                             Functions and Variables for Polynomials.
                                                             (line 1458)
* return:                                Functions and Variables for Program Flow.
                                                             (line  540)
* reveal:                                Functions and Variables for Expressions.
                                                             (line  963)
* reverse:                               Functions and Variables for Lists.
                                                             (line  396)
* revert:                                Functions and Variables for Series.
                                                             (line  318)
* revert2:                               Functions and Variables for Series.
                                                             (line  319)
* rgb2level:                             Functions and Variables for pictures.
                                                             (line   93)
* rhs:                                   Functions and Variables for Equations.
                                                             (line  693)
* ric:                                   Functions and Variables for ctensor.
                                                             (line 1205)
* ricci:                                 Functions and Variables for ctensor.
                                                             (line  212)
* riem:                                  Functions and Variables for ctensor.
                                                             (line 1188)
* riemann:                               Functions and Variables for ctensor.
                                                             (line  256)
* rinvariant:                            Functions and Variables for ctensor.
                                                             (line  300)
* risch:                                 Functions and Variables for Integration.
                                                             (line  541)
* rk:                                    Functions and Variables for dynamics.
                                                             (line  128)
* rmxchar:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1125)
* rncombine:                             Functions and Variables for Miscellaneous Options.
                                                             (line   84)
* romberg:                               Functions and Variables for romberg.
                                                             (line    7)
* rombergabs:                            Functions and Variables for romberg.
                                                             (line  108)
* rombergit:                             Functions and Variables for romberg.
                                                             (line  122)
* rombergmin:                            Functions and Variables for romberg.
                                                             (line  134)
* rombergtol:                            Functions and Variables for romberg.
                                                             (line  144)
* room:                                  Functions and Variables for Runtime Environment.
                                                             (line   36)
* rootsconmode:                          Functions and Variables for Equations.
                                                             (line  735)
* rootscontract:                         Functions and Variables for Equations.
                                                             (line  742)
* rootsepsilon:                          Functions and Variables for Equations.
                                                             (line  804)
* round:                                 Functions for Numbers.
                                                             (line  231)
* row:                                   Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1120)
* rowop:                                 Functions and Variables for linearalgebra.
                                                             (line  580)
* rowswap:                               Functions and Variables for linearalgebra.
                                                             (line  597)
* rreduce:                               Functions and Variables for Sets.
                                                             (line  812)
* run_testsuite:                         Functions and Variables for Bug Detection and Reporting.
                                                             (line    7)
* run_viewer:                            Plotting Options.   (line  219)
* save:                                  Functions and Variables for File Input and Output.
                                                             (line  342)
* savedef:                               Functions and Variables for Function Definition.
                                                             (line 1041)
* savefactors:                           Functions and Variables for Polynomials.
                                                             (line 1480)
* saving:                                Functions and Variables for finance.
                                                             (line  188)
* scalar:                                Functions and Variables for Properties.
                                                             (line  604)
* scalarmatrixp:                         Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1134)
* scalarp:                               Functions and Variables for Properties.
                                                             (line  611)
* scaled_bessel_i:                       Bessel Functions.   (line  156)
* scaled_bessel_i0:                      Bessel Functions.   (line  166)
* scaled_bessel_i1:                      Bessel Functions.   (line  170)
* scalefactors:                          Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1148)
* scanmap:                               Functions and Variables for Program Flow.
                                                             (line  546)
* scatterplot:                           Functions and Variables for statistical graphs.
                                                             (line  340)
* scatterplot_description:               Functions and Variables for statistical graphs.
                                                             (line  341)
* schur2comp:                            Functions and Variables for Symmetries.
                                                             (line  205)
* sconcat:                               Functions and Variables for Strings.
                                                             (line   46)
* scopy:                                 Functions and Variables for strings.
                                                             (line   65)
* scsimp:                                Functions and Variables for Simplification.
                                                             (line  593)
* scurvature:                            Functions and Variables for ctensor.
                                                             (line  230)
* sdowncase:                             Functions and Variables for strings.
                                                             (line   69)
* sec:                                   Functions and Variables for Trigonometric.
                                                             (line  231)
* sech:                                  Functions and Variables for Trigonometric.
                                                             (line  235)
* second:                                Functions and Variables for Lists.
                                                             (line  402)
* sequal:                                Functions and Variables for strings.
                                                             (line   76)
* sequalignore:                          Functions and Variables for strings.
                                                             (line   81)
* set_draw_defaults:                     Functions and Variables for draw.
                                                             (line  170)
* set_edge_weight:                       Functions and Variables for graphs.
                                                             (line 1050)
* set_partitions:                        Functions and Variables for Sets.
                                                             (line  931)
* set_plot_option:                       Functions and Variables for Plotting.
                                                             (line  361)
* set_random_state:                      Random Numbers.     (line   27)
* set_tex_environment:                   Functions and Variables for TeX Output.
                                                             (line  210)
* set_tex_environment_default:           Functions and Variables for TeX Output.
                                                             (line  250)
* set_up_dot_simplifications:            Functions and Variables for Affine.
                                                             (line   44)
* set_vertex_label:                      Functions and Variables for graphs.
                                                             (line 1064)
* setcheck:                              Functions and Variables for Debugging.
                                                             (line   25)
* setcheckbreak:                         Functions and Variables for Debugging.
                                                             (line   55)
* setdifference:                         Functions and Variables for Sets.
                                                             (line  854)
* setelmx:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1166)
* setequalp:                             Functions and Variables for Sets.
                                                             (line  879)
* setify:                                Functions and Variables for Sets.
                                                             (line  895)
* setp:                                  Functions and Variables for Sets.
