build::libcerf::src::libcerf::man::voigt_hwhm(3)                libcerf manual


NAME
       voigt_hwhm - Half width at half maximum of Voigt's function

SYNOPSIS
       #include <cerf.h>

       double voigt_hwhm ( double sigma, double gamma );

DESCRIPTION
       The function voigt_hwhm returns the half width at half maximum of the
       function voigt.

       The function voigt returns Voigt's convolution

         voigt(x,sigma,gamma) = integral G(t,sigma) L(x-t,gamma) dt

       of a Gaussian

         G(x,sigma) = 1/sqrt(2*pi)/|sigma| * exp(-x^2/2/sigma^2)

       and a Lorentzian

         L(x,gamma) = |gamma| / pi / ( x^2 + gamma^2 ),

       with the integral extending from -infinity to +infinity.

ACCURACY
       Errors are given in units of eps = 2^-53 = 1.1e-16, as relative
       deviations from high-precision reference values.

       For 0.25 <= gamma/sigma < 16, voigt_hwhm uses a piecewise Chebyshev
       approximation generated by ppapp, whose error analysis (Wuttke and
       Kleinsorge, see REFERENCES) bounds the relative error by 2.2 eps; the
       largest deviation found at the 576 generated test points in this range
       is 1.8 eps. Below and above that range, series expansions around the
       Gaussian and the Lorentzian limit are used, for which no theoretical
       bound exists yet; the relative error was found below 2.0 eps at 1152
       generated test points for 0.03 <= gamma/sigma < 128 and at random
       points for 1e-8 < gamma/sigma < 1e8. Rescaling both arguments by a
       common factor adds at most one rounding.

SEE ALSO
       voigt(3)

       Homepage: https://jugit.fz-juelich.de/mlz/lib/cerf

REFERENCES
       Joachim Wuttke, "Power series for the half width of the Voigt function,
       rederived", J. Numer. Anal. Approx. Theo. 54, 345-356 (2025),
       https://doi.org/10.33993/jnaat542-1640, derives the series expansions
       used for small and large gamma/sigma.

       Joachim Wuttke and Alexander Kleinsorge, "Algorithm 1062: Code
       generation for piecewise Chebyshev approximation", ACM Trans. Math.
       Softw. 52, 13 (2026), https://doi.org/10.1145/3805698, explains the
       piecewise polynomial approximation used for intermediate gamma/sigma
       and derives its error bound.

AUTHOR
       Joachim Wuttke, Forschungszentrum Juelich.

CONTACT
       Please report bugs to the maintainer:

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

COPYING
       Copyright (c) 2018-25 Forschungszentrum Juelich GmbH

       Software: MIT License.

       This documentation: Creative Commons Attribution Share Alike.

perl v5.42.2                      2026-09-05
                              build::libcerf::src::libcerf::man::voigt_hwhm(3)