Class SpecialFunctions

java.lang.Object
org.mariuszgromada.math.mxparser.mathcollection.SpecialFunctions
public final class SpecialFunctions extends Object
SpecialFunctions - special (non-elementary functions).
Version:
5.1.0

  • Field Summary

    Fields
    Modifier and Type
    Field
    Description
    private static final double
    doublePrecision
     
    private static final int
    doubleWidth
     
    private static final double
    EI_DBL_EPSILON
    Constants for Exponential integral function Ei(x) calculation
    private static final double
    EI_EPSILON
     
    private static final double
    GSL_DBL_EPSILON
     

  • Constructor Summary

    Constructors
    Constructor
    Description
    SpecialFunctions()
     

  • Method Summary

    Modifier and Type
    Method
    Description
    private static double
    argumentAdditionSeriesEi(double x)
    Supporting function while Exponential integral function Ei(x) calculation
    static double
    beta(double x, double y)
    Beta special function
    private static double
    continuedFractionEi(double x)
    Supporting function while Exponential integral function Ei(x) calculation
    static double
    diGamma(double x)
    Digamma function as the logarithmic derivative of the Gamma special function
    static double
    erf(double x)
    Calculates the error function
    static double
    erfc(double x)
    Calculates the complementary error function.
    static double
    erfcInv(double z)
    Calculates the complementary inverse error function evaluated at x.
    private static double
    erfImp(double z, boolean invert)
    Calculates the inverse error function evaluated at x.
    static double
    erfInv(double x)
    Calculates the inverse error function evaluated at x.
    private static double
    erfInvImpl(double p, double q, double s)
    The implementation of the inverse error function.
    static double
    exponentialIntegralEi(double x)
    Exponential integral function Ei(x)
    static double
    gamma(double x)
    Real valued Gamma function
    private static double
    gammaInt(long n)
    Gamma function for the integers
    private static double
    halleyIteration(double x, double wInitial, int maxIter)
    Halley's iteration used in Lambert-W approximation
    static double
    hypergeometricF(double a, double b, double c, double z, double maxIterations, double precision)
    The Gaussian or ordinary hypergeometric special function
    private static double
    hypergeometricFDirect(double a, double b, double c, double z, double maxIterations, double precision)
    The Gaussian or ordinary hypergeometric special function - direct
    static double
    incompleteBeta(double a, double b, double x)
    Log Incomplete Beta special function
    static double
    incompleteGammaLower(double s, double x)
    Incomplete lower gamma function
    static double
    incompleteGammaUpper(double s, double x)
    Incomplete upper gamma function
    static double
    inverseRegularizedBeta(double a, double b, double p)
    Inerse regularized incomplete Beta special function
    static double
    inverseRegularizedGammaLowerP(double a, double p)
    Inverse of regularized lower gamma function 'P'
    static double
    lambertW(double x, double branch)
    Real-valued Lambert-W function approximation.
    private static double
    lambertW0(double x)
    W0 - Principal branch of Lambert-W function
    private static double
    lambertW1(double x)
    Minus 1 branch of Lambert-W function Analytical approximations for real values of the Lambert W-function - D.A.
    static double
    lanchosGamma(double x)
    Gamma function implementation based on Lanchos approximation algorithm
    static double
    logarithmicIntegralLi(double x)
    Logarithmic integral function li(x)
    static double
    logBeta(double x, double y)
    Log Beta special function
    static double
    logGamma(double x)
    Real valued log gamma function.
    static double
    offsetLogarithmicIntegralLi(double x)
    Offset logarithmic integral function Li(x)
    private static double
    powerSeriesEi(double x)
    Supporting function while Exponential integral function Ei(x) calculation
    static double
    regularizedBeta(double a, double b, double x)
    Regularized incomplete Beta special function
    static double
    regularizedGammaLowerP(double s, double x)
    Regularized lower gamma function 'P'
    static double
    regularizedGammaUpperQ(double s, double x)
    Regularized upper gamma function 'Q'
    private static double
    seriesEval(double r)
    Private method used in Lambert-W approximation - near zero
    static double
    sgnGamma(double x)
    Signum from the real valued gamma function.

