Convergence of restarted Krylov subspace methods for Stieltjes functions of matrices

Andreas, Frommer and Stefan, Güttel and Marcel, Schweitzer (2014) Convergence of restarted Krylov subspace methods for Stieltjes functions of matrices. [MIMS Preprint]

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Abstract

To approximate f(A)b---the action of a matrix function on a vector---by a Krylov subspace method, restarts may become mandatory due to storage requirements for the Arnoldi basis or due to the growing computational complexity of evaluating f on a Hessenberg matrix of growing size. A number of restarting methods have been proposed in the literature in recent years and there has been substantial algorithmic advancement concerning their stability and computational efficiency. However, the question under which circumstances convergence of these methods can be guaranteed has remained largely unanswered. In this paper we consider the class of Stieltjes functions and a related class, which contains important functions like the (inverse) square root and the matrix logarithm. For these classes of functions we present new theoretical results which prove convergence for Hermitian positive definite matrices A and arbitrary restart lengths. We also propose a modification of the Arnoldi approximation which guarantees convergence for the same classes of functions and any restart length if A is not necessarily Hermitian but positive real.

Item Type: MIMS Preprint
Additional Information: This item is also available from the maths preprint server of the Bergische Universit�¤t Wuppertal: http://www.imacm.uni-wuppertal.de/imacm/research/preprints/
Uncontrolled Keywords: matrix functions, Krylov subspace methods, restarted Arnoldi method, conjugate gradient method, shifted linear systems, shifted GMRES method, harmonic Ritz values
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Stefan Güttel
Date Deposited: 01 Oct 2014
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2185

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