cerf, cerfc - complex error functions
#include <cerf.h>
double complex cerf ( double complex z );
double complex cerfc ( double complex z );
The data type double complex is defined in the header <complex.h>, which C99 introduced. Since C11 it is optional: an implementation may define __STDC_NO_COMPLEX__ and then provide neither the header nor the type, and Microsoft's C compiler does not support the arithmetic operators for it. As a fallback, use the C++ variant of this library, libcerfcpp, in which double complex is replaced by std::complex<double> from the header <complex>.
The function cerf is the complex version of the error function: erf(z) = 2/sqrt(pi) * integral from 0 to z of exp(-t*t) dt.
The complementary complex error function cerfc is defined as erfc(z) = 1-cerf(z).
Errors are given in units of eps = 2^-53 = 1.1e-16, as relative deviations of the modulus from high-precision reference values.
Both functions are computed from Faddeeva's function w_of_z(3) through the factor exp(-z^2), as erfc(z) = exp(-z^2) w(iz) and erf(z) = 1 - erfc(z), with the exponent -z^2 carried as an unevaluated sum of two doubles. Where exp(-z^2) overflows although the product with w does not, the modulus of the factor is taken in scaled form and the scale applied to the product, so that both functions cover their full range and return an infinity only where the value itself exceeds the double format.
The error is not controlled. For Re z < 0, w(iz) is obtained by reflection, and cancellation must be expected (see w_of_z(3)); near the zeros of erfc, the first of which lie at z = -1.3548 +- 1.9915i, the relative error of cerfc is unbounded. cerf shares the error of cerfc in absolute terms, so that its relative error is unbounded near the zeros of erf, the first of which lie at z = +-1.4506 +- 1.8809i, and reaches about 120 eps close to the origin, for 0.01 < |z| < 0.1 off the coordinate axes, where the Maclaurin series is not used.
Joachim Wuttke, "libcerf, complex error function and related functions reimplemented with relative accuracy guarantees" (unpublished manuscript, available upon request) documents the algorithms of this library and derives their error bounds.
The computations are based on Faddeeva's function w_of_z(3).
Other complex error functions in libcerf: w_of_z(3), dawson(3), voigt(3), erfcx(3), erfi(3).
The real error function comes with recent versions of glibc, as requested by the C99 standard: erf(3)
Homepage: https://jugit.fz-juelich.de/mlz/lib/cerf
Steven G. Johnson, Massachusetts Institute of Technology, wrote this function as part of the MIT Faddeeva package.
Joachim Wuttke, Forschungszentrum Juelich, reorganized the code into a library, and wrote this man page.
Please report bugs to the maintainer:
Joachim Wuttke <j.wuttke@fz-juelich.de>
Copyright (c) 2012 Massachusetts Institute of Technology
Copyright (c) 2013 Forschungszentrum Juelich GmbH
Software: MIT License.
This documentation: Creative Commons Attribution Share Alike.