Package org.apache.commons.numbers.gamma

Class RegularizedBeta

java.lang.Object

org.apache.commons.numbers.gamma.RegularizedBeta

public final class RegularizedBeta
extends Object

Regularized Beta function.

\[ I_x(a,b) = \frac{1}{B(a, b)} \int_0^x t^{a-1}\,(1-t)^{b-1}\,dt \]

where \( B(a, b) \) is the beta function.

This code has been adapted from the Boost c++ implementation <boost/math/special_functions/beta.hpp>.

See Also:

Boost C++ Incomplete Beta functions

Method Summary

Modifier and Type	Method	Description
static double	complement(double x, double a, double b)	Computes the complement of the regularized beta function I(x, a, b).
static double	complement(double x, double a, double b, double epsilon, int maxIterations)	Computes the complement of the regularized beta function I(x, a, b).
static double	derivative(double x, double a, double b)	Computes the derivative of the regularized beta function I(x, a, b).
static double	value(double x, double a, double b)	Computes the value of the regularized beta function I(x, a, b).
static double	value(double x, double a, double b, double epsilon, int maxIterations)	Computes the value of the regularized beta function I(x, a, b).

Methods inherited from class java.lang.Object

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

Method Detail

value

public static double value(double x,
                           double a,
                           double b)

Computes the value of the regularized beta function I(x, a, b).

\[ I_x(a,b) = \frac{1}{B(a, b)} \int_0^x t^{a-1}\,(1-t)^{b-1}\,dt \]

where \( B(a, b) \) is the beta function.

Parameters:

x - Value.

a - Parameter a.

b - Parameter b.

Returns:

the regularized beta function \( I_x(a, b) \).

Throws:

ArithmeticException - if the series evaluation fails to converge.

value

public static double value(double x,
                           double a,
                           double b,
                           double epsilon,
                           int maxIterations)

Computes the value of the regularized beta function I(x, a, b).

\[ I_x(a,b) = \frac{1}{B(a, b)} \int_0^x t^{a-1}\,(1-t)^{b-1}\,dt \]

where \( B(a, b) \) is the beta function.

Parameters:

x - the value.

a - Parameter a.

b - Parameter b.

epsilon - Tolerance in series evaluation.

maxIterations - Maximum number of iterations in series evaluation.

Returns:

the regularized beta function \( I_x(a, b) \).

Throws:

ArithmeticException - if the series evaluation fails to converge.

complement

public static double complement(double x,
                                double a,
                                double b)

Computes the complement of the regularized beta function I(x, a, b).

\[ 1 - I_x(a,b) = I_{1-x}(b, a) \]

Parameters:

x - Value.

a - Parameter a.

b - Parameter b.

Returns:

the complement of the regularized beta function \( 1 - I_x(a, b) \).

Throws:

ArithmeticException - if the series evaluation fails to converge.

Since:

1.1

complement

public static double complement(double x,
                                double a,
                                double b,
                                double epsilon,
                                int maxIterations)

Computes the complement of the regularized beta function I(x, a, b).

\[ 1 - I_x(a,b) = I_{1-x}(b, a) \]

Parameters:

x - the value.

a - Parameter a.

b - Parameter b.

epsilon - Tolerance in series evaluation.

maxIterations - Maximum number of iterations in series evaluation.

Returns:

the complement of the regularized beta function \( 1 - I_x(a, b) \).

Throws:

ArithmeticException - if the series evaluation fails to converge.

Since:

1.1

derivative

public static double derivative(double x,
                                double a,
                                double b)

Computes the derivative of the regularized beta function I(x, a, b).

\[ \frac{\delta}{\delta x} I_x(a,b) = \frac{(1-x)^{b-1} x^{a-1}}{B(a, b)} \]

where \( B(a, b) \) is the beta function.

This function has uses in some statistical distributions.

Parameters:

x - Value.

a - Parameter a.

b - Parameter b.

Returns:

the derivative of the regularized beta function \( I_x(a, b) \).

Throws:

ArithmeticException - if the series evaluation fails to converge.

Since:

1.1