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

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

public class PowerLawSingularity
extends Substitution

This transformation is good for an integral which diverges at one of the end points.

For singularity at the lower limit, we have (x - a)-γ diverging, near x = a, 0 ≤ γ < 1.

 /b          /(b-a)1-γ   1
 | f(x) dx = |         ----- tγ/(1-γ)f(t1/(1-γ) + a)dt, b > a
 /a          /0         1-γ
 

For singularity at the upper limit, we have (x - b)-γ diverging, near x = b, 0 ≤ γ < 1.

 /b          /(b-a)1-γ   1
 | f(x) dx = |         ----- tγ/(1-γ)f(b - t1/(1-γ))dt, b > a
 /a          /0         1-γ
 

A common case is when γ = 0.5.

Nested Class Summary

static class

PowerLawSingularity.Type the type of end point divergence

Field Summary

double	a the lower limit
double	b the upper limit
double	gamma gamma
PowerLawSingularity.Type	type

Fields inherited from class com.numericalmethod.suanshu.analysis.integration.univariate.riemann.substitution.Substitution

dx, x

Constructor Summary

PowerLawSingularity(PowerLawSingularity.Type type, double gamma, double a, double b)
Construct an instance of the PowerLawSingularity substitution rule.

Method Summary

double	ta() Get the lower limit of the integral.
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

type

public final PowerLawSingularity.Type type

gamma

public final double gamma

gamma

a

public final double a

the lower limit

b

public final double b

the upper limit

Constructor Detail

PowerLawSingularity

public PowerLawSingularity(PowerLawSingularity.Type type,
                           double gamma,
                           double a,
                           double b)

Construct an instance of the PowerLawSingularity substitution rule.

Parameters:

type - the type of end point divergence

gamma - γ, the exponential

a - the lower limit

b - the upper limit

Method Detail

ta

public double ta()

Description copied from class: Substitution

Get the lower limit of the integral.

Specified by:

ta in class Substitution

Returns:

the lower limit

tb

public double tb()

Description copied from class: Substitution

Get the upper limit of the integral.

Specified by:

tb in class Substitution

Returns:

the upper limit