Numerical stability of Lanczos methods

Collaboration, The UKQCD, Cahill, Eamonn, Irving, Alan, Johnston, Christopher and Sexton, James (2000) Numerical stability of Lanczos methods Nuclear and Particle Physics Proceedings, 83. pp. 825-827. ISSN 2405-6014

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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 ψ† · g(A) · ψ. We show that this quadrature evaluation is numerically stable (accurate to machine precision) 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
Additional Information:	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.
Uncontrolled Keywords:	46 Information and Computing Sciences, 40 Engineering, 4601 Applied Computing
Subjects:	?? QC ??
Divisions:	Faculty of Science & Engineering > School of Physical Sciences > Mathematical Sciences
Depositing User:	Symplectic Admin
Date Deposited:	28 Nov 2008 10:46
Last Modified:	22 May 2026 20:28
DOI:	10.1016/s0920-5632(00)91816-4
Related Websites:	Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/614
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