Computation of the complex error function using modified trapezoidal rules

In this paper we propose a method for computing the Faddeeva function $w(z) := \re^{-z^2}\erfc(-\ri\,z)$ via truncated modified trapezoidal rule approximations to integrals on the real line. Our starting point is the method due to Matta and Reichel ({\em Math.\ Comp.} {\bf 25} (1971), pp.~339--344) and Hunter and Regan ({\em Math.\ Comp.} {\bf 26} (1972), pp.~339--541). Addressing shortcomings flagged by Weideman ({\em SIAM.\ J.\ Numer.\ Anal. } {\bf 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 $z$