Krylov subspace methods for eigenvalues with special properties and their analysis for normal matrices

Abstract

In this paper we propose a general approach by which eigenvalues with a special property of a given matrix A can be obtained. In this approach we first determine a scalar function ψ: C → C whose modulus is maximized by the eigenvalues that have the special property. Next, we compute the generalized power iterations uinj + 1 = ψ( A) u j , j = 0, 1,…, where u 0 is an arbitrary initial vector. Finally, we apply known Krylov subspace methods, such as the Arnoldi and Lanczos methods, to the vector u n for some sufficiently large n. We can also apply the simultaneous iteration method to the subspace span{ x ( n) 1,…, x ( n) k } with some sufficiently large n, where x ( j+1) m = ψ( A) x ( j) m , j = 0, 1,…, m = 1,…, k. In all cases the resulting Ritz pairs are approximations to the eigenpairs of A with the special property. We provide a rather thorough convergence analysis of the approach involving all three methods as n → ∞ for the case in which A is a normal matrix. We also discuss the connections and similarities of our approach with the existing methods and approaches in the literature.