A new version of successive approximations method for solving Sylvester matrix equations A. Kaabi * , A. Kerayechian, F. Toutounian Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran Abstract This paper presents a new version of the successive approximations method for solving Sylvester equations AX  XB = C, where A and B are symmetric negative and positive definite matrices, respectively. This method is based on the block GMRES-Sylvester method. We also discuss the convergence of the new method. Some numerical experiments for obtaining the numerical solution of Sylvester equations are given. Numerical experiments show that the solution of Sylvester equations can be obtained with high accuracy and the new algorithm is a robust technique for solving Sylvester equations. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Sylvester matrix equations; Field of values; Block GMRES Sylvester method; Successive approximations method 1. Introduction Linear control theory provides an important application for Sylvester equations [3,12,14,22]. In this paper, we focus on the numerical solution of Sylvester equations S ðX Þ¼ C; ð1Þ where S ðX Þ¼ AX  XB and A 2 R nn , B 2 R pp , C 2 R np , with n  p. The necessary and sufficient condition for (1) to have a unique solution is that S ðAÞ\ S ðBÞ¼;; where S(A) and S(B) are the spectrum of A and B, respectively [7,8]. When the size n of A is small, direct meth- ods based on Schur factorization of A and the Hessenberg factorization of B have been proposed, respectively by Bartels–Stewart [2] and by Enright [5], Golub et al. [7,8]. When n is large, we can use the Krylov-subspaces Galerkin and minimal residual algorithms which have been presented by Hu and Reichel [13]. In [19], 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.08.007 * Corresponding author. E-mail address: kaabi@pgu.ac.ir (A. Kaabi). Applied Mathematics and Computation 186 (2007) 638–645 www.elsevier.com/locate/amc