Package org.apache.commons.numbers.gamma

Class BoostGamma.Lanczos

java.lang.Object
org.apache.commons.numbers.gamma.BoostGamma.Lanczos
Enclosing class:
BoostGamma
static final class BoostGamma.Lanczos extends Object
53-bit precision implementation of the Lanczos approximation.

This implementation is in partial fraction form with the leading constant of \( \sqrt{2\pi} \) absorbed into the sum.

It is related to the Gamma function by the following equation \[ \Gamma(z) = \frac{(z + g - 0.5)^{z - 0.5}}{e^{z + g - 0.5}} \mathrm{lanczos}(z) \] where \( g \) is the Lanczos constant.

Warning

This is not a substitute for LanczosApproximation. The approximation is written in partial fraction form with the leading constants absorbed by the coefficients in the sum.

See Also:

  • Field Summary

    Fields
    Modifier and Type
    Field
    Description
    private static final int[]
    DENOM
    Common denominator used for the rational evaluation.
    (package private) static final double
    G
    Lanczos constant G.
    (package private) static final double
    GMH
    Lanczos constant G - half.

  • Constructor Summary

    Constructors
    Modifier
    Constructor
    Description
    private
    Lanczos()
    Private constructor.

  • Method Summary

    Modifier and Type
    Method
    Description
    private static double
    evaluateRational(double[] a, int[] b, double x)
    Evaluate the rational number as two polynomials.
    (package private) static double
    lanczosSum(double z)
    Computes the Lanczos approximation.
    (package private) static double
    lanczosSumExpGScaled(double z)
    Computes the Lanczos approximation scaled by exp(g).

    Methods inherited from class java.lang.Object

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

  • Field Details

    • G

      static final double G
      Lanczos constant G.
      See Also:

    • GMH

      static final double GMH
      Lanczos constant G - half.

      Note: The form (g - 0.5) is used when computing the gamma function.

      See Also:

    • DENOM

      private static final int[] DENOM
      Common denominator used for the rational evaluation.

  • Constructor Details

    • Lanczos

      private Lanczos()
      Private constructor.

  • Method Details

    • lanczosSum

      static double lanczosSum(double z)
      Computes the Lanczos approximation.
      Parameters:
      z - Argument.
      Returns:
      the Lanczos approximation.

    • lanczosSumExpGScaled

      static double lanczosSumExpGScaled(double z)
      Computes the Lanczos approximation scaled by exp(g).
      Parameters:
      z - Argument.
      Returns:
      the scaled Lanczos approximation.

    • evaluateRational

      private static double evaluateRational(double[] a, int[] b, double x)
      Evaluate the rational number as two polynomials.

      Adapted from boost/math/tools/detail/rational_horner3_13.hpp. Note: There are 3 variations of the unrolled rational evaluation. These methods change the order based on the sign of x. This should be used for the Lanczos code as this comment in boost/math/tools/rational.hpp notes:

      However, there are some tricks we can use to prevent overflow that might otherwise occur in polynomial evaluation, if z is large. This is important in our Lanczos code for example.
      Parameters:
      a - Coefficients of the numerator polynomial
      b - Coefficients of the denominator polynomial
      x - Value
      Returns:
      the rational number