Class InternalGamma

java.lang.Object

org.apache.commons.rng.sampling.distribution.InternalGamma

final class InternalGamma
extends Object

Adapted and stripped down copy of class "org.apache.commons.math4.special.Gamma".

This is a utility class that provides computation methods related to the Γ (Gamma) family of functions.

Field Summary

Fields

Modifier and Type	Field	Description
private static final double	HALF_LOG_2_PI	Avoid repeated computation of log(2*PI) / 2 in logGamma.
private static final double[]	LANCZOS_COEFFICIENTS	Lanczos coefficients.
(package private) static final double	LANCZOS_G	Constant \( g = \frac{607}{128} \) in the Lanczos approximation.

Constructor Summary

Constructors

Modifier	Constructor	Description
private	InternalGamma()	Class contains only static methods.

Method Summary

Modifier and Type	Method	Description
private static double	lanczos(double x)	Computes the Lanczos approximation used to compute the gamma function.
(package private) static double	logGamma(double x)	Computes the function \( \ln \Gamma(x) \) for \( x \geq 8 \).

Methods inherited from class Object

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

Field Details

LANCZOS_G

static final double LANCZOS_G

Constant \( g = \frac{607}{128} \) in the Lanczos approximation.

See Also:

• Constant Field Values

LANCZOS_COEFFICIENTS

private static final double[] LANCZOS_COEFFICIENTS

Lanczos coefficients.

HALF_LOG_2_PI

private static final double HALF_LOG_2_PI

Avoid repeated computation of log(2*PI) / 2 in logGamma.

See Also:

• Constant Field Values

Constructor Details

InternalGamma

private InternalGamma()

Class contains only static methods.

Method Details

logGamma

static double logGamma(double x)

Computes the function \( \ln \Gamma(x) \) for \( x \geq 8 \).

The implementation is based on

• Gamma Function, equation (28).
• Lanczos Approximation, equations (1) through (5).
• Paul Godfrey, A note on the computation of the convergent Lanczos complex Gamma approximation

Warning: This method is not accurate when x is less than 8.

Parameters:

x - Argument (must be above 8).

Returns:

\( \ln \Gamma(x) \)

lanczos

private static double lanczos(double x)

Computes the Lanczos approximation used to compute the gamma function.

The Lanczos approximation is related to the Gamma function by the following equation \[ \Gamma(x) = \sqrt{2\pi} \, \frac{(g + x + \frac{1}{2})^{x + \frac{1}{2}} \, e^{-(g + x + \frac{1}{2})} \, \mathrm{lanczos}(x)} {x} \] where \(g\) is the Lanczos constant.

Parameters:

x - Argument.

Returns:

The Lanczos approximation.

See Also:

• Lanczos Approximation equations (1) through (5), and Paul Godfrey's Note on the computation of the convergent Lanczos complex Gamma approximation