A Lanczos-type method for multiple starting vectors
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- by J. I. Aliaga, D. L. Boley, R. W. Freund and V. Hernández PDF
- Math. Comp. 69 (2000), 1577-1601
Abstract:
Given a square matrix and single right and left starting vectors, the classical nonsymmetric Lanczos process generates two sequences of biorthogonal basis vectors for the right and left Krylov subspaces induced by the given matrix and vectors. In this paper, we propose a Lanczos-type algorithm that extends the classical Lanczos process for single starting vectors to multiple starting vectors. Given a square matrix and two blocks of right and left starting vectors, the algorithm generates two sequences of biorthogonal basis vectors for the right and left block Krylov subspaces induced by the given data. The algorithm can handle the most general case of right and left starting blocks of arbitrary sizes, while all previously proposed extensions of the Lanczos process are restricted to right and left starting blocks of identical sizes. Other features of our algorithm include a built-in deflation procedure to detect and delete linearly dependent vectors in the block Krylov sequences, and the option to employ look-ahead to remedy the potential breakdowns that may occur in nonsymmetric Lanczos-type methods.References
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Additional Information
- J. I. Aliaga
- Affiliation: Departamento de Informática, Universidad Jaume I, Campus de Riu Sec, 12071 Castellón, Spain
- Email: aliaga@inf.uji.es
- D. L. Boley
- Affiliation: Department of Computer Science and Engineering, University of Minnesota, 4-192 EE/CSci Building, 200 Union Street S.E., Minneapolis, Minnesota 55455-0159
- Email: boley@cs.umn.edu
- R. W. Freund
- Affiliation: Bell Laboratories, Lucent Technologies, Room 2C-420, 700 Mountain Avenue, Murray Hill, New Jersey 07974-0636
- Email: freund@research.bell-labs.com
- V. Hernández
- Affiliation: Departamento de Sistemas Informáticos y Computación, Universidad Politécnica de Valencia, Apartado de Correos 22012, 46071 Valencia, Spain
- Email: vhernand@dsic.upv.es
- Received by editor(s): October 10, 1996
- Received by editor(s) in revised form: September 22, 1998
- Published electronically: May 20, 1999
- Additional Notes: The first and the last author were supported in part by the European ESPRIT III Basic Research Project GEPPCOM #9072. The second author was supported in part by the U.S. NSF Grants #CCR-9405380 and #CCR-9628786.
- © Copyright 2000 Lucent Technologies
- Journal: Math. Comp. 69 (2000), 1577-1601
- MSC (1991): Primary 65F10, 65F15; Secondary 65F25, 65F30
- DOI: https://doi.org/10.1090/S0025-5718-99-01163-1
- MathSciNet review: 1665942