The Voigt and complex error function: Huml\'i\v{c}ek's rational approximation generalized

TL;DR

This paper generalizes Humlíček's rational approximation for the Voigt and complex error functions, achieving higher accuracy with more terms and proposing optimized combinations for efficient computation in physics applications.

Contribution

The authors extend Humlíček's 12-term approximation to 16 and 20 terms, improving accuracy and efficiency in computing the Voigt and complex error functions.

Findings

16-term approximation has <10^{-5} accuracy across most of the complex plane.

20-term approximation achieves <10^{-6} accuracy.

Proposed combination with asymptotic approximation enhances computational efficiency.

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.