Restarted simpler GMRES augmented with harmonic Ritz vectors and approximate errors

In this paper we propose an acceleration strategy for the Simpler GMRES (SGMRES) method that can considerably improve the convergence of SGMRES method and its variants. The idea is to use the augmentation of approximate errors in SGMRES-E so as to correct the alternative direction behavior of the residual vectors, which hampers the convergence. We show that the skip angles of the residual vectors are increased by the augmentation, which implies acceleration of the iterative method. Finally, we demonstrate the advantage of the strategy by several academic examples.