Evaluation of complex-valued error-like functions by the exponentially-convergent trapezoidal rule

TL;DR

This paper introduces an efficient, accurate algorithm for evaluating complex error-like functions using the exponentially convergent trapezoidal rule, implemented in the erflike library, outperforming existing methods in accuracy and speed for complex arguments.

Contribution

It presents a novel application of the exponentially convergent trapezoidal rule to evaluate the Faddeeva function and related functions, with an open-source implementation that improves accuracy and efficiency.

Findings

The erflike library achieves better accuracy than existing methods.

It provides more regular relative error behavior over the complex plane.

The library outperforms Faddeeva package in speed for complex arguments.

Abstract

The exponentially convergent trapezoidal rule is applied to a suitable integral representation of the Faddeeva function to derive a simple formula for its evaluation. I describe its properties, strategies for maximising its efficiency, and its coupling with other evaluation methods (asymptotic expansions and Maclaurin series). From knowledge of the values of the Faddeeva function, all other complex-valued error-like functions such as $\rm erf$ and $\rm erfc$ can be easily obtained. The resulting algorithm has been implemented in a publicly-available C/C++ library named $\texttt{erflike}$ in IEEE double precision arithmetic, and tested against more widespread valuation methods based on Taylor series and continued fractions, as provided by the widely used Faddeeva package. It is found that the algorithm presented here and its implementation achieve better accuracy and a more regular behaviour of the relative error over vast regions of the complex plane. In terms of speed of evaluation the $\texttt{erflike}$ library also outperforms the Faddeeva package for complex valued arguments, although not for real-valued ones.