                                                             (line  912)
* setunits:                              Functions and Variables for Units.
                                                             (line    7)
* setup_autoload:                        Functions and Variables for Miscellaneous Options.
                                                             (line  101)
* setval:                                Functions and Variables for Debugging.
                                                             (line   68)
* seventh:                               Functions and Variables for Lists.
                                                             (line  407)
* sexplode:                              Functions and Variables for strings.
                                                             (line   85)
* sf:                                    Functions and Variables for atensor.
                                                             (line   97)
* shortest_path:                         Functions and Variables for graphs.
                                                             (line 1078)
* shortest_weighted_path:                Functions and Variables for graphs.
                                                             (line 1089)
* show:                                  Functions and Variables for itensor.
                                                             (line  137)
* show_edge_color:                       Functions and Variables for graphs.
                                                             (line 1607)
* show_edge_type:                        Functions and Variables for graphs.
                                                             (line 1615)
* show_edge_width:                       Functions and Variables for graphs.
                                                             (line 1611)
* show_edges:                            Functions and Variables for graphs.
                                                             (line 1602)
* show_id:                               Functions and Variables for graphs.
                                                             (line 1516)
* show_label:                            Functions and Variables for graphs.
                                                             (line 1522)
* show_vertex_color:                     Functions and Variables for graphs.
                                                             (line 1572)
* show_vertex_size:                      Functions and Variables for graphs.
                                                             (line 1568)
* show_vertex_type:                      Functions and Variables for graphs.
                                                             (line 1562)
* show_vertices:                         Functions and Variables for graphs.
                                                             (line 1556)
* show_weight:                           Functions and Variables for graphs.
                                                             (line 1535)
* showcomps:                             Functions and Variables for itensor.
                                                             (line  294)
* showratvars:                           Functions and Variables for Polynomials.
                                                             (line 1489)
* showtime:                              Functions and Variables for Command Line.
                                                             (line  605)
* sierpinskiale:                         Definitions for IFS fractals.
                                                             (line   20)
* sierpinskimap:                         Definitions for Peano maps.
                                                             (line   20)
* sign:                                  Functions and Variables for Facts.
                                                             (line  409)
* signum:                                Functions for Numbers.
                                                             (line  237)
* similaritytransform:                   Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1174)
* simp:                                  Functions and Variables for Simplification.
                                                             (line  603)
* simp_inequality:                       Functions and Variables for to_poly_solve.
                                                             (line  694)
* simple_linear_regression:              Functions and Variables for stats.
                                                             (line  825)
* simplified_output:                     Functions and Variables for zeilberger.
                                                             (line  103)
* simplify_products:                     Functions and Variables for solve_rec.
                                                             (line   55)
* simplify_sum:                          Functions and Variables for solve_rec.
                                                             (line   64)
* simplode:                              Functions and Variables for strings.
                                                             (line   90)
* simpmetderiv:                          Functions and Variables for itensor.
                                                             (line  809)
* simpsum:                               Sums and Products.  (line  107)
* simtran:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1175)
* sin:                                   Functions and Variables for Trigonometric.
                                                             (line  239)
* sinh:                                  Functions and Variables for Trigonometric.
                                                             (line  243)
* sinnpiflag:                            Functions and Variables for Fourier series.
                                                             (line   57)
* sinsert:                               Functions and Variables for strings.
                                                             (line  103)
* sinvertcase:                           Functions and Variables for strings.
                                                             (line  115)
* sixth:                                 Functions and Variables for Lists.
                                                             (line  412)
* skewness:                              Functions and Variables for descriptive statistics.
                                                             (line  428)
* skewness_bernoulli:                    Functions and Variables for discrete distributions.
                                                             (line  315)
* skewness_beta:                         Functions and Variables for continuous distributions.
                                                             (line  938)
* skewness_binomial:                     Functions and Variables for discrete distributions.
                                                             (line  150)
* skewness_chi2:                         Functions and Variables for continuous distributions.
                                                             (line  387)
* skewness_continuous_uniform:           Functions and Variables for continuous distributions.
                                                             (line 1000)
* skewness_discrete_uniform:             Functions and Variables for discrete distributions.
                                                             (line  460)
* skewness_exp:                          Functions and Variables for continuous distributions.
                                                             (line  700)
* skewness_f:                            Functions and Variables for continuous distributions.
                                                             (line  555)
* skewness_gamma:                        Functions and Variables for continuous distributions.
                                                             (line  865)
* skewness_general_finite_discrete:      Functions and Variables for discrete distributions.
                                                             (line   75)
* skewness_geometric:                    Functions and Variables for discrete distributions.
                                                             (line  399)
* skewness_gumbel:                       Functions and Variables for continuous distributions.
                                                             (line 1512)
* skewness_hypergeometric:               Functions and Variables for discrete distributions.
                                                             (line  534)
* skewness_laplace:                      Functions and Variables for continuous distributions.
                                                             (line 1419)
* skewness_logistic:                     Functions and Variables for continuous distributions.
                                                             (line 1060)
* skewness_lognormal:                    Functions and Variables for continuous distributions.
                                                             (line  799)
* skewness_negative_binomial:            Functions and Variables for discrete distributions.
                                                             (line  608)
* skewness_noncentral_chi2:              Functions and Variables for continuous distributions.
                                                             (line  480)
* skewness_noncentral_student_t:         Functions and Variables for continuous distributions.
                                                             (line  246)
* skewness_normal:                       Functions and Variables for continuous distributions.
                                                             (line   62)
* skewness_pareto:                       Functions and Variables for continuous distributions.
