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

Class DoubleExponential

java.lang.Object

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

All Implemented Interfaces:

SubstitutionRule, java.io.Serializable

Direct Known Subclasses:

DoubleExponential4HalfRealLine, DoubleExponential4RealLine, MixedRule

public class DoubleExponential
extends java.lang.Object
implements SubstitutionRule

This transformation speeds up the convergence of the Trapezoidal Rule exponentially. It applies to a finite integral region [a, b]. The tricky part of using this transformation is to figure out a good range for t. If there is information about the integrand available, SubstitutionRule.ta() and SubstitutionRule.tb() should be overridden. The substitution is

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

See Also:

Wikipedia: Tanh-sinh quadrature, Serialized Form

Constructor Summary

Constructors

Constructor and Description
DoubleExponential(UnivariateRealFunction f, double a, double b, double c) Construct a DoubleExponential substitution rule by trying to automatically determine the substitution rule.

Method Summary

Modifier and Type	Method and Description
UnivariateRealFunction	dx() the first order derivative of the transformation: x'(t) = dx(t)/dt
double	ta() Get the lower limit of the integral.
double	tb() Get the upper limit of the integral.
UnivariateRealFunction	x() the transformation: x(t)

Methods inherited from class java.lang.Object

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

Constructor Detail

DoubleExponential

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

Construct a DoubleExponential substitution rule by trying to automatically determine the substitution rule.

Parameters:

f - the integrand

a - the lower limit

b - the upper limit

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

Method Detail

x

public UnivariateRealFunction x()

Description copied from interface: SubstitutionRule

the transformation: x(t)

Specified by:

x in interface SubstitutionRule

Returns:

x(t)

dx

public UnivariateRealFunction dx()

Description copied from interface: SubstitutionRule

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

Specified by:

dx in interface SubstitutionRule

Returns:

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

ta

public double ta()

Get the lower limit of the integral. The accuracy of using the double exponential transformation lies in choosing the correct region of t for integration. It is very hard to determine the region without knowing about the function, such as where the singularities are. If the region of t is known, override this function.

Specified by:

ta in interface SubstitutionRule

Returns:

the lower limit

tb

public double tb()

Get the upper limit of the integral. The accuracy of using the double exponential transformation lies in choosing the correct region of t for integration. It is very hard to determine the region without knowing about the function, such as where the singularities are. If the region of t is known, override this function.

Specified by:

tb in interface SubstitutionRule

Returns:

the upper limit