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A preconditioned block Arnoldi method for large Sylvester matrix equations


Correspondence to: K. Jbilou, Université de Lille Nord de France, L.M.P.A, ULCO, 50 rue F. Buisson BP699, F-62228 Calais-Cedex, France.



In 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.