org.apache.commons.math3.special
Class Beta

java.lang.Object
  extended by org.apache.commons.math3.special.Beta

public class Beta
extends java.lang.Object

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

Implementation of logBeta(double, double) is based on the algorithms described in

and implemented in the NSWC Library of Mathematical Functions, available here. This library is "approved for public release", and the Copyright guidance indicates that unless otherwise stated in the code, all FORTRAN functions in this library are license free. Since no such notice appears in the code these functions can safely be ported to Commons-Math.

Version:
$Id: Beta.java 1420669 2012-12-12 13:40:35Z erans $

Field Summary
private static double DEFAULT_EPSILON
          Maximum allowed numerical error.
private static double[] DELTA
           The coefficients of the series expansion of the Δ function.
private static double HALF_LOG_TWO_PI
          The constant value of ½log 2π.
 
Constructor Summary
private Beta()
          Default constructor.
 
Method Summary
private static double deltaMinusDeltaSum(double a, double b)
          Returns the value of Δ(b) - Δ(a + b), with 0 ≤ a ≤ b and b ≥ 10.
static double logBeta(double p, double q)
          Returns the value of log B(p, q) for 0 ≤ x ≤ 1 and p, q > 0.
static double logBeta(double a, double b, double epsilon, int maxIterations)
          Deprecated. as of version 3.1, this method is deprecated as the computation of the beta function is no longer iterative; it will be removed in version 4.0. Current implementation of this method internally calls logBeta(double, double).
private static double logGammaMinusLogGammaSum(double a, double b)
          Returns the value of log[Γ(b) / Γ(a + b)] for a ≥ 0 and b ≥ 10.
private static double logGammaSum(double a, double b)
          Returns the value of log Γ(a + b) for 1 ≤ a, b ≤ 2.
static double regularizedBeta(double x, double a, double b)
          Returns the regularized beta function I(x, a, b).
static double regularizedBeta(double x, double a, double b, double epsilon)
          Returns the regularized beta function I(x, a, b).
static double regularizedBeta(double x, double a, double b, double epsilon, int maxIterations)
          Returns the regularized beta function I(x, a, b).
static double regularizedBeta(double x, double a, double b, int maxIterations)
          Returns the regularized beta function I(x, a, b).
private static double sumDeltaMinusDeltaSum(double p, double q)
          Returns the value of Δ(p) + Δ(q) - Δ(p + q), with p, q ≥ 10.
 
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Field Detail

DEFAULT_EPSILON

private static final double DEFAULT_EPSILON
Maximum allowed numerical error.

See Also:
Constant Field Values

HALF_LOG_TWO_PI

private static final double HALF_LOG_TWO_PI
The constant value of ½log 2π.

See Also:
Constant Field Values

DELTA

private static final double[] DELTA

The coefficients of the series expansion of the Δ function. This function is defined as follows

Δ(x) = log Γ(x) - (x - 0.5) log a + a - 0.5 log 2π,

see equation (23) in Didonato and Morris (1992). The series expansion, which applies for x ≥ 10, reads

                 14
                ====
             1  \                2 n
     Δ(x) = ---  >    d  (10 / x)
             x  /      n
                ====
                n = 0
 

Constructor Detail

Beta

private Beta()
Default constructor. Prohibit instantiation.

Method Detail

regularizedBeta

public static double regularizedBeta(double x,
                                     double a,
                                     double b)
Returns the regularized beta function I(x, a, b).

Parameters:
x - Value.
a - Parameter a.
b - Parameter b.
Returns:
the regularized beta function I(x, a, b).
Throws:
MaxCountExceededException - if the algorithm fails to converge.

regularizedBeta

public static double regularizedBeta(double x,
                                     double a,
                                     double b,
                                     double epsilon)
Returns the regularized beta function I(x, a, b).

Parameters:
x - Value.
a - Parameter a.
b - Parameter b.
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.
Returns:
the regularized beta function I(x, a, b)
Throws:
MaxCountExceededException - if the algorithm fails to converge.

regularizedBeta

public static double regularizedBeta(double x,
                                     double a,
                                     double b,
                                     int maxIterations)
Returns the regularized beta function I(x, a, b).

