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# Numerical stability of Lanczos methods

Cahill, Eamonn; Irving, Alan; Johnston, Christopher and Sexton, James (2000) Numerical stability of Lanczos methods. Nuclear Physics B - Proceedings Supplements, 83-84 . pp. 825-827. ISSN 0920-5632

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## Abstract

The Lanczos algorithm for matrix tridiagonalisation suffers from strong numerical instability in finite precision arithmetic when applied to evaluate matrix eigenvalues. The mechanism by which this instability arises is well documented in the literature. A recent application of the Lanczos algorithm proposed by Bai, Fahey and Golub allows quadrature evaluation of inner products of the form $\psi^\dagger g(A) \psi$. We show that this quadrature evaluation is numerically stable and explain how the numerical errors which are such a fundamental element of the finite precision Lanczos tridiagonalisation procedure are automatically and exactly compensated in the Bai, Fahey and Golub algorithm. In the process, we shed new light on the mechanism by which roundoff error corrupts the Lanczos procedure.

Item Type: Article LTH 457. arXiv Number: arXiv:hep-lat/9909131v1. Available online 23 July 2003. Issue: April 2000. Proceedings of the XVIIth International Symposium on Lattice Field Theory. Lanczos algorithm; matrix tridiagonalisation; Bai, Fahey and Golub; Lanczos procedure Q Science > QC Physics Academic Faculties, Institutes and Research Centres > Faculty of Science > Department of Mathematical Sciences 10.1016/S0920-5632(00)91816-4 http://arxiv.org/abs/hep-lat/9909131 No Published 614 28 Nov 2008 10:46 20 May 2011 09:40

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