On Some Recurrent Theorems Concerning Krylov Subspace Methods

Oliver Ernst

UMIACS University of Maryland College Park, MD 20742


Abstract

The recent development of Krylov subspace methods for solving linear systems of equations has shown that two fundamental approaches underlie the most commonly used algorithms: the Minimal Residual (MR) and Orthogonal Residual (OR) approaches. We show that these two approaches can be formulated as techniques for solving an approximation problem on a nested sequence of subspaces of a Hilbert space. Many well-known relations among the iterates and residuals of OR/MR pairs are shown to hold even in this abstract setting. When applied to the solution of linear systems of equations and when these subspaces are specialized to be a sequence of Krylov spaces, familiar OR/MR pairs result, among these CR/CG, GMRES/FOM and their specializations MINRES/CG to the Hermitian case, QMR/BCG as well as TFQMR/CGS. Further, we show that a common error analysis for these methods involving the canonical angles between subspaces allows many of the recently developed error bounds to be derived in a simple and uniform manner. An application of this analysis to operator equations involving compact perturbations of the identity shows that OR/MR pairs of Krylov subspace methods converge q-superlinearly for this class of problems.