A note on the Davison–Man method for Sylvester matrix equations

Abstract

In this paper, we introduce a modernized and improved version of the Davison–Man method for the numerical resolution of Sylvester matrix equations. In the case of moderate size problems, we give some background facts about this iterative method, addressing the problem of stagnation and we propose an iterative refinement technique to improve its accuracy. Although not designed to solve large-scale Sylvester equations, we propose an approach combining the extended block Krylov algorithm and the Davison–Man method which gives interesting results in terms of accuracy and shows to be competitive with the classical Krylov subspaces methods. Numerical examples are given to illustrate the efficiency of this approach.