Computation of the Complex Error Function using Modified Trapezoidal Rules

10/12/2020
by   Mohammad Al Azah, et al.
0

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

Please sign up or login with your details

Forgot password? Click here to reset