Efficient and stable Arnoldi restarts for matrix functions based on quadrature

Frommer, Andreas and Güttel, Stefan and Schweitzer, Marcel (2013) Efficient and stable Arnoldi restarts for matrix functions based on quadrature. [MIMS Preprint]

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Abstract

When using the Arnoldi method for approximating f(A)b, the action of a matrix function on a vector, the maximum number of iterations that can be performed is often limited by the storage requirements of the full Arnoldi basis. As a remedy, different restarting algorithms have been proposed in the literature, none of which was universally applicable, efficient, and stable at the same time. We utilize an integral representation for the error of the iterates in the Arnoldi method which then allows us to develop an efficient quadrature-based restarting algorithm suitable for a large class of functions, including the so-called Stieltjes functions and the exponential function. Our method is applicable for functions of Hermitian and non-Hermitian matrices, requires no a-priori spectral information, and runs with essentially constant computational work per restart cycle. We comment on the relation of this new restarting approach to other existing algorithms and illustrate its efficiency and numerical stability by various numerical experiments.

Item Type: MIMS Preprint
Uncontrolled Keywords: matrix function, Krylov subspace approximation, restarted Arnoldi/Lanczos method, deflated restarting, polynomial interpolation, Gaussian quadrature, Padé approximation
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Stefan Güttel
Date Deposited: 20 Jan 2014
Last Modified: 08 Nov 2017 18:18
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2090

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