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Sharpness in rates of convergence for the symmetric Lanczos method

Author: Ren-Cang Li
Journal: Math. Comp. 79 (2010), 419-435
MSC (2000): Primary 65F10
Published electronically: May 14, 2009
MathSciNet review: 2552233
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Abstract: The Lanczos method is often used to solve a large and sparse symmetric matrix eigenvalue problem. There is a well-established convergence theory that produces bounds to predict the rates of convergence good for a few extreme eigenpairs. These bounds suggest at least linear convergence in terms of the number of Lanczos steps, assuming there are gaps between individual eigenvalues. In practice, often superlinear convergence is observed. The question is ``do the existing bounds tell the correct convergence rate in general?''. An affirmative answer is given here for the two extreme eigenvalues by examples whose Lanczos approximations have errors comparable to the error bounds for all Lanczos steps.

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Additional Information

Ren-Cang Li
Affiliation: Department of Mathematics, University of Texas at Arlington, P.O. Box 19408, Arlington, Texas 76019-0408.

Keywords: Lanczos method, Krylov subspace, rate of convergence, Chebyshev polynomial
Received by editor(s): July 31, 2006
Received by editor(s) in revised form: January 14, 2008, and January 6, 2009
Published electronically: May 14, 2009
Additional Notes: This work was supported in part by the National Science Foundation under Grant No. DMS-0702335 and DMS-0810506.
Article copyright: © Copyright 2009 American Mathematical Society