Restarted Simpler GMRES augmented with harmonic Ritz vectors

Abstract

We describe an augmented version of Simpler GMRES (SGMRES) for problems with small eigenvalues. The advantage is that the SGMRES variant requires a lesser amount of work than augmented GMRES. We show that the problem of finding harmonic Ritz vectors for augmenting the Krylov subspace can be easily reduced to an eigenvalue problem by computing recurrences for a transformation matrix. Numerical examples are described to illustrate the better performance of the augmented version of SGMRES.