Abstract

The recent development of Krylov subspace methods for the solution of operator equations has shown that two basic construction principles, the orthogonal residual (OR) and minimal residual (MR) approaches, underlie the most commonly used algorithms. It is shown that these can both be formulated as techniques for solving an approximation problem on a sequence of nested subspaces of a Hilbert space, a problem not necessarily related to an operator equation. Most of the familiar Krylov subspace algorithms result when these subspaces form a Krylov sequence. The well-known relations among the iterates and residuals of OR/MR pairs are shown to hold also in this rather general setting. We further 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 manner. An application of this analysis to compact perturbations of the identity shows that OR/MR pairs of Krylov subspace methods converge q-superlinearly when applied to such operator equations.

Citations