Parameters:
x - the value.
a - Parameter a.
b - Parameter b.
maxIterations - Maximum number of "iterations" to complete.
Returns:
the regularized beta function I(x, a, b)
Throws:
MaxCountExceededException - if the algorithm fails to converge.

regularizedBeta

public static double regularizedBeta(double x,
                                     double a,
                                     double b,
                                     double epsilon,
                                     int maxIterations)
Returns the regularized beta function I(x, a, b). The implementation of this method is based on:

Parameters:
x - the value.
a - Parameter a.
b - Parameter b.
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 beta function I(x, a, b)
Throws:
MaxCountExceededException - if the algorithm fails to converge.

logBeta

@Deprecated
public static double logBeta(double a,
                                        double b,
                                        double epsilon,
                                        int maxIterations)
Deprecated. as of version 3.1, this method is deprecated as the computation of the beta function is no longer iterative; it will be removed in version 4.0. Current implementation of this method internally calls logBeta(double, double).

Returns the natural logarithm of the beta function B(a, b). The implementation of this method is based on:

Parameters:
a - Parameter a.
b - Parameter b.
epsilon - This parameter is ignored.
maxIterations - This parameter is ignored.
Returns:
log(B(a, b)).

logGammaSum

private static double logGammaSum(double a,
                                  double b)
                           throws OutOfRangeException
Returns the value of log Γ(a + b) for 1 ≤ a, b ≤ 2. Based on the NSWC Library of Mathematics Subroutines double precision implementation, DGSMLN. In BetaTest#testLogGammaSum(), this private method is accessed through reflection.

Parameters:
a - First argument.
b - Second argument.
Returns:
the value of log(Gamma(a + b)).
Throws:
OutOfRangeException - if a or b is lower than 1.0 or greater than 2.0.

logGammaMinusLogGammaSum

private static double logGammaMinusLogGammaSum(double a,
                                               double b)
                                        throws NumberIsTooSmallException
Returns the value of log[Γ(b) / Γ(a + b)] for a ≥ 0 and b ≥ 10. Based on the NSWC Library of Mathematics Subroutines double precision implementation, DLGDIV. In BetaTest#testLogGammaMinusLogGammaSum(), this private method is accessed through reflection.

Parameters:
a - First argument.
b - Second argument.
Returns:
the value of log(Gamma(b) / Gamma(a + b)).
Throws:
NumberIsTooSmallException - if a < 0.0 or b < 10.0.

deltaMinusDeltaSum

private static double deltaMinusDeltaSum(double a,
                                         double b)
                                  throws OutOfRangeException,
                                         NumberIsTooSmallException
Returns the value of Δ(b) - Δ(a + b), with 0 ≤ a ≤ b and b ≥ 10. Based on equations (26), (27) and (28) in Didonato and Morris (1992).

Parameters:
a - First argument.
b - Second argument.
Returns:
the value of Delta(b) - Delta(a + b)
Throws:
OutOfRangeException - if a < 0 or a > b
NumberIsTooSmallException - if b < 10

sumDeltaMinusDeltaSum

private static double sumDeltaMinusDeltaSum(double p,
                                            double q)
Returns the value of Δ(p) + Δ(q) - Δ(p + q), with p, q ≥ 10. Based on the NSWC Library of Mathematics Subroutines double precision implementation, DBCORR. In BetaTest#testSumDeltaMinusDeltaSum(), this private method is accessed through reflection.

Parameters:
p - First argument.
q - Second argument.
Returns:
the value of Delta(p) + Delta(q) - Delta(p + q).
Throws:
NumberIsTooSmallException - if p < 10.0 or q < 10.0.

logBeta

public static double logBeta(double p,
                             double q)
Returns the value of log B(p, q) for 0 ≤ x ≤ 1 and p, q > 0. Based on the NSWC Library of Mathematics Subroutines implementation, DBETLN.

Parameters:
p - First argument.
q - Second argument.
Returns:
the value of log(Beta(p, q)), NaN if p <= 0 or q <= 0.


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