Implicitly restarted global FOM and GMRES for nonsymmetric matrix equations and Sylvester equations

The global GMRES and global FOM algorithm are recently proposed by Jbilou et al. for solving the linear equations with multiple right-hand sides [K. Jbilou, A. Messaoudi, H. Sados, Global FOM and GMRES algorithms for matrix equations, Appl. Numer. Math. 31 (1999) 49-63]. Like GMRES for the linear equations, they generally uses restarting, which slows the convergence. However, some information can be retained at the time of the restart and used in the next cycle. We present algorithms that use implicit restarting in order to retain this information as Morgan have proposed recently [R.B. Morgan, Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations, SIAM J. Matrix anal. Appl. 21 (2000) 1112-1135]. At the same time, we prove that global GMRES and global FOM methods for matrix equations are equivalent with the corresponding methods for linear equations and propose implicitly restarted global FOM and GMRES. Numerical examples show that our methods are efficient.