org.apache.commons.math3.special

Class Gamma

java.lang.Object

org.apache.commons.math3.special.Gamma

public class Gamma
extends java.lang.Object

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

Field Summary

Fields

Modifier and Type	Field and Description
static double	GAMMA Euler-Mascheroni constant

Method Summary

Modifier and Type	Method and Description
static double	digamma(double x) Computes the digamma function of x.
static double	logGamma(double x) Returns the natural logarithm of the gamma function Γ(x).
static double	regularizedGammaP(double a, double x) Returns the regularized gamma function P(a, x).
static double	regularizedGammaP(double a, double x, double epsilon, int maxIterations) Returns the regularized gamma function P(a, x).
static double	regularizedGammaQ(double a, double x) Returns the regularized gamma function Q(a, x) = 1 - P(a, x).
static double	regularizedGammaQ(double a, double x, double epsilon, int maxIterations) Returns the regularized gamma function Q(a, x) = 1 - P(a, x).
static double	trigamma(double x) Computes the trigamma function of x.

Methods inherited from class java.lang.Object

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

Field Detail

GAMMA

public static final double GAMMA

Euler-Mascheroni constant

Since:

2.0

See Also:

Constant Field Values

Method Detail

logGamma

public static double logGamma(double x)

Returns the natural logarithm of the gamma function Γ(x). The implementation of this method 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

Parameters:

x - Value.

Returns:

log(Γ(x))

regularizedGammaP

public static double regularizedGammaP(double a,
                                       double x)

Returns the regularized gamma function P(a, x).

Parameters:

a - Parameter.

x - Value.

Returns:

the regularized gamma function P(a, x).

Throws:

MaxCountExceededException - if the algorithm fails to converge.

regularizedGammaP

public static double regularizedGammaP(double a,
                                       double x,
                                       double epsilon,
                                       int maxIterations)

Returns the regularized gamma function P(a, x). The implementation of this method is based on:

• Regularized Gamma Function, equation (1)
• Incomplete Gamma Function, equation (4).
• Confluent Hypergeometric Function of the First Kind, equation (1).

Parameters:

a - the a parameter.

x - the value.

epsilon - When the absolute value of the nth item in the series is less than epsilon the approximation ceases to calculate further elements in the series.

maxIterations - Maximum number of "iterations" to complete.

Returns:

the regularized gamma function P(a, x)

Throws:

MaxCountExceededException - if the algorithm fails to converge.

regularizedGammaQ

public static double regularizedGammaQ(double a,
                                       double x)

Returns the regularized gamma function Q(a, x) = 1 - P(a, x).

Parameters:

a - the a parameter.

x - the value.

Returns:

the regularized gamma function Q(a, x)

Throws:

MaxCountExceededException - if the algorithm fails to converge.

regularizedGammaQ

public static double regularizedGammaQ(double a,
                                       double x,
                                       double epsilon,
                                       int maxIterations)

Returns the regularized gamma function Q(a, x) = 1 - P(a, x). The implementation of this method is based on:

• Regularized Gamma Function, equation (1).
• Regularized incomplete gamma function: Continued fraction representations (formula 06.08.10.0003)

Parameters:

a - the a parameter.

x - the value.

epsilon - When the absolute value of the nth item in the series is less than epsilon the approximation ceases to calculate further elements in the series.

maxIterations - Maximum number of "iterations" to complete.

Returns:

the regularized gamma function P(a, x)

Throws:

MaxCountExceededException - if the algorithm fails to converge.

digamma

public static double digamma(double x)

Computes the digamma function of x.

This is an independently written implementation of the algorithm described in Jose Bernardo, Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976.

Some of the constants have been changed to increase accuracy at the moderate expense of run-time. The result should be accurate to within 10^-8 absolute tolerance for x >= 10^-5 and within 10^-8 relative tolerance for x > 0.

Performance for large negative values of x will be quite expensive (proportional to |x|). Accuracy for negative values of x should be about 10^-8 absolute for results less than 10^5 and 10^-8 relative for results larger than that.

Parameters:

x - Argument.

Returns:

digamma(x) to within 10-8 relative or absolute error whichever is smaller.

Since:

2.0

See Also:

Digamma, Bernardo's original article

trigamma

public static double trigamma(double x)

Computes the trigamma function of x. This function is derived by taking the derivative of the implementation of digamma.

Parameters:

x - Argument.

Returns:

trigamma(x) to within 10-8 relative or absolute error whichever is smaller

Since:

2.0

See Also:

Trigamma, digamma(double)