This project contains Matlab code for evaluation of the Faddeeva function w(z), for complex z, where
w(z) := exp(-z^2) erfc(-i z),
erfc is the standard complementary error function, and i = sqrt(-1). The methods used are based on representations for w(z) as an integral on the real line [1, (3)], and the evaluation of this integral by trapezoidal rules modified to take into account poles of the integrand near the real axis.
The specific approximation w_N(z) implemented is the approximation obtained when the modified trapezoidal rule is used with N+1 quadrature points. This new method is attractive because it is provably exponentially convergent at a rapid rate, the accuracy improving by a factor exp(pi) = 23.1... for each extra quadrature point. (The results in [1, Figure 2] demonstrate that this convergence is achieved in practice in double precision arithmetic.)
Precisely [1, Theorem 1.1] proves that |w(z)-w_N(z)| \leq 0.7 exp(-pi N) for all non-negative integers N and all complex z (in exact arithmetic), and a similar bound holds on the relative error in the upper complex half-plane where w(z) is non-zero.
See [1] and its Supplementary Materials [2] for more details. In particular [1, Table 1] makes comparisons of accuracies and timings of this new method with those of previously published methods and their associated Matlab implementations.
The key Matlab codes in this project are:
wTrap.m Matlab function, with inputs z and N, that evaluates w_N(z) using N+1 quadrature points in the case when z=x+iy with x,y >= 0. (N = 11 is recommended to achieve absolute and relative errors no larger than 1.4e-15)
wTrapWCP.m Matlab function, with inputs z and N, that evaluates w_N(z) using N+1 quadrature points for arbitrary complex z, by calling wTrap and using symmetries of w(z). (Again, N = 11 is recommended.)
test_wTrapWCP.m Matlab script file to run to test that wTrapWCP (and wTrap which it calls) are working. This should produce the output ErrorPS = 1.132209773400735e-15, indicating that the maximum difference between wTrapWCP(z,11) and w(z) computed using the power series of [3,(2.16)] with N = 12 is < 1.2e-15 in the square z = x+iy, -1/2 <= x < = 1/2, -1/2 <= y <= 1/2.
The remaining Matlab codes in this project are the files: wTrap_Q.m, a quadruple precision version of wTrap.m (which implements quadruple precision using the ADVANPIX Multiprecision Computing Toolbox for Matlab (https://www.advanpix.com/)); test_time.m which times the execution of wTrap.m against other published Matlab codes for evaluating w(z); test_accuracy_Q.m which tests the accuracy of wTrap.m and the accuracy of the same other publicly available Matlab codes against the quadruple precision version wTrap_Q.m.
For more details of all these codes see [2].
Mohammad Al Azah (Al Hussein Technical University, Amman, Jordan) and Simon Chandler-Wilde (University of Reading, UK)
[1] M. Al Azah and S. N. Chandler-Wilde, Computation of the complex error function using modified trapezoidal rules, 2021, https://arxiv.org/abs/2010.05659 . To appear in SIAM Journal on Numerical Analysis.
[2] M. Al Azah and S. N. Chandler-Wilde, Supplementary Materials: Computation of the complex error function using modified trapezoidal rules, 2021, http://www.personal.rdg.ac.uk/~sms03snc/#respre . To appear (as supplementary materials to [1]) in SIAM Journal on Numerical Analysis.
[3] G. Poppe and C. Wijers, More efficient computation of the complex error function, ACM Trans. Math. Software, 16 (1990), pp. 38–46.