Restarted Simpler GMRES Augmented with Harmonic Ritz Vectors

Abstract

We describe a method for improving the convergence of the Simpler GMRES method for problems with small eigenvalues. We augment the Krylov subspace with harmonic Ritz vectors corresponding to the smallest harmonic Ritz values. The advantage over augmented GMRES is that the problem of finding the minimal residual solution reduces to an upper triangular least-squares problem instead of an upper-Hessenberg least-squares problem. A second advantage is that harmonic Ritz pairs can be cheaply computed. Numerical tests indicate that augmented Simpler GMRES(m) is superior to Simpler GMRES(m) and requires a lesser amount of work than augmented GMRES(m).