The Voigt and complex error function: Humlíček’s rational approximation generalized

Abstract

Accurate yet efficient computation of the Voigt and complex error function is a challenge since decades in astrophysics and other areas of physics. Rational approximations have attracted considerable attention and are used in many codes, often in combination with other techniques. The 12-term code "cpf12" of Huml\'i\v{c}ek (1979) achieves an accuracy of five to six significant digits throughout the entire complex plane. Here we generalize this algorithm to a larger (even) number of terms. The $n=16$ approximation has a relative accuracy better than $10^{-5}$ for almost the entire complex plane except for very small imaginary values of the argument even without the correction term required for the cpf12 algorithm. With 20 terms the accuracy is better than $10^{-6}$. In addition to the accuracy assessment we discuss methods for optimization and propose a combination of the 16-term approximation with the asymptotic approximation of Huml\'i\v{c}ek (1982) for high efficiency.