com.numericalmethod.suanshu.analysis.integration.univariate.riemann.substitution
Class Substitution

java.lang.Object
  com.numericalmethod.suanshu.analysis.integration.univariate.riemann.substitution.Substitution

Direct Known Subclasses:

DoubleExponential, Exponential, InvertingVariable, NoChangeOfVariable, PowerLawSingularity, StandardInterval

public abstract class Substitution
extends java.lang.Object

This class specifies a substitution rule. All specific substitution rule extends this class.

Change of variable (or substitution) can easy the computation of some integrals, such as improper integrals. The idea is to transform a dependent variable to another so that the "new" integral is easier to compute.

We set

 x = x(t)
 t = x-1(x) = t(x)
 

 / b           /t(b)
 |   f(x) dx = | f(x(t)) * x'(t) dt
 / a           /t(a)
 

This class is the specification of x(t) and x'(t).

See Also:

Wikipedia: Integration by substitution

Field Summary

UnivariateRealFunction	dx the first order derivative of the transformation x'(t) = dx(t)/dt
UnivariateRealFunction	x the transformation x(t)

Constructor Summary

Substitution(UnivariateRealFunction x, UnivariateRealFunction dx)
Create a substitution by specifying the transformation rule.

Method Summary

abstract double	ta() Get the lower limit of the integral.
abstract double	tb() Get the upper limit of the integral.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

x

public final UnivariateRealFunction x

the transformation

x(t)

dx

public final UnivariateRealFunction dx

the first order derivative of the transformation

x'(t) = dx(t)/dt

Constructor Detail

Substitution

public Substitution(UnivariateRealFunction x,
                    UnivariateRealFunction dx)

Create a substitution by specifying the transformation rule.

Parameters:

x - x(t)

dx - x'(t) = dx(t)/dt

Method Detail

ta

public abstract double ta()

Get the lower limit of the integral.

Returns:

the lower limit

tb

public abstract double tb()

Get the upper limit of the integral.

Returns:

the upper limit