Complementary cycles of restarted GMRES

Restarted GMRES is one of the most popular methods for solving large nonsymmetric linear systems. The algorithm GMRES(m) restarts every m iterations. It is generally thought the information of previous GMRES cycles is lost at the time of a restart, so that each cycle contributes to the global convergence individually. However, this is not the full story. In this paper, we shed light on the relationship between different GMRES(m) cycles. It is shown that some important information of the previous cycles may be saved by the iteration approximates automatically, with which successive GMRES(m) cycles can complement one another harmoniously. These groups of cycles, called complementary cycles of GMRES(m), are defined and studied. This helps to present a new point of view on this algorithm.