Restarted GMRES for Shifted Linear Systems

Shifted matrices, which differ by a multiple of the identity only, generate the same Krylov subspaces with respect to any fixed vector. This fact has been exploited in Lanczos-based methods like CG, QMR, and BiCG to simultaneously solve several shifted linear systems at the expense of only one matrix--vector multiplication per iteration. Here, we develop a variant of the restarted GMRES method exhibiting the same advantage and we investigate its convergence for positive real matrices in some detail. We apply our method to speed up "multiple masses" calculations arising in lattice gauge computations in quantum chromodynamics, one of the most time-consuming supercomputer applications.