Abstract
LExpRes is a k-step explicit restart variant of the Lanczos algorithm. In this method a periodic/selective reorthogonalization strategy is adopted in order to dampen the affects of instability incurred by a loss of orthogonality among the Lanczos vectors. Despite this however, round-off error still effects its performance particularly on the estimated Lanczos error bounds that are considerably different from the true error bound. In this paper a detailed analysis of the Lanczos error bound is presented and a new more realistic error proposed for determining a ‘good’ Ritz vector. A scheme is also proposed that uses this bound to derive start vectors for subsequent eigenpairs. The performance of the implementation of this scheme on the CRAY-T3E is critically assessed and some conclusions are drawn.
This work was carried out using the facilities of the University of Manchester Parallel Computing Centre
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References
Duff, I.S., Grimes, R.G., and Lewis, J.G., User’s Guide for the Harwell-Boeing Sparse Matrix Collection (release I) 1992, available online ftp orion.cerfacs.fr.
Golub, G., and Van Loan C.F., Matrix Computations, John Hopkins University Press, 1989, London.
Parlett, B.N., The Symmetric Eigenvalue Problem, Society for Industrial and Applied Mathematics, 1998.
Simon, H.,D., Analysis of the Symmetric Lanczos Algorithm with Reorthogonaliza-tion Methods, Linear Algebra and its Appl., 1984 (61), 101–131.
Lehoucq, R., Sorensen, D.C., and Vu, P.A., SSAUPD: Fortran subroutines for solving large-scale eigenvalue problems, Release 2.1, 1994, available from netlib@ornl.gov in the scalapack directory.
Szularz, M., Weston, J., and Clint, M., Explicitly restarted Lanczos algorithms in an MPP environment, Parallel Computing, 1999 (25), 613–631.
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© 2002 Springer-Verlag Berlin Heidelberg
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Cooper, A., Szularz, M., Weston, J. (2002). Analysis of the Lanczos Error Bounds and Its Application to the Explicitly Restarted Lanczos Algorithm. In: Wyrzykowski, R., Dongarra, J., Paprzycki, M., Waśniewski, J. (eds) Parallel Processing and Applied Mathematics. PPAM 2001. Lecture Notes in Computer Science, vol 2328. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48086-2_46
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DOI: https://doi.org/10.1007/3-540-48086-2_46
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