GMRES with Deflated Restarting

A modification is given of the GMRES iterative method for nonsymmetric systems of linear equations. The new method deflates eigenvalues using Wu and Simon's thick restarting approach [SIAM J. Matrix Anal. Appl., 22 (2000), pp. 602--616]. It has the efficiency of implicit restarting but is simpler and does not have the same numerical concerns. The deflation of small eigenvalues can greatly improve the convergence of restarted GMRES. Also, it is demonstrated that using harmonic Ritz vectors is important because then the whole subspace is a Krylov subspace that contains certain important smaller subspaces.