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

AbstractIn 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)uj, j = 0, 1,…, where u0 is an arbitrary initial vector. Finally, we apply known Krylov subspace methods, such as the Arnoldi and Lanczos methods, to the vector un 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