A convergence analysis of GMRES and FOM methods for Sylvester equations

We discuss convergence properties of the GMRES and FOM methods for solving large Sylvester equations of the form AX-XB=C. In particular we show the importance of the separation between the fields of values of A and B on the convergence behavior of GMRES. We also discuss the stagnation phenomenon in GMRES and its consequence on FOM. We generalize the issue of breakdown in the block-Arnoldi algorithm and explain its consequence on FOM and GMRES methods. Several numerical tests illustrate the theoretical results