org.apache.commons.math3.special

## Class Gamma

- java.lang.Object
- - org.apache.commons.math3.special.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

  - Didonato and Morris (1986), Computation of the Incomplete Gamma Function Ratios and their Inverse, TOMS 12(4), 377-393,
  - Didonato and Morris (1992), Algorithm 708: Significant Digit Computation of the Incomplete Beta Function Ratios, TOMS 18(3), 360-373,
  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.

- - ### Field Summary

    Fields
    Modifier and Type | Field and Description
    --- | ---
    `static double` | `GAMMA` Euler-Mascheroni constant
    `static double` | `LANCZOS_G` The value of the `g` constant in the Lanczos approximation, see `lanczos(double)`.

  - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `static double` | `digamma(double x)` Computes the digamma function of x.
    `static double` | `gamma(double x)` Returns the value of Γ(x).
    `static double` | `invGamma1pm1(double x)` Returns the value of 1 / Γ(1 + x) - 1 for -0.5 ≤ x ≤ 1.5.
    `static double` | `lanczos(double x)` Returns the Lanczos approximation used to compute the gamma function.
    `static double` | `logGamma(double x)` Returns the value of log Γ(x) for x > 0.
    `static double` | `logGamma1p(double x)` Returns the value of log Γ(1 + x) for -0.5 ≤ x ≤ 1.5.
    `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

    - #### LANCZOS_G

          public static final double LANCZOS_G

      The value of the `g` constant in the Lanczos approximation, see `lanczos(double)`.
      Since:
        3.1
      See Also:
        Constant Field Values

  - ### Method Detail

    - #### logGamma

          public static double logGamma(double x)

      Returns the value of log Γ(x) for x > 0.

      For x ≤ 8, the implementation is based on the double precision implementation in the NSWC Library of Mathematics Subroutines, `DGAMLN`. For x > 8, the implementation 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` - Argument.
      Returns:
        the value of `log(Gamma(x))`, `Double.NaN` if `x <= 0.0`.

    - #### 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:
        `MaxCountExceededException` - 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:
      - 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:
        `MaxCountExceededException` - 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:
        `MaxCountExceededException` - 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:
      - 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:
        `MaxCountExceededException` - 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.
      Since:
        2.0
      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
      Since:
        2.0
      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) = sqrt(2 * pi) / x * (x + g + 0.5) ^ (x + 0.5) * exp(-x - g - 0.5) * lanczos(x)`,
      where `g` is the Lanczos constant.
      Parameters:
        `x` - Argument.
      Returns:
        The Lanczos approximation.
      Since:
        3.1
      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:
        `NumberIsTooSmallException` - if `x < -0.5`
        `NumberIsTooLargeException` - if `x > 1.5`
      Since:
        3.1

    - #### logGamma1p

          public static double logGamma1p(double x)
                                   throws NumberIsTooSmallException,
                                          NumberIsTooLargeException

      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:
        `NumberIsTooSmallException` - if `x < -0.5`.
        `NumberIsTooLargeException` - if `x > 1.5`.
      Since:
        3.1

    - #### 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)`.
      Since:
        3.1

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