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(-z2) 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. Cornp., 25 (1971), pp. 339-344] and Hunter and Regan [Math. Comp., 26 (1972), pp. 339-541]. Addressing shortcomings flagged by Weideman [SIAM. T. 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 x 10(-15) in the computation of w(z) throughout the complex plane are achieved with N = 11; this is confirmed by computations. These approximations, moreover, provably achieve small relative errors throughout the upper complex half-plane where w(z) is nonzero. Numerical tests suggest that this new method is competitive, in accuracy and computation times, with existing methods for computing w(z) for complex z.