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A Restarted Lanczos Approximation to Functions of a Symmetric Matrix

Turner, Ian W., Ilic, Milos, & Simpson, Daniel P. (2007) A Restarted Lanczos Approximation to Functions of a Symmetric Matrix. IMA Journal of Numerical Analysis, pp. 1044-1061.

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Abstract

In this paper, we investigate a method for restarting the Lanczos method for approximating the matrix-vector product $f(A)b$, where $A in mathbb{R}^{n imes n }$ is a symmetric matrix. For analytic $f$ we derive a novel restart function that identifies the error in the Lanczos approximation. The restart procedure is then generated by a restart formula using a sequence of these restart functions. We present an error bound for the proposed restart scheme. We also present an error bound for the restarted Lanczos approximation of $f(A)b$ for symmetric positive definite $A$ when $f$ is in a particular class of completely monotone functions. We illustrate for some important matrix function applications the usefulness of these bounds for terminating the restart process once the desired accuracy in the matrix function approximation has been achieved.

Impact and interest:

7 citations in Scopus
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357 since deposited on 07 Jun 2007
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ID Code: 8011
Item Type: Journal Article
Keywords: Krylov subspace methods, matrix functions, error bounds, Stieltjes transforms, Gaussian Markov random fields, fractional Poisson equation
DOI: http://dx.doi.org/10.1093/imanum/drp003
ISSN: 0272-4979
Subjects: Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > NUMERICAL AND COMPUTATIONAL MATHEMATICS (010300) > Numerical Analysis (010301)
Divisions: Past > QUT Faculties & Divisions > Faculty of Science and Technology
Copyright Owner: Copyright 2007 the authors.
Deposited On: 07 Jun 2007
Last Modified: 30 Jun 2014 11:36

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