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

Class ChangeOfVariable

java.lang.Object

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

All Implemented Interfaces:

Integrator, java.io.Serializable

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, x, to another variable, t, so that the "new" integral is easier to compute.

We set /[ x = x(t) t = x^{-1}(x) = t(x) /] such that, /[ \int_{a}^{b} f(x)\,dx = \int_{t(a)}^{t(b)} f(x)x'(t)\, dt /]

See Also:

Wikipedia: Integration by substitution, Serialized Form

Constructor Summary

Constructors

Constructor and Description
ChangeOfVariable(SubstitutionRule change, Integrator integrator) Construct an integrator that uses change of variable to do integration.

Method Summary

Modifier and Type	Method and Description
UnivariateRealFunction	fdx(UnivariateRealFunction f) Get the integrand in the "transformed" integral, g(t) = f(x(t)) * x'(t).
double	getPrecision() Get the convergence threshold.
double	integrate(UnivariateRealFunction f, double a, double b) Integrate function f from a to b, \[ \int_a^b\! f(x)\, dx \]

Methods inherited from class java.lang.Object

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

Constructor Detail

ChangeOfVariable

public ChangeOfVariable(SubstitutionRule change,
                        Integrator 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 such as Trapezoidal.

Method Detail

integrate

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

Description copied from interface: Integrator

Integrate function f from a to b, \[ \int_a^b\! f(x)\, dx \]

Specified by:

integrate in interface Integrator

Parameters:

f - a univariate function

a - the lower limit

b - the upper limit

Returns:

\(\int_a^b\! f(x)\, dx\)

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

getPrecision

public double getPrecision()

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:

getPrecision in interface Integrator

Returns:

the precision