Pseudospectra Computation of Large Matrices

Transfer functions have been shown to provide monotonic approximations to the resolvent 2-norm of A, R(z) = (A − zI), when associated with a sequence of nested spaces. This paper addresses the open question of the effectiveness of the transfer function scheme for the computation of the pseudospectrum of large matrices. It is shown that the scheme can be combined with certain Krylov type linear solvers, such as restarted fom, for the efficient solution of shifted linear systems of the form (A − zkI) b, for a large number of shifts zk. Extensive numerical experiments illustrate the performance of the methods developed in this paper. Tools for the effective combination of the transfer function framework with path following methods are developed. A hybrid method is proposed that combines transfer functions with iterative solvers and path following and is shown to be a powerful and cost effective scheme for computing pseudospectra of very large matrices.