                                                             (line 1118)
* skewness_poisson:                      Functions and Variables for discrete distributions.
                                                             (line  215)
* skewness_rayleigh:                     Functions and Variables for continuous distributions.
                                                             (line 1323)
* skewness_student_t:                    Functions and Variables for continuous distributions.
                                                             (line  137)
* skewness_weibull:                      Functions and Variables for continuous distributions.
                                                             (line 1176)
* slength:                               Functions and Variables for strings.
                                                             (line  126)
* smake:                                 Functions and Variables for strings.
                                                             (line  130)
* small_rhombicosidodecahedron_graph:    Functions and Variables for graphs.
                                                             (line  314)
* small_rhombicuboctahedron_graph:       Functions and Variables for graphs.
                                                             (line  318)
* smax:                                  Functions and Variables for descriptive statistics.
                                                             (line  201)
* smin:                                  Functions and Variables for descriptive statistics.
                                                             (line  184)
* smismatch:                             Functions and Variables for strings.
                                                             (line  137)
* snowmap:                               Definitions for Koch snowflakes.
                                                             (line    7)
* snub_cube_graph:                       Functions and Variables for graphs.
                                                             (line  322)
* snub_dodecahedron_graph:               Functions and Variables for graphs.
                                                             (line  326)
* solve:                                 Functions and Variables for Equations.
                                                             (line  812)
* solve_rec:                             Functions and Variables for solve_rec.
                                                             (line   87)
* solve_rec_rat:                         Functions and Variables for solve_rec.
                                                             (line  152)
* solvedecomposes:                       Functions and Variables for Equations.
                                                             (line  970)
* solveexplicit:                         Functions and Variables for Equations.
                                                             (line  977)
* solvefactors:                          Functions and Variables for Equations.
                                                             (line  985)
* solvenullwarn:                         Functions and Variables for Equations.
                                                             (line  993)
* solveradcan:                           Functions and Variables for Equations.
                                                             (line 1002)
* solvetrigwarn:                         Functions and Variables for Equations.
                                                             (line 1011)
* some:                                  Functions and Variables for Sets.
                                                             (line  984)
* somrac:                                Functions and Variables for Symmetries.
                                                             (line  582)
* sort:                                  Functions and Variables for Lists.
                                                             (line  417)
* space:                                 Functions and Variables for characters.
                                                             (line   95)
* sparse:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1200)
* sparse6_decode:                        Functions and Variables for graphs.
                                                             (line 1382)
* sparse6_encode:                        Functions and Variables for graphs.
                                                             (line 1387)
* sparse6_export:                        Functions and Variables for graphs.
                                                             (line 1392)
* sparse6_import:                        Functions and Variables for graphs.
                                                             (line 1397)
* specint:                               Functions and Variables for Special Functions.
                                                             (line    7)
* spherical:                             Functions and Variables for draw.
                                                             (line 3389)
* spherical_bessel_j:                    Functions and Variables for orthogonal polynomials.
                                                             (line  187)
* spherical_bessel_y:                    Functions and Variables for orthogonal polynomials.
                                                             (line  194)
* spherical_hankel1:                     Functions and Variables for orthogonal polynomials.
                                                             (line  201)
* spherical_hankel2:                     Functions and Variables for orthogonal polynomials.
                                                             (line  207)
* spherical_harmonic:                    Functions and Variables for orthogonal polynomials.
                                                             (line  213)
* spherical_to_xyz:                      Functions and Variables for Plotting.
                                                             (line  391)
* splice:                                Macros.             (line  179)
* split:                                 Functions and Variables for strings.
                                                             (line  149)
* sposition:                             Functions and Variables for strings.
                                                             (line  165)
* spring_embedding_depth:                Functions and Variables for graphs.
                                                             (line 1650)
* sprint:                                Functions and Variables for input and output.
                                                             (line  134)
* sqfr:                                  Functions and Variables for Polynomials.
                                                             (line 1496)
* sqrt:                                  Root Exponential and Logarithmic Functions.
                                                             (line  240)
* sqrtdenest:                            Package sqdnst.     (line    7)
* sqrtdispflag:                          Functions and Variables for Display.
                                                             (line  592)
* sremove:                               Functions and Variables for strings.
                                                             (line  172)
* sremovefirst:                          Functions and Variables for strings.
                                                             (line  188)
* sreverse:                              Functions and Variables for strings.
                                                             (line  195)
* ssearch:                               Functions and Variables for strings.
                                                             (line  203)
* ssort:                                 Functions and Variables for strings.
                                                             (line  215)
* sstatus:                               Functions and Variables for Runtime Environment.
                                                             (line   49)
* ssubst:                                Functions and Variables for strings.
                                                             (line  232)
* ssubstfirst:                           Functions and Variables for strings.
                                                             (line  250)
* staircase:                             Functions and Variables for dynamics.
                                                             (line  170)
* standardize_inverse_trig:              Functions and Variables for to_poly_solve.
                                                             (line  718)
* stardisp:                              Functions and Variables for Display.
                                                             (line  599)
* starplot:                              Functions and Variables for statistical graphs.
                                                             (line  398)
* starplot_description:                  Functions and Variables for statistical graphs.
                                                             (line  399)
* stats_numer:                           Functions and Variables for stats.
                                                             (line    7)
* status:                                Functions and Variables for Runtime Environment.
                                                             (line   61)
* std:                                   Functions and Variables for descriptive statistics.
                                                             (line   80)
* std1:                                  Functions and Variables for descriptive statistics.
                                                             (line   99)
* std_bernoulli:                         Functions and Variables for discrete distributions.
                                                             (line  299)
* std_beta:                              Functions and Variables for continuous distributions.
