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.IX Title "build::libcerf::src::libcerf::man::voigt_hwhm 3"
.TH build::libcerf::src::libcerf::man::voigt_hwhm 3 2026-09-05 "perl v5.42.2" "libcerf manual"
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.SH NAME
voigt_hwhm \- Half width at half maximum of Voigt\*(Aqs function
.SH SYNOPSIS
.IX Header "SYNOPSIS"
\&\fB#include <cerf.h\fR>
.PP
\&\fBdouble voigt_hwhm ( double sigma, double gamma );\fR
.SH DESCRIPTION
.IX Header "DESCRIPTION"
The function \fBvoigt_hwhm\fR returns the half width at half maximum of the function \fBvoigt\fR.
.PP
The function \fBvoigt\fR returns Voigt\*(Aqs convolution
.PP
.Vb 1
\&  voigt(x,sigma,gamma) = integral G(t,sigma) L(x\-t,gamma) dt
.Ve
.PP
of a Gaussian
.PP
.Vb 1
\&  G(x,sigma) = 1/sqrt(2*pi)/|sigma| * exp(\-x^2/2/sigma^2)
.Ve
.PP
and a Lorentzian
.PP
.Vb 1
\&  L(x,gamma) = |gamma| / pi / ( x^2 + gamma^2 ),
.Ve
.PP
with the integral extending from \-infinity to +infinity.
.SH ACCURACY
.IX Header "ACCURACY"
Errors are given in units of eps = 2^\-53 = 1.1e\-16, as relative deviations
from high\-precision reference values.
.PP
For 0.25 <= gamma/sigma < 16, \fBvoigt_hwhm\fR uses a piecewise Chebyshev
approximation generated by \fBppapp\fR, 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.
.SH "SEE ALSO"
.IX Header "SEE ALSO"
\&\fBvoigt\|(3)\fR
.PP
Homepage: https://jugit.fz\-juelich.de/mlz/lib/cerf
.SH REFERENCES
.IX Header "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.
.PP
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.
.SH AUTHOR
.IX Header "AUTHOR"
Joachim Wuttke, Forschungszentrum Juelich.
.SH CONTACT
.IX Header "CONTACT"
Please report bugs to the maintainer:
.PP
Joachim Wuttke <j.wuttke@fz\-juelich.de>
.SH COPYING
.IX Header "COPYING"
Copyright (c) 2018\-25 Forschungszentrum Juelich GmbH
.PP
Software: MIT License.
.PP
This documentation: Creative Commons Attribution Share Alike.