Preconditioned Galerkin and minimal residual methods for solving Sylvester equations

This paper presents preconditioned Galerkin and minimal residual algorithms for the solution of Sylvester equations AX − XB = C. Given two good preconditioner matrices M and N for matrices A and B, respectively, we solve the Sylvester equations MAXN − MXBN = MCN. The algorithms use the Arnoldi process to generate orthonormal bases of certain Krylov subspaces and simultaneously reduce the order of Sylvester equations. Numerical experiments show that the solution of Sylvester equations can be obtained with high accuracy by using the preconditioned versions of Galerkin and minimal residual algorithms and this versions are more robust and more efficient than those without preconditioning.