Computation of the Complex Error Function using Modified Trapezoidal Rules
share
In this paper we propose a method for computing the Faddeeva function w(z) := e^-z^2erfc(-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× 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 show that this new method is competitive, in accuracy and computation times, with existing methods for computing w(z) for complex z.
READ FULL TEXT