mnt::share::cygpkgs::libcerf::libcerf.x86_64::src::libcerf-3.7::man::cerf(3)

NAME
cerf, cerfc - complex error functions

SYNOPSIS
#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>.

DESCRIPTION
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).

ACCURACY
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). The exponent -z^2 = (y-x)(x+y) - 2ixy is carried as an unevaluated sum of two doubles, so that the factor is obtained to about 4 eps whatever |z| (Wuttke, see REFERENCES). At 20000 random points where the exponential neither over- nor underflows, |z| reaching 1e8, the relative error of cerfc stays below 5.5 eps. Before libcerf-3.7 the exponent was rounded into a single double, and the relative error grew in proportion to |z|^2, reaching 1080 eps at |z| = 26 and leaving no correct digit at all where x^2-y^2 stays small up to |z| = 1e8.

Where exp(-z^2) overflows although the product with w does not, |w| <= 1 leaving room of a factor sqrt(pi) |z|, the modulus of the factor is taken in scaled form and the scale applied to the product. Both functions therefore cover their full range, and return an infinity only where the value itself exceeds the double format.

cerf shares this error in absolute terms; where |erf(z)| is small, it is amplified accordingly: without bound near the zeros of erf, the first of which lie at z = +-1.4506 +- 1.8809i, and up to about 120 eps close to the origin, for 0.01 < |z| < 0.1 off the coordinate axes, where the Maclaurin series is not used. Likewise, cerfc loses accuracy without bound near its zeros, the first of which lie at z = -1.3548 +- 1.9915i.

For Im z >= 0 and |z| < 7, generated tests at 37670 points, up to 20 per tile of the disc, hold the relative error of cerf below 129 eps and that of cerfc below 86 eps; the largest deviations found are 126 eps, next to a zero of erf, and 82 eps.

REFERENCES
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.

SEE ALSO
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

AUTHORS
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.

CONTACT
Please report bugs to the maintainer:

Joachim Wuttke <j.wuttke@fz-juelich.de>

COPYING
Copyright (c) 2012 Massachusetts Institute of Technology

Copyright (c) 2013 Forschungszentrum Juelich GmbH

Software: MIT License.

This documentation: Creative Commons Attribution Share Alike.