Package org.hipparchus.special

Class Gamma

  • public class Gamma
extends Object

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

    Implementation of invGamma1pm1(double) and logGamma1p(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 Hipparchus.

    • Method Detail

      • 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:
        MathIllegalStateException - if the algorithm fails to converge.

      • regularizedGammaP

        public static <T extends CalculusFieldElement<T>> T regularizedGammaP​(T a,
                                                                      T x)
        Returns the regularized gamma function P(a, x).
        Type Parameters:
        T - Type of the field elements.
        Parameters:
        a - Parameter.
        x - Value.
        Returns:
        the regularized gamma function P(a, x).
        Throws:
        MathIllegalStateException - 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:

        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:
        MathIllegalStateException - if the algorithm fails to converge.

      • regularizedGammaP

        public static <T extends CalculusFieldElement<T>> T regularizedGammaP​(T a,
                                                                      T x,
                                                                      double epsilon,
                                                                      int maxIterations)
        Returns the regularized gamma function P(a, x).

        The implementation of this method is based on:

        Type Parameters:
        T - Type of the field elements.
        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:
        MathIllegalStateException - 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:
        MathIllegalStateException - if the algorithm fails to converge.

      • regularizedGammaQ

        public static <T extends CalculusFieldElement<T>> T regularizedGammaQ​(T a,
                                                                      T x)
        Returns the regularized gamma function Q(a, x) = 1 - P(a, x).
        Type Parameters:
        T - Type of the field elements.
        Parameters:
        a - the a parameter.
        x - the value.
        Returns:
        the regularized gamma function Q(a, x)
        Throws:
        MathIllegalStateException - 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:

        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:
        MathIllegalStateException - 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.
        See Also:
        Digamma, Bernardo's original article

      • digamma

        public static <T extends CalculusFieldElement<T>> T digamma​(T 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.

        Type Parameters:
        T - Type of the field elements.
        Parameters:
        x - Argument.
        Returns:
        digamma(x) to within 10-8 relative or absolute error whichever is smaller.
        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
        See Also:
        Trigamma, digamma(double)

      • trigamma

        public static <T extends CalculusFieldElement<T>> T trigamma​(T x)
        Computes the trigamma function of x. This function is derived by taking the derivative of the implementation of digamma.
        Type Parameters:
        T - Type of the field elements.
        Parameters:
        x - Argument.
        Returns:
        trigamma(x) to within 10-8 relative or absolute error whichever is smaller
        See Also:
        Trigamma, digamma(double)

      • lanczos

        public static double lanczos​(double x)

        Returns the Lanczos approximation used to compute the gamma function. The Lanczos approximation is related to the Gamma function by the following equation \[ \Gamma(x) = \frac{\sqrt{2\pi}}{x} \times (x + g + \frac{1}{2}) ^ (x + \frac{1}{2}) \times e^{-x - g - 0.5} \times \mathrm{lanczos}(x) \] where g is the Lanczos constant.

        Parameters:
        x - Argument.
        Returns:
        The Lanczos approximation.
        See Also:
        Lanczos Approximation equations (1) through (5), and Paul Godfrey's Note on the computation of the convergent Lanczos complex Gamma approximation

      • lanczos

        public static <T extends CalculusFieldElement<T>> T lanczos​(T x)

        Returns the Lanczos approximation used to compute the gamma function. The Lanczos approximation is related to the Gamma function by the following equation \[ \Gamma(x) = \frac{\sqrt{2\pi}}{x} \times (x + g + \frac{1}{2}) ^ (x + \frac{1}{2}) \times e^{-x - g - 0.5} \times \mathrm{lanczos}(x) \] where g is the Lanczos constant.

        Type Parameters:
        T - Type of the field elements.
        Parameters:
        x - Argument.
        Returns:
        The Lanczos approximation.
        See Also:
        Lanczos Approximation equations (1) through (5), and Paul Godfrey's Note on the computation of the convergent Lanczos complex Gamma approximation

      • invGamma1pm1

        public static double invGamma1pm1​(double x)
        Returns the value of 1 / Γ(1 + x) - 1 for -0.5 ≤ x ≤ 1.5. This implementation is based on the double precision implementation in the NSWC Library of Mathematics Subroutines, DGAM1.
        Parameters:
        x - Argument.
        Returns:
        The value of 1.0 / Gamma(1.0 + x) - 1.0.
        Throws:
        MathIllegalArgumentException - if x < -0.5
        MathIllegalArgumentException - if x > 1.5

      • invGamma1pm1

        public static <T extends CalculusFieldElement<T>> T invGamma1pm1​(T x)
        Returns the value of 1 / Γ(1 + x) - 1 for -0.5 ≤ x ≤ 1.5. This implementation is based on the double precision implementation in the NSWC Library of Mathematics Subroutines, DGAM1.
        Type Parameters:
        T - Type of the field elements.
        Parameters:
        x - Argument.
        Returns:
        The value of 1.0 / Gamma(1.0 + x) - 1.0.
        Throws:
        MathIllegalArgumentException - if x < -0.5
        MathIllegalArgumentException - if x > 1.5

      • logGamma1p

        public static double logGamma1p​(double x)
                         throws MathIllegalArgumentException
        Returns the value of log Γ(1 + x) for -0.5 ≤ x ≤ 1.5. This implementation is based on the double precision implementation in the NSWC Library of Mathematics Subroutines, DGMLN1.
        Parameters:
        x - Argument.
        Returns:
        The value of log(Gamma(1 + x)).
        Throws:
        MathIllegalArgumentException - if x < -0.5.
        MathIllegalArgumentException - if x > 1.5.

      • gamma

        public static double gamma​(double x)
        Returns the value of Γ(x). Based on the NSWC Library of Mathematics Subroutines double precision implementation, DGAMMA.
        Parameters:
        x - Argument.
        Returns:
        the value of Gamma(x).

      • gamma

        public static <T extends CalculusFieldElement<T>> T gamma​(T x)
        Returns the value of Γ(x). Based on the NSWC Library of Mathematics Subroutines double precision implementation, DGAMMA.
        Type Parameters:
        T - Type of the field elements.
        Parameters:
        x - Argument.
        Returns:
        the value of Gamma(x).