Class SpecialMathUtils

java.lang.Object

org.episteme.core.mathematics.util.SpecialMathUtils

public final class SpecialMathUtils
extends Object

The special function math library. This class cannot be subclassed or instantiated because all methods are static.

Version:

1.1

Author:

Mark Hale

Field Summary

Fields

Modifier and Type	Field	Description
static final double	GAMMA_X_MAX_VALUE	The largest argument for which gamma(x) is representable in the machine.
static final double	LOG_GAMMA_X_MAX_VALUE	The largest argument for which logGamma(x) is representable in the machine.

Method Summary

Modifier and Type	Method	Description
static double	airy(double x)	Airy function.
static double	besselFirstOne(double x)	Bessel function of first kind, order one.
static double	besselFirstZero(double x)	Bessel function of first kind, order zero.
static double	besselSecondOne(double x)	Bessel function of second kind, order one.
static double	besselSecondZero(double x)	Bessel function of second kind, order zero.
static double	beta(double p, double q)	Beta function.
static double	chebyshev(double x, double[] series)	Evaluates a Chebyshev series.
static double	complementaryError(double x)	Complementary error function.
static double	error(double x)	Error function.
static double	gamma(double x)	Gamma function.
static double	incompleteBeta(double x, double p, double q)	Incomplete beta function.
static double	incompleteGamma(double a, double x)	Incomplete gamma function.
static double	logBeta(double p, double q)	The natural logarithm of the beta function.
static double	logGamma(double x)	The natural logarithm of the gamma function.
static double	modBesselFirstOne(double x)	Modified Bessel function of first kind, order one.
static double	modBesselFirstZero(double x)	Modified Bessel function of first kind, order zero.

Methods inherited from class Object

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

Field Details

GAMMA_X_MAX_VALUE

public static final double GAMMA_X_MAX_VALUE

The largest argument for which gamma(x) is representable in the machine.

See Also:

• Constant Field Values

LOG_GAMMA_X_MAX_VALUE

public static final double LOG_GAMMA_X_MAX_VALUE

The largest argument for which logGamma(x) is representable in the machine.

See Also:

• Constant Field Values

Method Details

chebyshev

public static double chebyshev(double x,
 double[] series)

Evaluates a Chebyshev series.

Parameters:

x - value at which to evaluate series

series - the coefficients of the series

airy

public static double airy(double x)

Airy function. Based on the NETLIB Fortran function ai written by W. Fullerton.

besselFirstZero

public static double besselFirstZero(double x)

Bessel function of first kind, order zero. Based on the NETLIB Fortran function besj0 written by W. Fullerton.

modBesselFirstZero

public static double modBesselFirstZero(double x)

Modified Bessel function of first kind, order zero. Based on the NETLIB Fortran function besi0 written by W. Fullerton.

besselFirstOne

public static double besselFirstOne(double x)

Bessel function of first kind, order one. Based on the NETLIB Fortran function besj1 written by W. Fullerton.

modBesselFirstOne

public static double modBesselFirstOne(double x)

Modified Bessel function of first kind, order one. Based on the NETLIB Fortran function besi1 written by W. Fullerton.

besselSecondZero

public static double besselSecondZero(double x)

Bessel function of second kind, order zero. Based on the NETLIB Fortran function besy0 written by W. Fullerton.

besselSecondOne

public static double besselSecondOne(double x)

Bessel function of second kind, order one. Based on the NETLIB Fortran function besy1 written by W. Fullerton.

gamma

public static double gamma(double x)

Gamma function. Based on public domain NETLIB (Fortran) code by W. J. Cody and L. Stoltz
Applied Mathematics Division
Argonne National Laboratory
Argonne, IL 60439

References:

1. "An Overview of Software Development for Special Functions", W. J. Cody, Lecture Notes in Mathematics, 506, Numerical Analysis Dundee, 1975, G. A. Watson (ed.), Springer Verlag, Berlin, 1976.
2. Computer Approximations, Hart, Et. Al., Wiley and sons, New York, 1968.