                                                             (line  932)
* std_binomial:                          Functions and Variables for discrete distributions.
                                                             (line  144)
* std_chi2:                              Functions and Variables for continuous distributions.
                                                             (line  369)
* std_continuous_uniform:                Functions and Variables for continuous distributions.
                                                             (line  994)
* std_discrete_uniform:                  Functions and Variables for discrete distributions.
                                                             (line  454)
* std_exp:                               Functions and Variables for continuous distributions.
                                                             (line  680)
* std_f:                                 Functions and Variables for continuous distributions.
                                                             (line  549)
* std_gamma:                             Functions and Variables for continuous distributions.
                                                             (line  859)
* std_general_finite_discrete:           Functions and Variables for discrete distributions.
                                                             (line   68)
* std_geometric:                         Functions and Variables for discrete distributions.
                                                             (line  393)
* std_gumbel:                            Functions and Variables for continuous distributions.
                                                             (line 1506)
* std_hypergeometric:                    Functions and Variables for discrete distributions.
                                                             (line  527)
* std_laplace:                           Functions and Variables for continuous distributions.
                                                             (line 1413)
* std_logistic:                          Functions and Variables for continuous distributions.
                                                             (line 1054)
* std_lognormal:                         Functions and Variables for continuous distributions.
                                                             (line  793)
* std_negative_binomial:                 Functions and Variables for discrete distributions.
                                                             (line  602)
* std_noncentral_chi2:                   Functions and Variables for continuous distributions.
                                                             (line  474)
* std_noncentral_student_t:              Functions and Variables for continuous distributions.
                                                             (line  239)
* std_normal:                            Functions and Variables for continuous distributions.
                                                             (line   56)
* std_pareto:                            Functions and Variables for continuous distributions.
                                                             (line 1112)
* std_poisson:                           Functions and Variables for discrete distributions.
                                                             (line  209)
* std_rayleigh:                          Functions and Variables for continuous distributions.
                                                             (line 1301)
* std_student_t:                         Functions and Variables for continuous distributions.
                                                             (line  131)
* std_weibull:                           Functions and Variables for continuous distributions.
                                                             (line 1170)
* stemplot:                              Functions and Variables for statistical graphs.
                                                             (line  461)
* stirling:                              Functions and Variables for stirling.
                                                             (line    8)
* stirling1:                             Functions and Variables for Sets.
                                                             (line 1054)
* stirling2:                             Functions and Variables for Sets.
                                                             (line 1113)
* strim:                                 Functions and Variables for strings.
                                                             (line  255)
* striml:                                Functions and Variables for strings.
                                                             (line  266)
* strimr:                                Functions and Variables for strings.
                                                             (line  270)
* string:                                Functions and Variables for Strings.
                                                             (line   54)
* stringdisp:                            Functions and Variables for Strings.
                                                             (line   62)
* stringout:                             Functions and Variables for File Input and Output.
                                                             (line  399)
* stringp:                               Functions and Variables for strings.
                                                             (line    7)
* strong_components:                     Functions and Variables for graphs.
                                                             (line 1106)
* structures:                            Functions and Variables for Structures.
                                                             (line    7)
* struve_h:                              Struve Functions.   (line   10)
* struve_l:                              Struve Functions.   (line   14)
* style:                                 Plotting Options.   (line  227)
* sublis:                                Functions and Variables for Expressions.
                                                             (line 1026)
* sublis_apply_lambda:                   Functions and Variables for Expressions.
                                                             (line 1042)
* sublist:                               Functions and Variables for Lists.
                                                             (line  550)
* sublist_indices:                       Functions and Variables for Lists.
                                                             (line  562)
* submatrix:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1209)
* subnumsimp:                            Functions and Variables for Expressions.
                                                             (line 1050)
* subsample:                             Functions and Variables for data manipulation.
                                                             (line   65)
* Subscript operator:                    Functions and Variables for Lists.
                                                             (line    8)
* subset:                                Functions and Variables for Sets.
                                                             (line 1183)
* subsetp:                               Functions and Variables for Sets.
                                                             (line 1202)
* subst:                                 Functions and Variables for Expressions.
                                                             (line 1069)
* subst_parallel:                        Functions and Variables for to_poly_solve.
                                                             (line  730)
* substinpart:                           Functions and Variables for Expressions.
                                                             (line 1120)
* substpart:                             Functions and Variables for Expressions.
                                                             (line 1177)
* substring:                             Functions and Variables for strings.
                                                             (line  274)
* Subtraction:                           Arithmetic operators.
                                                             (line    8)
* subvar:                                Functions and Variables for Arrays.
                                                             (line  428)
* subvarp:                               Functions and Variables for Arrays.
                                                             (line  453)
* sum:                                   Sums and Products.  (line  117)
* sumcontract:                           Sums and Products.  (line  218)
* sumexpand:                             Sums and Products.  (line  231)
* summand_to_rec:                        Functions and Variables for solve_rec.
                                                             (line  185)
* sumsplitfact:                          Combinatorial Functions.
                                                             (line  181)
* supcase:                               Functions and Variables for strings.
                                                             (line  289)
* supcontext:                            Functions and Variables for Facts.
                                                             (line  419)
* surface_hide:                          Functions and Variables for draw.
                                                             (line 1584)
* symbolp:                               Functions and Variables for Expressions.
                                                             (line 1206)
* symmdifference:                        Functions and Variables for Sets.
                                                             (line 1215)
* symmetric:                             Functions and Variables for Simplification.
                                                             (line  642)
* symmetricp:                            Functions and Variables for ctensor.
