A Hybrid GMRES Algorithm for Nonsymmetric Linear Systems

Abstract
A new hybrid iterative algorithm is proposed for solving large nonsymmetric systems of linear equations. Unlike other hybrid algorithms, which first estimate eigenvalues and then apply this knowledge in further iterations, this algorithm avoids eigenvalue estimates. Instead, it runs GMRES until the residual norm drops by a certain factor, then re-applies the polynomial implicitly constructed by GMRES via a Richardson iteration with Leja ordering. Preliminary experiments suggest that the new algorithm frequently outperforms the restarted GMRES algorithm.