libcerf

Documentation

Overview

Package libcerf implements functions from libcerf by Steven G. Johnson

Index

• func Erfcx(x float64) float64
• func ErfcxY100(y100 float64) float64

Functions

func Erfcx

func Erfcx(x float64) float64

Erfcx computes the scaled complementary error function erfcx(x) = exp(x*x) * erfc(x).

The implementation based on Steven G. Johnson and Joachim Wuttke algorithm.

This function combines a few different ideas. First, for x > 50, it uses a continued-fraction expansion (same as for the Faddeeva function, but with algebraic simplifications for z=i*x).

Second, for 0 <= x <= 50, it uses Chebyshev polynomial approximations, but with two twists:

a) It maps x to y = 4 / (4+x) in [0,1]. This simple transformation, inspired by a similar transformation in the octave-forge/specfun erfcx by Soren Hauberg, results in much faster Chebyshev convergence than other simple transformations I have examined.

b) Instead of using a single Chebyshev polynomial for the entire [0,1] y interval, we break the interval up into 100 equal subintervals, with a switch/lookup table, and use much lower degree Chebyshev polynomials in each subinterval. This greatly improves performance in my tests.

For x < 0, we use the relationship erfcx(-x) = 2 exp(x^2) - erfc(x), with the usual checks for overflow etcetera.

Performance-wise, it seems to be substantially faster than either the SLATEC DERFC function [or an erfcx function derived therefrom] or Cody's CALERF function (from netlib.org/specfun), while retaining near machine precision in accuracy.

func ErfcxY100

func ErfcxY100(y100 float64) float64

ErfcxY100 uses y100=100*y, where y = 4/(4+x) for x >= 0 to compute erfc(x).