                                                             (line  839)
* system:                                Functions and Variables for Runtime Environment.
                                                             (line   84)
* t:                                     Plotting Options.   (line  265)
* tab:                                   Functions and Variables for characters.
                                                             (line   99)
* take_channel:                          Functions and Variables for pictures.
                                                             (line   98)
* take_inference:                        Functions and Variables for inference_result.
                                                             (line   74)
* tan:                                   Functions and Variables for Trigonometric.
                                                             (line  247)
* tanh:                                  Functions and Variables for Trigonometric.
                                                             (line  251)
* taylor:                                Functions and Variables for Series.
                                                             (line  360)
* taylor_logexpand:                      Functions and Variables for Series.
                                                             (line  555)
* taylor_order_coefficients:             Functions and Variables for Series.
                                                             (line  572)
* taylor_simplifier:                     Functions and Variables for Series.
                                                             (line  582)
* taylor_truncate_polynomials:           Functions and Variables for Series.
                                                             (line  587)
* taylordepth:                           Functions and Variables for Series.
                                                             (line  518)
* taylorinfo:                            Functions and Variables for Series.
                                                             (line  526)
* taylorp:                               Functions and Variables for Series.
                                                             (line  551)
* taytorat:                              Functions and Variables for Series.
                                                             (line  597)
* tcl_output:                            Functions and Variables for Miscellaneous Options.
                                                             (line  135)
* tcontract:                             Functions and Variables for Symmetries.
                                                             (line  279)
* tellrat:                               Functions and Variables for Polynomials.
                                                             (line 1513)
* tellsimp:                              Functions and Variables for Rules and Patterns.
                                                             (line  610)
* tellsimpafter:                         Functions and Variables for Rules and Patterns.
                                                             (line  668)
* tensorkill:                            Functions and Variables for ctensor.
                                                             (line 1271)
* tentex:                                Functions and Variables for itensor.
                                                             (line 1548)
* tenth:                                 Functions and Variables for Lists.
                                                             (line  599)
* terminal <1>:                          Functions and Variables for draw.
                                                             (line 1602)
* terminal:                              Functions and Variables for graphs.
                                                             (line 1657)
* test_mean:                             Functions and Variables for stats.
                                                             (line   16)
* test_means_difference:                 Functions and Variables for stats.
                                                             (line  125)
* test_normality:                        Functions and Variables for stats.
                                                             (line  791)
* test_proportion:                       Functions and Variables for stats.
                                                             (line  388)
* test_proportions_difference:           Functions and Variables for stats.
                                                             (line  487)
* test_rank_sum:                         Functions and Variables for stats.
                                                             (line  718)
* test_sign:                             Functions and Variables for stats.
                                                             (line  580)
* test_signed_rank:                      Functions and Variables for stats.
                                                             (line  637)
* test_variance:                         Functions and Variables for stats.
                                                             (line  241)
* test_variance_ratio:                   Functions and Variables for stats.
                                                             (line  314)
* testsuite_files:                       Functions and Variables for Bug Detection and Reporting.
                                                             (line   49)
* tex:                                   Functions and Variables for TeX Output.
                                                             (line   12)
* tex1:                                  Functions and Variables for TeX Output.
                                                             (line   66)
* texput:                                Functions and Variables for TeX Output.
                                                             (line   76)
* third:                                 Functions and Variables for Lists.
                                                             (line  604)
* throw:                                 Functions and Variables for Program Flow.
                                                             (line  586)
* time:                                  Functions and Variables for Runtime Environment.
                                                             (line  104)
* timedate:                              Functions and Variables for Runtime Environment.
                                                             (line  115)
* timer:                                 Functions and Variables for Debugging.
                                                             (line   78)
* timer_devalue:                         Functions and Variables for Debugging.
                                                             (line  119)
* timer_info:                            Functions and Variables for Debugging.
                                                             (line  131)
* title:                                 Functions and Variables for draw.
                                                             (line 1673)
* tldefint:                              Functions and Variables for Integration.
                                                             (line  564)
* tlimit:                                Functions and Variables for Limits.
                                                             (line   67)
* tlimswitch:                            Functions and Variables for Limits.
                                                             (line   74)
* to_lisp:                               Functions and Variables for Command Line.
                                                             (line  617)
* to_poly:                               Functions and Variables for to_poly_solve.
                                                             (line  785)
* to_poly_solve:                         Functions and Variables for to_poly_solve.
                                                             (line  844)
* todd_coxeter:                          Functions and Variables for Groups.
                                                             (line    8)
* toeplitz:                              Functions and Variables for linearalgebra.
                                                             (line  602)
* tokens:                                Functions and Variables for strings.
                                                             (line  300)
* topological_sort:                      Functions and Variables for graphs.
                                                             (line 1118)
* totaldisrep:                           Functions and Variables for Polynomials.
                                                             (line 1567)
* totalfourier:                          Functions and Variables for Fourier series.
                                                             (line   85)
* totient:                               Functions and Variables for Number Theory.
                                                             (line  448)
* tpartpol:                              Functions and Variables for Symmetries.
                                                             (line  285)
* tr:                                    Functions and Variables for ctensor.
                                                             (line 1253)
* tr_array_as_ref:                       Functions and Variables for Function Definition.
                                                             (line 1184)
* tr_bound_function_applyp:              Functions and Variables for Function Definition.
                                                             (line 1196)
* tr_file_tty_messagesp:                 Functions and Variables for Function Definition.
                                                             (line 1208)
* tr_float_can_branch_complex:           Functions and Variables for Function Definition.
                                                             (line 1218)
* tr_function_call_default:              Functions and Variables for Function Definition.
