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IMA Journal of Numerical Analysis Advance Access published online on June 17, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drp003
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A restarted Lanczos approximation to functions of a symmetric matrix

M. Ilic, I. W. Turner and D. P. Simpson{dagger}

School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane 4001, Australia

{dagger} Corresponding author. Email: dp.simpson{at}qut.edu.au

Received on 30 May 2007. Accepted for publication 5 January 2009.


  Abstract

In this paper the error term in the ‘first’ method of Eiermann & Ernst(2006, SIAM J. Numer. Anal., 44, 2481–2504) for restarting the Lanczos approximation of the matrix–vector product f(A)b, where A isin Rn x n is symmetric, is re-derived and expressed as an explicit partial fraction expansion of the divided differences. The partial fraction representation makes the new variant slightly more stable (albeit still unstable) than the former method because it requires fewer finite-difference evaluations. We then 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 and 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. Finally, in an attempt to overcome the inherent instability of our restart procedure, we propose a simple heuristic that identifies when to halt the iterations.

Key Words: Krylov subspace methods; matrix functions; error bounds; Stieltjes transforms; Gaussian Markov random fields; fractional Poisson equation


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