A refined shift-and-invert arnoldi algorithm for large unsymmetric generalized eigenproblems

The shift-and-invert Arnoldi method has been popularly used for computing a number of eigenvalues close to a given shift and/or the associated eigenvectors of a large unsymmetric matrix pair, but there is no guarantee for the approximate eigenvectors, Ritz vectors, obtained by this method to converge even though the subspace is good enough. In order to correct this problem, a refined shift-and-invert Arnoldi method is proposed that uses certain refined Ritz vectors to approximate the desired eigenvectors. The refined Ritz vectors can be computed cheaply and reliably by small-sized singular value decompositions. It is shown that the refined method converges. A refined shift-and-invert Arnoldi algorithm is developed, and several numerical examples are reported. Comparisons are drawn on the refined algorithm and the shift-and-invert Arnoldi algorithm, indicating that the former is considerably more efficient than the latter.