From the original documentation:

This routine calculates the GAMMA function for a real argument X. Computation is based on an algorithm outlined in reference 1. The program uses rational functions that approximate the GAMMA function to at least 20 significant decimal digits. Coefficients for the approximation over the interval (1,2) are unpublished. Those for the approximation for X .GE. 12 are from reference 2. The accuracy achieved depends on the arithmetic system, the compiler, the intrinsic functions, and proper selection of the machine-dependent constants.

Error returns:
The program returns the value XINF for singularities or when overflow would occur. The computation is believed to be free of underflow and overflow.

Returns:

Double.MAX_VALUE if overflow would occur, i.e. if abs(x) > 171.624

logGamma

public static double logGamma(double x)

The natural logarithm of the gamma function. Based on public domain NETLIB (Fortran) code by W. J. Cody and L. Stoltz
Applied Mathematics Division
Argonne National Laboratory
Argonne, IL 60439

References:

1. W. J. Cody and K. E. Hillstrom, 'Chebyshev Approximations for the Natural Logarithm of the Gamma Function,' Math. Comp. 21, 1967, pp. 198-203.
2. K. E. Hillstrom, ANL/AMD Program ANLC366S, DGAMMA/DLGAMA, May, 1969.
3. Hart, Et. Al., Computer Approximations, Wiley and sons, New York, 1968.

From the original documentation:

This routine calculates the LOG(GAMMA) function for a positive real argument X. Computation is based on an algorithm outlined in references 1 and 2. The program uses rational functions that theoretically approximate LOG(GAMMA) to at least 18 significant decimal digits. The approximation for X > 12 is from reference 3, while approximations for X invalid input: '<' 12.0 are similar to those in reference 1, but are unpublished. The accuracy achieved depends on the arithmetic system, the compiler, the intrinsic functions, and proper selection of the machine-dependent constants.

Error returns:
The program returns the value XINF for X .LE. 0.0 or when overflow would occur. The computation is believed to be free of underflow and overflow.

Returns:

Double.MAX_VALUE for x invalid input: '<' 0.0 or when overflow would occur, i.e. x > 2.55E305

incompleteGamma

public static double incompleteGamma(double a,
 double x)

Incomplete gamma function. The computation is based on approximations presented in Numerical Recipes, Chapter 6.2 (W.H. Press et al, 1992).

Parameters:

a - require a>=0

x - require x>=0

Returns:

0 if xinvalid input: '<'0, ainvalid input: '<'=0 or a>2.55E305 to avoid errors and over/underflow

beta

public static double beta(double p,
 double q)

Beta function.

Parameters:

p - require p>0

q - require q>0

Returns:

0 if pinvalid input: '<'=0, qinvalid input: '<'=0 or p+q>2.55E305 to avoid errors and over/underflow

logBeta

public static double logBeta(double p,
 double q)

The natural logarithm of the beta function.

Parameters:

p - require p>0

q - require q>0

Returns:

0 if pinvalid input: '<'=0, qinvalid input: '<'=0 or p+q>2.55E305 to avoid errors and over/underflow

incompleteBeta

public static double incompleteBeta(double x,
 double p,
 double q)

Incomplete beta function. The computation is based on formulas from Numerical Recipes, Chapter 6.4 (W.H. Press et al, 1992).

Parameters:

x - require 0invalid input: '<'=xinvalid input: '<'=1

p - require p>0

q - require q>0

Returns:

0 if xinvalid input: '<'0, pinvalid input: '<'=0, qinvalid input: '<'=0 or p+q>2.55E305 and 1 if x>1 to avoid errors and over/underflow

error

public static double error(double x)

Error function. Based on C-code for the error function developed at Sun Microsystems.

complementaryError

public static double complementaryError(double x)

Complementary error function. Based on C-code for the error function developed at Sun Microsystems.