Combining Rational Approximation with Asymptotic Wiener–Hopf Factorization Algorithm for Matrix Functions: Implementation and Testing

Rougerie T and Kisil A

Abstract

This paper discusses how an asymptotic Wiener–Hopf factorization can be implemented for a wide class of functions. Asymptotic Wiener–Hopf factorization was discussed [19] and the convergence for matrices sufficiently close to the identity matrix is shown. We demonstrate how the algorithm can be successfully implemented with the help of the rational approximations. The idea is to simplify the matrix first by rationally approximating and then perform the approximate factorisation. There is no compromise in accuracy since the factorisation is approximate anyway and rational approximations are very precise usually. There is also a mapping of real line discussed and implemented to make the rational approximation more optimal. The code is tested against some easy examples which are calculated by hand. The use of this code is illustrated with some more complicated matrix functions motivated by applications. The method has been implemented for 2 × 2 and 4 × 4 matrices but can easy be adapted for any size matrix. The code will be made available with the publication of this paper. We note that to date there are very few implemented Wiener–Hopf factorisation available due to instabilities, so this paper will make an important contribution to this area.