Package nl.tudelft.simulation.jstats.math

Class ErfPrecision

  • public final class ErfPrecision
extends Object
    Erf function precision test.

    Copyright (c) 2021-2023 Delft University of Technology, Jaffalaan 5, 2628 BX Delft, the Netherlands. All rights reserved. See for project information DSOL Manual. The DSOL project is distributed under a three-clause BSD-style license, which can be found at DSOL License.

    Author:
    Alexander Verbraeck

    • Method Summary

      All Methods Static Methods Concrete Methods 
      Modifier and Type Method Description
      (package private) static double erf​(double z)
      Calculate erf(z), where three different algorithms are used for small, medium, and large values.
      (package private) static double erfBig​(double z)
      Calculate erf(z) for large values using the Taylor series:
          erf(z) = 1 - (exp(-z2) / √π) Σ [ (-1)n (2n - 1)!! z-(2n + 1) / 2n]
      where the !! operator is the 'double factorial' operator which is (n).(n-2)...8.4.2 for even n, and (n).(n-2)...3.5.1 for odd n.
      (package private) static double erfInv​(double y)
      Approximates erf-1(p) using a Taylor series.
      The Taylor series for erf-1(z) is:
          erf-1(p) = √π Σ [ 1/2 p + 1/24 π p3 + 7/960 π2 p5 + 127/80640 π3 p7 + ...
      (package private) static double erfSmall​(double z)
      The Taylor series for erf(z) for abs(z) < 0.5 that is used is:
          erf(z) = (exp(-z2) / √π) Σ [ 2z2n + 1 / (2n + 1)!!]
      where the !! operator is the 'double factorial' operator which is (n).(n-2)...8.4.2 for even n, and (n).(n-2)...3.5.1 for odd n.
      (package private) static double erfTaylor​(double z)
      Calculate erf(z) using the Taylor series:
          erf(z) = (2/√π) (z - z3/3 + z5/10 - z7/42 + z9/216 - ...)
      The factors are given by https://oeis.org/A007680, which evaluates to a(n) = (2n+1)n!.
      (package private) static double inverseErf​(double y)
      Calculate based on http://www.naic.edu/~jeffh/inverse_cerf.c code.
      static void main​(String[] args)  

    • Method Detail

      • erf

        static double erf​(double z)
        Calculate erf(z), where three different algorithms are used for small, medium, and large values.
        Parameters:
        z - double; the value to calculate erf(z) for
        Returns:
        double; erf(z)

      • erfSmall

        static double erfSmall​(double z)
        The Taylor series for erf(z) for abs(z) < 0.5 that is used is:
            erf(z) = (exp(-z2) / √π) Σ [ 2z2n + 1 / (2n + 1)!!]
        where the !! operator is the 'double factorial' operator which is (n).(n-2)...8.4.2 for even n, and (n).(n-2)...3.5.1 for odd n. See https://mathworld.wolfram.com/Erf.html formula (9) and (10). This function would work well for z < 0.5.
        Parameters:
        z - double; the parameter
        Returns:
        double; erf(x)

      • erfBig

        static double erfBig​(double z)
        Calculate erf(z) for large values using the Taylor series:
            erf(z) = 1 - (exp(-z2) / √π) Σ [ (-1)n (2n - 1)!! z-(2n + 1) / 2n]
        where the !! operator is the 'double factorial' operator which is (n).(n-2)...8.4.2 for even n, and (n).(n-2)...3.5.1 for odd n. See https://mathworld.wolfram.com/Erf.html formula (18) to (20). This function would work well for z > 3.7.
        Parameters:
        z - double; the argument
        Returns:
        double; erf(z)

      • erfTaylor

        static double erfTaylor​(double z)
        Calculate erf(z) using the Taylor series:
            erf(z) = (2/√π) (z - z3/3 + z5/10 - z7/42 + z9/216 - ...)
        The factors are given by https://oeis.org/A007680, which evaluates to a(n) = (2n+1)n!. See https://en.wikipedia.org/wiki/Error_function. This works pretty well on the interval [0.5,3.7].
        Parameters:
        z - double; the argument
        Returns:
        double; erf(z)

      • erfInv

        static double erfInv​(double y)
        Approximates erf-1(p) using a Taylor series.
        The Taylor series for erf-1(z) is:
            erf-1(p) = √π Σ [ 1/2 p + 1/24 π p3 + 7/960 π2 p5 + 127/80640 π3 p7 + ... ]
        See https://mathworld.wolfram.com/InverseErf.html.
        The factors are given by https://oeis.org/A002067 which evaluates to and https://oeis.org/A122551
        Parameters:
        y - double; the cumulative probability to calculate the inverse error function for
        Returns:
        erf-1(y)

      • inverseErf

        static double inverseErf​(double y)
        Calculate based on http://www.naic.edu/~jeffh/inverse_cerf.c code.
        Parameters:
        y - double; value to calculate erf-1(y) for
        Returns:
        double; erf-1(x)

      • main

        public static void main​(String[] args)
        Parameters:
        args - not used