                                                             (line 1233)
* tr_numer:                              Functions and Variables for Function Definition.
                                                             (line 1248)
* tr_optimize_max_loop:                  Functions and Variables for Function Definition.
                                                             (line 1255)
* tr_semicompile:                        Functions and Variables for Function Definition.
                                                             (line 1264)
* tr_state_vars:                         Functions and Variables for Function Definition.
                                                             (line 1272)
* tr_warn_bad_function_calls:            Functions and Variables for Function Definition.
                                                             (line 1290)
* tr_warn_fexpr:                         Functions and Variables for Function Definition.
                                                             (line 1298)
* tr_warn_meval:                         Functions and Variables for Function Definition.
                                                             (line 1306)
* tr_warn_mode:                          Functions and Variables for Function Definition.
                                                             (line 1313)
* tr_warn_undeclared:                    Functions and Variables for Function Definition.
                                                             (line 1320)
* tr_warn_undefined_variable:            Functions and Variables for Function Definition.
                                                             (line 1327)
* tr_warnings_get:                       Functions and Variables for Function Definition.
                                                             (line 1285)
* trace:                                 Functions and Variables for Debugging.
                                                             (line  152)
* trace_options:                         Functions and Variables for Debugging.
                                                             (line  179)
* tracematrix:                           Package functs.     (line   28)
* transcompile:                          Functions and Variables for Function Definition.
                                                             (line 1053)
* transform:                             Functions and Variables for draw.
                                                             (line 1689)
* transform_xy:                          Plotting Options.   (line  271)
* translate:                             Functions and Variables for Function Definition.
                                                             (line 1065)
* translate_file:                        Functions and Variables for Function Definition.
                                                             (line 1119)
* transparent:                           Functions and Variables for draw.
                                                             (line 1730)
* transpose:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1215)
* transrun:                              Functions and Variables for Function Definition.
                                                             (line 1176)
* tree_reduce:                           Functions and Variables for Sets.
                                                             (line 1243)
* treefale:                              Definitions for IFS fractals.
                                                             (line   32)
* treillis:                              Functions and Variables for Symmetries.
                                                             (line  496)
* treinat:                               Functions and Variables for Symmetries.
                                                             (line  505)
* triangle:                              Functions and Variables for draw.
                                                             (line 3410)
* triangularize:                         Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1228)
* trigexpand:                            Functions and Variables for Trigonometric.
                                                             (line  255)
* trigexpandplus:                        Functions and Variables for Trigonometric.
                                                             (line  292)
* trigexpandtimes:                       Functions and Variables for Trigonometric.
                                                             (line  301)
* triginverses:                          Functions and Variables for Trigonometric.
                                                             (line  310)
* trigrat:                               Functions and Variables for Trigonometric.
                                                             (line  360)
* trigreduce:                            Functions and Variables for Trigonometric.
                                                             (line  327)
* trigsign:                              Functions and Variables for Trigonometric.
                                                             (line  341)
* trigsimp:                              Functions and Variables for Trigonometric.
                                                             (line  349)
* trivial_solutions:                     Functions and Variables for zeilberger.
                                                             (line  132)
* true:                                  Functions and Variables for Constants.
                                                             (line  111)
* trunc:                                 Functions and Variables for Series.
                                                             (line  603)
* truncated_cube_graph:                  Functions and Variables for graphs.
                                                             (line  330)
* truncated_dodecahedron_graph:          Functions and Variables for graphs.
                                                             (line  334)
* truncated_icosahedron_graph:           Functions and Variables for graphs.
                                                             (line  338)
* truncated_tetrahedron_graph:           Functions and Variables for graphs.
                                                             (line  342)
* ttyoff:                                Functions and Variables for Display.
                                                             (line  606)
* tube:                                  Functions and Variables for draw.
                                                             (line 3441)
* tube_extremes:                         Functions and Variables for draw.
                                                             (line 1748)
* tutte_graph:                           Functions and Variables for graphs.
                                                             (line  346)
* ueivects:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1252)
* ufg:                                   Functions and Variables for ctensor.
                                                             (line 1183)
* uforget:                               Functions and Variables for Units.
                                                             (line  115)
* ug:                                    Functions and Variables for ctensor.
                                                             (line 1218)
* ultraspherical:                        Functions and Variables for orthogonal polynomials.
                                                             (line  227)
* und:                                   Functions and Variables for Constants.
                                                             (line  116)
* Undefined:                             Functions and Variables for Constants.
                                                             (line  116)
* underlying_graph:                      Functions and Variables for graphs.
                                                             (line  350)
* undiff:                                Functions and Variables for itensor.
                                                             (line  673)
* union:                                 Functions and Variables for Sets.
                                                             (line 1278)
* unique:                                Functions and Variables for Lists.
                                                             (line  586)
* unit_step:                             Functions and Variables for orthogonal polynomials.
                                                             (line  219)
* unit_vectors:                          Functions and Variables for draw.
                                                             (line 1764)
* uniteigenvectors:                      Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1251)
* unitp:                                 Functions and Variables for ezunits.
                                                             (line  410)
* units:                                 Functions and Variables for ezunits.
                                                             (line  301)
* unitvector:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1280)
* unknown:                               Functions and Variables for Predicates.
                                                             (line  184)
* unless:                                Functions and Variables for Program Flow.
                                                             (line  593)
* unorder:                               Functions and Variables for Expressions.
                                                             (line 1214)
* unsum:                                 Functions and Variables for Series.
                                                             (line  620)
* untellrat:                             Functions and Variables for Polynomials.
                                                             (line 1578)
* untimer:                               Functions and Variables for Debugging.
