Inexact Shift-and-invert Arnoldi’s Method and Implicit Restarts with Preconditioning for Eigencomputations

We consider the computation of a few eigenvectors and corresponding eigen-values of a large sparse nonsymmetric matrix. In order to compute eigenvaluesin an isolated cluster around a given shift we apply shift-and-invert Arnoldi’smethod with and without implicit restarts. For the inner iterations we useGMRES as the iterative solver. The costs of the inexact solves are measuredby the number of inner iterations needed by the iterative solver at each outerstep of the algorithm.We first extend the relaxation strategy developed by Simoncini [2] to im-plicitely restarted Arnoldi’s method which yields an improvement in the overallcosts of the method.Secondly we apply a new preconditioning strategy to the inner solver. Weshow that small rank changes of the preconditioner can produce significantsavings in the total number of iterations. This property has been observed in[1]. Numerical experiments illustrate the theory. References[1] M.A. Freitag and A. Spence, Convergence rates for inexact inverse iteration withapplication to preconditioned iterative solves, to appear in BIT.[2] V. Simoncini, Variable accuracy of matrix-vector products in projection methods foreigencomputation, SIAM J. Numerical Analysis 43, 3 (2005), pp. 1155–1174.