Computation of the Complex Error Function using Modified Trapezoidal Rules

TL;DR

This paper introduces a modified trapezoidal rule-based method for computing the Faddeeva function, achieving high accuracy and exponential convergence, and demonstrating competitiveness with existing algorithms.

Contribution

It develops an improved, exponentially convergent approximation for the Faddeeva function based on modified trapezoidal rules, addressing previous shortcomings and providing rigorous error bounds.

Findings

Achieves machine precision accuracy with 11 quadrature points.

Provides exponential convergence and rigorous error bounds.

Demonstrates competitive performance with existing methods.

Abstract

In this paper we propose a method for computing the Faddeeva function $w(z) := e^{-z^2}\mathrm{erfc}(-i z)$ via truncated modified trapezoidal rule approximations to integrals on the real line. Our starting point is the method due to Matta and Reichel (Math. Comp. 25 (1971), pp. 339-344) and Hunter and Regan (Math. Comp. 26 (1972), pp. 339-541). Addressing shortcomings flagged by Weideman (SIAM. J. Numer. Anal. 31 (1994), pp. 1497-1518), we construct approximations which we prove are exponentially convergent as a function of $N+1$, the number of quadrature points, obtaining error bounds which show that accuracies of $2\times 10^{-15}$ in the computation of $w(z)$ throughout the complex plane are achieved with $N = 11$, this confirmed by computations. These approximations, moreover, provably achieve small relative errors throughout the upper complex half-plane where $w(z)$ is non-zero. Numerical tests suggest that this new method is competitive, in accuracy and computation times, with existing methods for computing $w(z)$ for complex $z$.