Algorihms for interval solvers, and guaranteed inclusions.
function ieval (f:string, x:interval scalar, n:integer=10) Better evaluation of the expression in f for the interval x. The interval is split into sub-intervals for more accuracy. See:
mxmieval (Maxima Functions for Euler),
ievalder (Interval Solvers and Guaranteed Solutions)
function ievalder (f:string, fd:string, xi:interval scalar, n:integer=10) Better evaluation of the expression in f for the interval x. The derivative is used to improve the interval accuracy. The interval is split into sub-intervals for more accuracy. See:
mxmieval (Maxima Functions for Euler)
function idgl (f:string, x:real vector, y0:interval scalar) Guaranteed inclusion of y'=f(t,y0;...) at points t with y(t[1])=y0. This is a quick inclusion for a differential equation, which avoids the use of any Taylor series. The inclusion is not narrow, however. The function uses a simple Euler method. The result is an interval vector of values. See:
mxmidgl (Maxima Functions for Euler),
idglder (Interval Solvers and Guaranteed Solutions)
function idglder (f:string, fx:string, fy:string, x:real vector, .. y0:interval scalar) Guaranteed inclusion the solution of y'=f(t,y0;...) at t with y(t[1])=y0. This function needs the partial derivatives of f to x and y. The result is an interval vector of values. f, fx and fy are functions in f(x,y), or expressions of x and y. Additional arguments are passed to the functions. See:
mxmidgl (Maxima Functions for Euler)
function isimpson (f:string, der:string, a:number, b:number, .. n:index=50) Interval Simpson integral of f from a to b. This function uses the Simpson method and its error estimate to get guaranteed inclusions of integrals. f : expression (must map to arguments and work for intervals) der : expression for fourth derivative (like f) a,b : interval bounds n : number of subintervals See:
mxmisimpson (Maxima Functions for Euler)
function ilgs (A:interval, b:interval, R="", steps=100) Guaranteed interval inclusion for the solution of A.x=b. This function uses an interval agorithm, and an exact residuum calculation. If the algorithm succeeds, the result is a guaranteed inclusion for the solution of the linear system. Note that the algorithm can only work for regular A, or interval matrices not containing singular A. A and b may be of interval or real type. The optional R is a provided pseudo inverse to A. See:
xlgs (Exact Computation)
function iinv (A:interval) Guaranteed interval inverse of the matrix A. See:
inv (Linear Algebra),
xinv (Exact Computation)
function ievalpoly (t:interval, p:interval vector) Guaranteed evaluation of a polynomial p(t). p contains the coefficients of a polynomial. Euler stores polynomials starting with the constant coefficient. See:
polyval (Euler Core),
xpolyval (Exact Computation)
function ipolyval (p:interval vector, t:interval) Guaranteed evaluation of a polynomial p(t). See:
ievalpoly (Interval Solvers and Guaranteed Solutions)
function ibisect (f:string, a:scalar, b:scalar=none, y:scalar=0) Interval bisection algorithm to solve f(x)=y See:
bisect (Numerical Algorithms),
inewton (Interval Solvers and Guaranteed Solutions)
function inewton (f:string, df:string , xi: interval, yi:real scalar="", y=0) Guaranteed interval inclusion of the zero of f. df must compute an inclusion of the derivative of f for intervals x. f and df must be functions of one scalar variable, or expressions in x. Additional parameters after the semicolon are passed to both functions. The initial interval x must already contain a zero. If x is a point, and not an interval, the function tries to get an initial interval with the usual Newton method. Returns {x0,f}: the solution and a flag, if the solution is guaranteed. See:
inewton2 (Interval Solvers and Guaranteed Solutions),
mxminewton (Maxima Functions for Euler),
inewton2 (Interval Solvers and Guaranteed Solutions)
function inewton2 (f:string, Df:string, x:interval, check:integer=false) Guaranteed inclusion of the zero of f, a function of several parameters. Works like newton2, starting from a interval vector x which already contains a solution. If x is no interval, the function tries to find such an interval. f and Df must be a function of a row vector x, or an expression in x. f must return a row vector, and Df the Jacobi matrix of f. Returns {x,valid}. If check is false, the result is not checked for a guaranteed inclusion. In this case the return value of valid can be checked to learn, if the inclusion is a guaranteed inclusion. If checked is true valid=0 will throw an error exception. See:
newton2 (Numerical Algorithms)
function plotintervals (r) Adds plots of two dimensional intervals to a given plot. r is an nx2 vector of intervals. See:
mxmibisectfxy (Maxima Functions for Euler)