org.apache.commons.math.special

Class Gamma

java.lang.Object

org.apache.commons.math.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.

Version:

$Revision: 1042510 $ $Date: 2010-12-06 02:54:18 +0100 (lun. 06 déc. 2010) $

Field Summary

Fields

Modifier and Type	Field and Description
private static double	C_LIMIT C limit.
private static double	DEFAULT_EPSILON Maximum allowed numerical error.
static double	GAMMA Euler-Mascheroni constant
private static double	HALF_LOG_2_PI Avoid repeated computation of log of 2 PI in logGamma
private static double[]	LANCZOS Lanczos coefficients
private static double	S_LIMIT S limit.

Constructor Summary

Constructors

Modifier	Constructor and Description
private	Gamma() Default constructor.

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

DEFAULT_EPSILON

private static final double DEFAULT_EPSILON

Maximum allowed numerical error.

See Also:

Constant Field Values

LANCZOS

private static final double[] LANCZOS

Lanczos coefficients

HALF_LOG_2_PI

private static final double HALF_LOG_2_PI

Avoid repeated computation of log of 2 PI in logGamma

C_LIMIT

private static final double C_LIMIT

C limit.

See Also:

Constant Field Values

S_LIMIT

private static final double S_LIMIT

S limit.

See Also:

Constant Field Values

Constructor Detail

Gamma

private Gamma()

Default constructor. Prohibit instantiation.

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 - the value.

Returns:

log(Γ(x))

regularizedGammaP

public static double regularizedGammaP(double a,
                                       double x)
                                throws MathException

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

Parameters:

a - the a parameter.

x - the value.

Returns:

the regularized gamma function P(a, x)

Throws:

MathException - if the algorithm fails to converge.

regularizedGammaP

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

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:

MathException - if the algorithm fails to converge.

regularizedGammaQ

public static double regularizedGammaQ(double a,
                                       double x)
                                throws MathException

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:

MathException - if the algorithm fails to converge.

regularizedGammaQ

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

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:

MathException - 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 - the argument

Returns:

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

Since:

2.0

See Also:

Digamma at wikipedia , 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 - the argument

Returns:

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

Since:

2.0

See Also:

Trigamma at wikipedia , digamma(double)