    Methods inherited from class Object

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

  • Field Details

    • EI_DBL_EPSILON

      private static final double EI_DBL_EPSILON
      Constants for Exponential integral function Ei(x) calculation

    • EI_EPSILON

      private static final double EI_EPSILON

    • doubleWidth

      private static final int doubleWidth
      See Also:

    • doublePrecision

      private static final double doublePrecision

    • GSL_DBL_EPSILON

      private static final double GSL_DBL_EPSILON
      See Also:

  • Constructor Details

    • SpecialFunctions

      public SpecialFunctions()

  • Method Details

    • exponentialIntegralEi

      public static double exponentialIntegralEi(double x)
      Exponential integral function Ei(x)
      Parameters:
      x - Point at which function will be evaluated.
      Returns:
      Exponential integral function Ei(x)

    • continuedFractionEi

      private static double continuedFractionEi(double x)
      Supporting function while Exponential integral function Ei(x) calculation

    • powerSeriesEi

      private static double powerSeriesEi(double x)
      Supporting function while Exponential integral function Ei(x) calculation

    • argumentAdditionSeriesEi

      private static double argumentAdditionSeriesEi(double x)
      Supporting function while Exponential integral function Ei(x) calculation

    • logarithmicIntegralLi

      public static double logarithmicIntegralLi(double x)
      Logarithmic integral function li(x)
      Parameters:
      x - Point at which function will be evaluated.
      Returns:
      Logarithmic integral function li(x)

    • offsetLogarithmicIntegralLi

      public static double offsetLogarithmicIntegralLi(double x)
      Offset logarithmic integral function Li(x)
      Parameters:
      x - Point at which function will be evaluated.
      Returns:
      Offset logarithmic integral function Li(x)

    • erf

      public static double erf(double x)
      Calculates the error function
      Parameters:
      x - Point at which function will be evaluated.
      Returns:
      Error function erf(x)

    • erfc

      public static double erfc(double x)
      Calculates the complementary error function.
      Parameters:
      x - Point at which function will be evaluated.
      Returns:
      Complementary error function erfc(x)

    • erfInv

      public static double erfInv(double x)
      Calculates the inverse error function evaluated at x.
      Parameters:
      x - Point at which function will be evaluated.
      Returns:
      Inverse error function erfInv(x)

    • erfImp

      private static double erfImp(double z, boolean invert)
      Calculates the inverse error function evaluated at x.
      Parameters:
      z -
      invert -
      Returns:

    • erfcInv

      public static double erfcInv(double z)
      Calculates the complementary inverse error function evaluated at x.
      Parameters:
      z - Point at which function will be evaluated.
      Returns:
      Inverse of complementary inverse error function erfcInv(x)

    • erfInvImpl

      private static double erfInvImpl(double p, double q, double s)
      The implementation of the inverse error function.
      Parameters:
      p -
      q -
      s -
      Returns:

    • gammaInt

      private static double gammaInt(long n)
      Gamma function for the integers
      Parameters:
      n - Integer number
      Returns:
      Returns Gamma function for the integers.

    • gamma

      public static double gamma(double x)
      Real valued Gamma function
      Parameters:
      x - Argument value
      Returns:
      Returns gamma function value.

    • lanchosGamma

      public static double lanchosGamma(double x)
      Gamma function implementation based on Lanchos approximation algorithm
      Parameters:
      x - Function parameter
      Returns:
      Gamma function value (Lanchos approx).

    • logGamma

      public static double logGamma(double x)
      Real valued log gamma function.
      Parameters:
      x - Argument value
      Returns:
      Returns log value from gamma function.

    • sgnGamma

      public static double sgnGamma(double x)
      Signum from the real valued gamma function.
      Parameters:
      x - Argument value
      Returns:
      Returns signum of the gamma(x)

    • regularizedGammaLowerP

      public static double regularizedGammaLowerP(double s, double x)
      Regularized lower gamma function 'P'
      Parameters:
      s - Argument value
      x - Argument value
      Returns:
      Value of the regularized lower gamma function 'P'.

    • inverseRegularizedGammaLowerP

      public static double inverseRegularizedGammaLowerP(double a, double p)
      Inverse of regularized lower gamma function 'P'
      Parameters:
      a - Argument value
      p - Argument value
      Returns:
      Value of the inverse regularized lower gamma function 'P'.

    • incompleteGammaLower

      public static double incompleteGammaLower(double s, double x)
      Incomplete lower gamma function
      Parameters:
      s - Argument value
      x - Argument value
      Returns:
      Value of the incomplete lower gamma function.

