Upper bounds for convergence rates of acceleration methods with initial iterations

GMRES(n,k), a version of GMRES for the solution of large sparse linear systems, is introduced. A cycle of GMRES(n,k) consists of n Richardson iterations followed by k iterations of GMRES. Such cycles can be repeated until convergence is achieved. The advantage in this approach is in the opportunity to use moderate k, which results in time and memory saving. Because the number of inner products among the vectors of iteration is about k2/2, using a moderate k is particularly attractive on message-passing parallel architectures, where inner products require expensive global communication. The present analysis provides tight upper bounds for the convergence rates of GMRES(n,k) for problems with diagonalizable coefficient matrices whose spectra lie in an ellipse in 0. The advantage of GMRES(n,k) over GMRES(k) is illustrated numerically.