A restarted Lanczos approximation to functions of a symmetric matrix

M. Ilić, I. W. Turner, D. P. Simpson

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22 Citations (Scopus)

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 ∈ ℝntimesn 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.

Original languageEnglish
Pages (from-to)1044-1061
Number of pages18
JournalIMA Journal of Numerical Analysis
Volume30
Issue number4
DOIs
Publication statusPublished - Oct 2010
Externally publishedYes

Keywords

  • error bounds
  • fractional Poisson equation
  • Gaussian Markov random fields
  • Krylov subspace methods
  • matrix functions
  • Stieltjes transforms

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