SHIFT-INVERT ARNOLDIS METHOD WITH PRECONDITIONEDITERATIVE SOLVES

摘要

We consider the computation of a few eigenvectors and corresponding eigenvalues ofa large sparse nonsymmetric matrix using shift-invert Arnoldi's method with and without implicitrestarts. For the inner iterations we use preconditioned GMRES as the inexact iterative solver. Thecosts of the solves are measured by the number of inner iterations needed by the iterative solver ateach outer step of the algorithm. We first extend the relaxation strategy developed by Simoncini[SIAM M J. Numer. Anal., 43 (2005), pp. 1155-1174] to implicitly restarted Arnoldi's method, whichyields an improvement in the overall costs of the method. Secondly, we apply a new preconditioningstrategy to the inner solver. We show that small rank changes to the preconditioner can producesignificant savings in the total number of iterations. The combination of the new preconditioner withthe relaxation strategy in implicitly restarted Arnoldi produces enhancement in the overall costs ofaround 50 percent in the examples considered here. Numerical experiments illustrate the theorythroughout the paper.