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

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

Direct Known Subclasses:

DoubleExponential4HalfRealLine, DoubleExponential4RealLine, MixedRule

public class DoubleExponential
extends Substitution

This transformation speeds up the convergence of the Trapezoidal Rule exponentially. It applies to a finite integral region [a, b].

The substitution is

 x = 0.5 * (b + a) + 0.5 * (b - a) * tanh(c * sinh(t))
 

The tricky part of using this transformation is to figure out a good range for t. The user should override Substitution.ta() and Substitution.tb() if there is information about the integrand available.

See Also:

Wikipedia: Tanh-sinh quadrature

Field Summary

double	a the lower limit
double	b the upper limit
double	c a constant; usually either 0 or 0.5 * PI
UnivariateRealFunction	f the original integrand

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

dx, x

Constructor Summary

DoubleExponential(UnivariateRealFunction f, double c, double a, double b)
Construct an instance of the DoubleExponential 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

f

public final UnivariateRealFunction f

the original integrand

a

public final double a

the lower limit

b

public final double b

the upper limit

c

public final double c

a constant; usually either 0 or 0.5 * PI

Constructor Detail

DoubleExponential

public DoubleExponential(UnivariateRealFunction f,
                         double c,
                         double a,
                         double b)

Construct an instance of the DoubleExponential substitution rule. We try to automatically determine the substitution rule.

Parameters:

f - the integrand; with f, we try to determine the lower and upper limits automatically

c - a constant; usually either 0 or 0.5 * PI

a - the lower limit

b - the upper limit

Method Detail

ta

public double ta()

Get the lower limit of the integral. The accuracy of using the Double Exponential transformation lies on choosing the region of t for integration. It is very hard to determine the appropriate region without knowing much about the function, such as where its singularities are.

If the user knows about t's region, s/he should override this function.

Specified by:

ta in class Substitution

Returns:

the lower limit

tb

public double tb()

Get the upper limit of the integral. The accuracy of using the Double Exponential transformation lies on choosing the region of t for integration. It is very hard to determine the appropriate region without knowing much about the function, such as where its singularities are.

If the user knows about t's region, s/he should override this function.

Specified by:

tb in class Substitution

Returns:

the upper limit