A refined variant of the inverse-free Krylov subspace method for symmetric generalized eigenvalue problems

A refined variant of the inverse-free Krylov subspace method is proposed in this paper. The new method retains the original Ritz value, and replaces the Ritz vector by a refined Ritz vector in each cycle of the iteration. Each refined Ritz vector is chosen in such a way that the norm of the residual vector formed with the Ritz value is minimized over the subspace involved, and it can be computed cheaply by solving a small sized SVD problem. The refined variant can overcome the irregular convergence behavior of the Ritz vectors which may happen in the inverse-free Krylov subspace method. An a priori error estimate for the refined Ritz vector is given, which shows that the refined Ritz vector converges once the deviation of the eigenvector from the trial Krylov subspace converges to zero. By using spectral transformation, this new method can be applied to compute an interior eigenvalue pair. Numerical experiments are given to show the efficiency of the new methods.