Matrices that Generate the same Krylov Residual Spaces

Abstract

Abstract. Given an n by n nonsingular matrix A and an n-vector v, we consider the spaces of the form AKk (A, v), k = 1, ... , n, where Kk(A, v) is the kth Krylov space, equal to span{ v, Av, ... , A k -1 v}. We characterize the set of matrices B that, with the given vector v, generate the same spaces; i.e., those matrices B for which BKk (B,v) = AKk(A,v), for all k = 1, ... ,n. It is shown that any such sequence of spaces can be generated by a unitary matrix. If zero is outside the field of values of A, then there is a Hermitian positive definite matrix that generates the same spaces, and, moreover, if A is close to Hermitian then there is a nearby Hermitian matrix that generates the same spaces. It is also shown that any such sequence of spaces can be generated by a matrix having any desired eigenvalues.Implications about the convergence rate of the GMRES method are discussed. A new proof is given that if zero is outside the field of values of A, then convergence of the GMRES algorithm is strictly monotonic. It is shown that if A is close to Hermitian, then the eigenvalues of A essentially determine the behavior of the GMRES iteration but that, in general, eigenvalue information alone is never sufficient to ensure rapid convergence of the GMRES algorithm.

Recent computational developments in Krylov subspace methods for linear systems

“…On the other hand, in the non-normal case, eigenvalues of A and thus their approximations, may not play a role in the convergence. In fact, it was shown by Greenbaum, Ptàk, and Strakoš [162], [164], that spectral information alone may provide misleading information in the non-normal case. These conflicting views indicate that the convergence analysis of Krylov subspace methods for general problems is a challenging area of research.…”

How to Make Simpler GMRES and GCR More Stable

“…[5,6] show that GCR (and hence any minimum residual method) does not stagnate if the symmetric part of A is positive definite, i.e., if the origin is not contained in the field of values of A. See also Greenbaum and Strakoš [10] for a different proof and Eiermann and Ernst [4]. Several other conditions can be found in Simoncini and Szyld [24] and the references therein.…”

A Preconditioned MINRES Method for Nonsymmetric Toeplitz Matrices

“…LSQR requires more matrix-vector products than the proposed MINRES method for all three problems and so it appears that the latter may be preferable in practice. It is clear that GMRES converges faster than MINRES for these examples; however, there are no theoretical guarantees of fast convergence and it is well known that clustered eigenvalues do not guarantee fast convergence [1,23]. Moreover, GMRES requires long recurrences, in contrast to the proposed MINRES method, so that each iteration becomes more expensive.…”