                                                             (line  104)
* untrace:                               Functions and Variables for Debugging.
                                                             (line  238)
* uppercasep:                            Functions and Variables for characters.
                                                             (line  103)
* uric:                                  Functions and Variables for ctensor.
                                                             (line 1209)
* uricci:                                Functions and Variables for ctensor.
                                                             (line  219)
* uriem:                                 Functions and Variables for ctensor.
                                                             (line 1201)
* uriemann:                              Functions and Variables for ctensor.
                                                             (line  294)
* use_fast_arrays:                       Functions and Variables for Arrays.
                                                             (line  458)
* user_preamble:                         Functions and Variables for draw.
                                                             (line 1785)
* usersetunits:                          Functions and Variables for Units.
                                                             (line  216)
* uvect:                                 Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1281)
* values:                                Functions and Variables for Command Line.
                                                             (line  646)
* vandermonde_matrix:                    Functions and Variables for linearalgebra.
                                                             (line  621)
* var:                                   Functions and Variables for descriptive statistics.
                                                             (line   34)
* var1:                                  Functions and Variables for descriptive statistics.
                                                             (line   55)
* var_bernoulli:                         Functions and Variables for discrete distributions.
                                                             (line  284)
* var_beta:                              Functions and Variables for continuous distributions.
                                                             (line  927)
* var_binomial:                          Functions and Variables for discrete distributions.
                                                             (line  138)
* var_chi2:                              Functions and Variables for continuous distributions.
                                                             (line  352)
* var_continuous_uniform:                Functions and Variables for continuous distributions.
                                                             (line  988)
* var_discrete_uniform:                  Functions and Variables for discrete distributions.
                                                             (line  448)
* var_exp:                               Functions and Variables for continuous distributions.
                                                             (line  659)
* var_f:                                 Functions and Variables for continuous distributions.
                                                             (line  544)
* var_gamma:                             Functions and Variables for continuous distributions.
                                                             (line  854)
* var_general_finite_discrete:           Functions and Variables for discrete distributions.
                                                             (line   61)
* var_geometric:                         Functions and Variables for discrete distributions.
                                                             (line  388)
* var_gumbel:                            Functions and Variables for continuous distributions.
                                                             (line 1501)
* var_hypergeometric:                    Functions and Variables for discrete distributions.
                                                             (line  520)
* var_laplace:                           Functions and Variables for continuous distributions.
                                                             (line 1408)
* var_logistic:                          Functions and Variables for continuous distributions.
                                                             (line 1049)
* var_lognormal:                         Functions and Variables for continuous distributions.
                                                             (line  788)
* var_negative_binomial:                 Functions and Variables for discrete distributions.
                                                             (line  596)
* var_noncentral_chi2:                   Functions and Variables for continuous distributions.
                                                             (line  469)
* var_noncentral_student_t:              Functions and Variables for continuous distributions.
                                                             (line  232)
* var_normal:                            Functions and Variables for continuous distributions.
                                                             (line   50)
* var_pareto:                            Functions and Variables for continuous distributions.
                                                             (line 1107)
* var_poisson:                           Functions and Variables for discrete distributions.
                                                             (line  204)
* var_rayleigh:                          Functions and Variables for continuous distributions.
                                                             (line 1279)
* var_student_t:                         Functions and Variables for continuous distributions.
                                                             (line  121)
* var_weibull:                           Functions and Variables for continuous distributions.
                                                             (line 1165)
* vect_cross:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1335)
* vector:                                Functions and Variables for draw.
                                                             (line 3466)
* vectorpotential:                       Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1290)
* vectorsimp:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1297)
* verbify:                               Functions and Variables for Expressions.
                                                             (line 1252)
* verbose:                               Functions and Variables for Series.
                                                             (line  650)
* vers:                                  Package functs.     (line  174)
* vertex_color:                          Functions and Variables for graphs.
                                                             (line 1552)
* vertex_coloring:                       Functions and Variables for graphs.
                                                             (line 1210)
* vertex_connectivity:                   Functions and Variables for graphs.
                                                             (line 1135)
* vertex_degree:                         Functions and Variables for graphs.
                                                             (line 1141)
* vertex_distance:                       Functions and Variables for graphs.
                                                             (line 1145)
* vertex_eccentricity:                   Functions and Variables for graphs.
                                                             (line 1158)
* vertex_in_degree:                      Functions and Variables for graphs.
                                                             (line 1168)
* vertex_out_degree:                     Functions and Variables for graphs.
                                                             (line 1188)
* vertex_partition:                      Functions and Variables for graphs.
                                                             (line 1576)
* vertex_size:                           Functions and Variables for graphs.
                                                             (line 1548)
* vertex_type:                           Functions and Variables for graphs.
                                                             (line 1541)
* vertices:                              Functions and Variables for graphs.
                                                             (line 1201)
* vertices_to_cycle:                     Functions and Variables for graphs.
                                                             (line 1685)
* vertices_to_path:                      Functions and Variables for graphs.
                                                             (line 1680)
* view:                                  Functions and Variables for draw.
                                                             (line 1808)
* warnings:                              Functions and Variables for zeilberger.
                                                             (line  117)
* Wedge product:                         Functions and Variables for itensor.
                                                             (line 1394)
* weyl:                                  Functions and Variables for ctensor.
                                                             (line 1222)
* wheel_graph:                           Functions and Variables for graphs.
                                                             (line  354)
* while:                                 Functions and Variables for Program Flow.
                                                             (line  592)
* wiener_index:                          Functions and Variables for graphs.
                                                             (line 1224)
* wired_surface:                         Functions and Variables for draw.