    • regularizedGammaUpperQ

      public static double regularizedGammaUpperQ(double s, double x)
      Regularized upper gamma function 'Q'
      Parameters:
      s - Argument value
      x - Argument value
      Returns:
      Value of the regularized upper gamma function 'Q'.

    • incompleteGammaUpper

      public static double incompleteGammaUpper(double s, double x)
      Incomplete upper gamma function
      Parameters:
      s - Argument value
      x - Argument value
      Returns:
      Value of the incomplete upper gamma function.

    • diGamma

      public static double diGamma(double x)
      Digamma function as the logarithmic derivative of the Gamma special function
      Parameters:
      x - Argument value
      Returns:
      Approximated value of the digamma function.

    • logBeta

      public static double logBeta(double x, double y)
      Log Beta special function
      Parameters:
      x - Argument value
      y - Argument value
      Returns:
      Return logBeta special function (for positive x and positive y)

    • beta

      public static double beta(double x, double y)
      Beta special function
      Parameters:
      x - Argument value
      y - Argument value
      Returns:
      Return Beta special function (for positive x and positive y)

    • incompleteBeta

      public static double incompleteBeta(double a, double b, double x)
      Log Incomplete Beta special function
      Parameters:
      a - Argument value
      b - Argument value
      x - Argument value
      Returns:
      Return incomplete Beta special function for positive a and positive b and x between 0 and 1

    • regularizedBeta

      public static double regularizedBeta(double a, double b, double x)
      Regularized incomplete Beta special function
      Parameters:
      a - Argument value
      b - Argument value
      x - Argument value
      Returns:
      Return incomplete Beta special function for positive a and positive b and x between 0 and 1

    • inverseRegularizedBeta

      public static double inverseRegularizedBeta(double a, double b, double p)
      Inerse regularized incomplete Beta special function
      Parameters:
      a - Argument value
      b - Argument value
      p - Argument value
      Returns:
      Return inverse incomplete Beta special function for positive a and positive b and x between 0 and 1

    • halleyIteration

      private static double halleyIteration(double x, double wInitial, int maxIter)
      Halley's iteration used in Lambert-W approximation
      Parameters:
      x - Point at which Halley iteration will be calculated
      wInitial - Starting point
      maxIter - Maximum number of iteration
      Returns:
      Halley's iteration value if successful, otherwise Double.NaN

    • seriesEval

      private static double seriesEval(double r)
      Private method used in Lambert-W approximation - near zero
      Parameters:
      r -
      Returns:
      Ner zero approximation

    • lambertW0

      private static double lambertW0(double x)
      W0 - Principal branch of Lambert-W function
      Parameters:
      x -
      Returns:
      Approximation of principal branch of Lambert-W function

    • lambertW1

      private static double lambertW1(double x)
      Minus 1 branch of Lambert-W function Analytical approximations for real values of the Lambert W-function - D.A. Barry Mathematics and Computers in Simulation 53 (2000) 95–103
      Parameters:
      x -
      Returns:
      Approxmiation of minus 1 branch of Lambert-W function

    • lambertW

      public static double lambertW(double x, double branch)
      Real-valued Lambert-W function approximation.
      Parameters:
      x - Point at which function will be approximated
      branch - Branch id, 0 for principal branch, -1 for the other branch
      Returns:
      Principal branch for x greater or equal than -1/e, otherwise Double.NaN. Minus 1 branch for x greater or equal than -1/e and lower than 0, otherwise Double.NaN.

    • hypergeometricF

      public static double hypergeometricF(double a, double b, double c, double z, double maxIterations, double precision)
      The Gaussian or ordinary hypergeometric special function
      Parameters:
      a - Argument value
      b - Argument value
      c - Argument value
      z - Argument value
      maxIterations - Stop condition
      precision - Stop condition
      Returns:
      Returns hypergeometric special function approximation

    • hypergeometricFDirect

      private static double hypergeometricFDirect(double a, double b, double c, double z, double maxIterations, double precision)
      The Gaussian or ordinary hypergeometric special function - direct
      Parameters:
      a - Argument value
      b - Argument value
      c - Argument value
      z - Argument value
      maxIterations - Stop condition
      precision - Stop condition
      Returns:
      Returns hypergeometric special function approximation - direct