A preconditioned block Arnoldi method for large Sylvester matrix equations

Abstract

SUMMARYIn this paper, we propose a block Arnoldi method for solving the continuous low‐rank Sylvester matrix equation AX + XB = EFT. We consider the case where both A and B are large and sparse real matrices, and E and F are real matrices with small rank. We first apply an alternating directional implicit preconditioner to our equation, turning it into a Stein matrix equation. We then apply a block Krylov method to the Stein equation to extract low‐rank approximate solutions. We give some theoretical results and report numerical experiments to show the efficiency of this method. Copyright © 2011 John Wiley & Sons, Ltd.