                                                             (line 1826)
* with_stdout:                           Functions and Variables for File Input and Output.
                                                             (line  445)
* write_binary_data:                     Functions and Variables for binary input and output.
                                                             (line   70)
* write_data:                            Functions and Variables for plain-text input and output.
                                                             (line   89)
* writefile:                             Functions and Variables for File Input and Output.
                                                             (line  474)
* wronskian:                             Package functs.     (line   16)
* x:                                     Plotting Options.   (line  281)
* x_voxel:                               Functions and Variables for draw.
                                                             (line 1844)
* xaxis:                                 Functions and Variables for draw.
                                                             (line 1852)
* xaxis_color:                           Functions and Variables for draw.
                                                             (line 1870)
* xaxis_secondary:                       Functions and Variables for draw.
                                                             (line 1889)
* xaxis_type:                            Functions and Variables for draw.
                                                             (line 1916)
* xaxis_width:                           Functions and Variables for draw.
                                                             (line 1935)
* xlabel <1>:                            Functions and Variables for draw.
                                                             (line 1954)
* xlabel:                                Plotting Options.   (line  290)
* xrange:                                Functions and Variables for draw.
                                                             (line 1973)
* xrange_secondary:                      Functions and Variables for draw.
                                                             (line 1994)
* xreduce:                               Functions and Variables for Sets.
                                                             (line 1306)
* xthru:                                 Functions and Variables for Simplification.
                                                             (line  650)
* xtics:                                 Functions and Variables for draw.
                                                             (line 2009)
* xtics_axis:                            Functions and Variables for draw.
                                                             (line 2070)
* xtics_rotate:                          Functions and Variables for draw.
                                                             (line 2081)
* xtics_rotate_secondary:                Functions and Variables for draw.
                                                             (line 2091)
* xtics_secondary:                       Functions and Variables for draw.
                                                             (line 2101)
* xtics_secondary_axis:                  Functions and Variables for draw.
                                                             (line 2110)
* xu_grid:                               Functions and Variables for draw.
                                                             (line 2121)
* xy_file:                               Functions and Variables for draw.
                                                             (line 2141)
* xyplane:                               Functions and Variables for draw.
                                                             (line 2152)
* y:                                     Plotting Options.   (line  300)
* y_voxel:                               Functions and Variables for draw.
                                                             (line 2170)
* yaxis:                                 Functions and Variables for draw.
                                                             (line 2178)
* yaxis_color:                           Functions and Variables for draw.
                                                             (line 2196)
* yaxis_secondary:                       Functions and Variables for draw.
                                                             (line 2215)
* yaxis_type:                            Functions and Variables for draw.
                                                             (line 2239)
* yaxis_width:                           Functions and Variables for draw.
                                                             (line 2258)
* ylabel <1>:                            Plotting Options.   (line  307)
* ylabel:                                Functions and Variables for draw.
                                                             (line 2277)
* yrange:                                Functions and Variables for draw.
                                                             (line 2296)
* yrange_secondary:                      Functions and Variables for draw.
                                                             (line 2318)
* ytics:                                 Functions and Variables for draw.
                                                             (line 2345)
* ytics_axis:                            Functions and Variables for draw.
                                                             (line 2354)
* ytics_rotate:                          Functions and Variables for draw.
                                                             (line 2365)
* ytics_rotate_secondary:                Functions and Variables for draw.
                                                             (line 2375)
* ytics_secondary:                       Functions and Variables for draw.
                                                             (line 2385)
* ytics_secondary_axis:                  Functions and Variables for draw.
                                                             (line 2394)
* yv_grid:                               Functions and Variables for draw.
                                                             (line 2405)
* z:                                     Plotting Options.   (line  316)
* z_voxel:                               Functions and Variables for draw.
                                                             (line 2425)
* zaxis:                                 Functions and Variables for draw.
                                                             (line 2433)
* zaxis_color:                           Functions and Variables for draw.
                                                             (line 2453)
* zaxis_type:                            Functions and Variables for draw.
                                                             (line 2474)
* zaxis_width:                           Functions and Variables for draw.
                                                             (line 2494)
* Zeilberger:                            Functions and Variables for zeilberger.
                                                             (line   73)
* zeroa:                                 Functions and Variables for Constants.
                                                             (line  127)
* zerob:                                 Functions and Variables for Constants.
                                                             (line  144)
* zerobern:                              Functions and Variables for Number Theory.
                                                             (line  453)
* zeroequiv:                             Functions and Variables for Predicates.
                                                             (line  189)
* zerofor:                               Functions and Variables for linearalgebra.
                                                             (line  627)
* zeromatrix:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1343)
* zeromatrixp:                           Functions and Variables for linearalgebra.
                                                             (line  640)
* zeta:                                  Functions and Variables for Number Theory.
                                                             (line  461)
* zeta%pi:                               Functions and Variables for Number Theory.
                                                             (line  494)
* zlabel <1>:                            Functions and Variables for draw.
                                                             (line 2514)
* zlabel:                                Plotting Options.   (line  321)
* zlange:                                Functions and Variables for lapack.
                                                             (line  278)
* zrange:                                Functions and Variables for draw.
                                                             (line 2534)
* ztics:                                 Functions and Variables for draw.
                                                             (line 2557)
* ztics_axis:                            Functions and Variables for draw.
                                                             (line 2566)
* ztics_rotate:                          Functions and Variables for draw.
                                                             (line 2577)
* |:                                     Functions and Variables for itensor.
                                                             (line 1428)
* ~:                                     Functions and Variables for itensor.
                                                             (line 1394)

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