Computation of the Complex Error Function using Modified Trapezoidal Rules

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 $z$.