Extended Arnoldi methods for large low-rank Sylvester matrix equations

In this paper, we present two iterative methods for the solution of the low-rank Sylvester equation A X + X B + E F T = 0 . These methods are projection methods that use the extended block Arnoldi (EBA) process and the extended global Arnoldi (EGA) process to generate orthonormal bases and F-orthonormal bases of extended Krylov subspaces. For each algorithm, we show how to stop the iterations by computing the residual norm or an upper bound without computing the approximate solution and without using expensive products with the matrices A and B. We also describe how to get the low rank solution of the Sylvester equation in a factored form. Finally, some numerical experiments are presented in order to show the efficiency and robustness of the proposed methods.