com.numericalmethod.suanshu.analysis.integration.univariate.riemann
Class ChangeOfVariable

java.lang.Object
  com.numericalmethod.suanshu.analysis.integration.univariate.riemann.ChangeOfVariable

All Implemented Interfaces:

Integrator

public class ChangeOfVariable
extends java.lang.Object
implements Integrator

Change of variable 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 implementation does not assume any particular integrator, other than that the integrator is an IterativeIntegrator. The user may combine a substitution rule with his favorite iterative integrator.

See Also:

Wikipedia: Integration by substitution

Constructor Summary

ChangeOfVariable(Substitution change, IterativeIntegrator integrator)
Construct an integrator that uses change of variable to do integration.

Method Summary

UnivariateRealFunction	fdx(UnivariateRealFunction f) Get the integrand in the "transformed" integral.
double	integrate(UnivariateRealFunction f, double a, double b) Integrate function f from a to b.
double	precision() Get the convergence threshold.

Methods inherited from class java.lang.Object

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

Constructor Detail

ChangeOfVariable

public ChangeOfVariable(Substitution change,
                        IterativeIntegrator integrator)

Construct an integrator that uses change of variable to do integration.

Parameters:

change - the substitution formula

integrator - the integrator. If there is a singularity at an endpoint, the integrator should use an OPEN formula such as Midpoint; otherwise, use an integrator with a CLOSED formula.

Method Detail

integrate

public double integrate(UnivariateRealFunction f,
                        double a,
                        double b)

Description copied from interface: Integrator

Integrate function f from a to b.

 / b
 |   f(x) dx
 / a
 

Specified by:

integrate in interface Integrator

Parameters:

f - a univariate function

a - lower limit

b - upper limit

Returns:

Σ(f(x))

fdx

public UnivariateRealFunction fdx(UnivariateRealFunction f)

Get the integrand in the "transformed" integral.

g(t) = f(x(t)) * x'(t)

Parameters:

f - the integrand in the original integral

Returns:

the integrand in the "transformed" integral

precision

public double precision()

Description copied from interface: Integrator

Get the convergence threshold. The usage depends on the specific integrator.

For example, for an IterativeIntegrator, the integral is considered converged if the relative error of two successive sums is less than the threshold.

Specified by:

precision in interface Integrator

Returns:

the precision