# Orthogonal polynomials and the Lanczos method

• Volume: 29, Issue: 1, page 19-33
• ISSN: 0137-6934

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## Abstract

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Lanczos method for solving a system of linear equations is well known. It is derived from a generalization of the method of moments and one of its main interests is that it provides the exact answer in at most n steps where n is the dimension of the system. Lanczos method can be implemented via several recursive algorithms known as Orthodir, Orthomin, Orthores, Biconjugate gradient,... In this paper, we show that all these procedures can be explained within the framework of formal orthogonal polynomials. This theory also provides a natural basis for curing breakdown and near-breakdown in these algorithms. The case of the conjugate gradient squared method can be treated similarly.

## How to cite

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Brezinski, C., Sadok, H., and Redivo Zaglia, M.. "Orthogonal polynomials and the Lanczos method." Banach Center Publications 29.1 (1994): 19-33. <http://eudml.org/doc/262795>.

@article{Brezinski1994,
abstract = {Lanczos method for solving a system of linear equations is well known. It is derived from a generalization of the method of moments and one of its main interests is that it provides the exact answer in at most n steps where n is the dimension of the system. Lanczos method can be implemented via several recursive algorithms known as Orthodir, Orthomin, Orthores, Biconjugate gradient,... In this paper, we show that all these procedures can be explained within the framework of formal orthogonal polynomials. This theory also provides a natural basis for curing breakdown and near-breakdown in these algorithms. The case of the conjugate gradient squared method can be treated similarly.},
author = {Brezinski, C., Sadok, H., Redivo Zaglia, M.},
journal = {Banach Center Publications},
keywords = {projection; biconjugate gradient; orthogonal polynomials; Lanczos method; Laczos method; biconjugate gradient method},
language = {eng},
number = {1},
pages = {19-33},
title = {Orthogonal polynomials and the Lanczos method},
url = {http://eudml.org/doc/262795},
volume = {29},
year = {1994},
}

TY - JOUR
AU - Brezinski, C.
AU - Redivo Zaglia, M.
TI - Orthogonal polynomials and the Lanczos method
JO - Banach Center Publications
PY - 1994
VL - 29
IS - 1
SP - 19
EP - 33
AB - Lanczos method for solving a system of linear equations is well known. It is derived from a generalization of the method of moments and one of its main interests is that it provides the exact answer in at most n steps where n is the dimension of the system. Lanczos method can be implemented via several recursive algorithms known as Orthodir, Orthomin, Orthores, Biconjugate gradient,... In this paper, we show that all these procedures can be explained within the framework of formal orthogonal polynomials. This theory also provides a natural basis for curing breakdown and near-breakdown in these algorithms. The case of the conjugate gradient squared method can be treated similarly.
LA - eng
KW - projection; biconjugate gradient; orthogonal polynomials; Lanczos method; Laczos method; biconjugate gradient method
UR - http://eudml.org/doc/262795
ER -

## References

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1. [1] C. Brezinski, Padé-type Approximation and General Orthogonal Polynomials, Internat. Ser. Number. Math. 50, Birkhäuser, Basel 1980. Zbl0418.41012
2. [2] C. Brezinski, The methods of Vorobyev and Lanczos, to appear. Zbl0923.65023
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4. [4] C. Brezinski and M. Redivo Zaglia, A new presentation of orthogonal polynomials with application to their computation, Numer. Algorithms 1 (1991), 207-221.
5. [5] C. Brezinski and M. Redivo Zaglia, Treatment of near-breakdown in the CGS algorithm, to appear.
6. [6] C. Brezinski, M. Redivo Zaglia and H. Sadok, A breakdown-free Lanczos type algorithm for solving linear systems, Numer. Math. 63 (1992), 29-38.
7. [7] C. Brezinski, M. Redivo Zaglia and H. Sadok, Avoiding breakdown and near-breakdown in Lanczos type algorithms, Numer. Algorithms 1 (1991), 261-284.
8. [8] C. Brezinski and H. Sadok, Lanczos type methods for solving systems of linear equations, Appl. Numer. Math., to appear.
9. [9] C. Brezinski and H. Sadok, Avoiding breakdown in the CGS algorithm, Numer. Algorithms 1 (1991), 199-206. Zbl0766.65024
10. [10] A. Draux, Polynômes Orthogonaux Formels - Applications, Lecture Notes in Math. 974, Springer, Berlin 1983. Zbl0504.42001
11. [11] R. Fletcher, Conjugate gradient methods for indefinite systems, in: Numerical Analysis, G. A. Watson (ed.), Lecture Notes in Math. 506, Springer, Berlin 1976, 73-89.
12. [12] M. H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms. Part I, SIAM J. Matrix Anal. Appl. 13 (1992), 594-639. Zbl0760.65039
13. [13] M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards 49 (1952), 409-439. Zbl0048.09901
14. [14] C. Lanczos, Solution of systems of linear equations by minimized iterations, ibid., 33-53.
15. [15] P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 10 (1989), 36-52. Zbl0666.65029
16. [16] Yu. V. Vorobyev, Method of Moments in Applied Mathematics, Gordon and Breach, New York 1965.
17. [17] D. M. Young and K. C. Jea, Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods, Linear Algebra Appl. 34 (1980), 159-194. Zbl0463.65025

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