Optimized implementations of rational approximations for the Voigt and complex error function

• Published in: Journal of quantitative spectroscopy & radiative transfer Vol. 112; no. 6; pp. 1010 - 1025
• Main Author: Schreier, Franz
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.04.2011
• Subjects: Atmospheric radiative transfer; Complex probability function; Fortran; Plasma dispersion function; Python; Atmospheric radiative transfer; Complex probability function; Plasma dispersion function; Python; Fortran
• ISSN: 0022-4073; 1879-1352
• DOI: 10.1016/j.jqsrt.2010.12.010

Rational functions are frequently used as efficient yet accurate numerical approximations for real and complex valued functions. For the complex error function w( x+i y), whose real part is the Voigt function K( x, y), code optimizations of rational approximations are investigated. An assessment of requirements for atmospheric radiative transfer modeling indicates a y range over many orders of magnitude and accuracy better than 10 −4. Following a brief survey of complex error function algorithms in general and rational function approximations in particular the problems associated with subdivisions of the x, y plane (i.e., conditional branches in the code) are discussed and practical aspects of Fortran and Python implementations are considered. Benchmark tests of a variety of algorithms demonstrate that programming language, compiler choice, and implementation details influence computational speed and there is no unique ranking of algorithms. A new implementation, based on subdivision of the upper half-plane in only two regions, combining Weideman's rational approximation for small | x | + y < 15 and Humlicek's rational approximation otherwise is shown to be efficient and accurate for all x, y. ► Function arguments spanning many orders of magnitude. ► No approximation good for all arguments, but two subregions sufficient. ► Line center: just a few evaluations, accuracy more important than speed. ► Combination of two rational approximations: Weideman and asymptotic Humlicek. ► Code performance is system dependent, no unique ranking of algorithms.