Geometric aspects of the theory of Krylov subspace methods

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Abstract
The development of Krylov subspace methods for the solution of operator equations has shown that two basic construction principles underlie the most commonly used algorithms: the orthogonal residual (OR) and minimal residual (MR) approaches. 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, an abstract problem not necessarily related to an operator equation. Essentially all Krylov subspace algorithms result when these subspaces form a Krylov sequence. The well-known relations among the iterates and residuals of MR/OR 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 known residual and error bounds to be derived in a simple and consistent manner. An application of this analysis to compact perturbations of the identity shows that MR/OR pairs of Krylov subspace methods converge q-superlinearly when applied to such operator equations.

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