Abstract. We show how the Arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted Krylov subspace methods for solving linear systems of equations. The resulting restarted algorithm reduces to other known algorithms for the reciprocal and the exponential functions. We further show that the restarted algorithm inherits the superlinear convergence property of its unrestarted counterpart for entire functions and present the results of numerical experiments.
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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