An alternative extended block Arnoldi method for solving low-rank Sylvester equations

Abstract

Projection methods based on Krylov subspaces are an effective tool for solving low-rank Sylvester matrix equations in the large scale cases. The initial problem is projected onto a sequence of nested subspaces. At each step of the process, a reduced Sylvester equation, whose coefficient matrices are restrictions of the original coefficient matrices to the projection subspaces, are solved by a direct method. In this paper, after recalling some important properties about the extended block Arnoldi process, we propose an alternative approach that enables us to determine approximate solutions by solving projected and reduced Sylvester equations whose coefficient matrices are restrictions of the inverse of the original matrices. Some numerical experiments are given in order to illustrate the performance of our approach.