# A unified approach to some strategies for the treatment of breakdown in Lanczos-type algorithms

Applicationes Mathematicae (1999)

- Volume: 26, Issue: 4, page 477-488
- ISSN: 1233-7234

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topEl Guennouni, A.. "A unified approach to some strategies for the treatment of breakdown in Lanczos-type algorithms." Applicationes Mathematicae 26.4 (1999): 477-488. <http://eudml.org/doc/219253>.

@article{ElGuennouni1999,

abstract = {The Lanczos method for solving systems of linear equations is implemented by using some recurrence relationships between polynomials of a family of formal orthogonal polynomials or between those of two adjacent families of formal orthogonal polynomials. A division by zero can occur in these relations, thus producing a breakdown in the algorithm which has to be stopped. In this paper, three strategies to avoid this drawback are discussed: the MRZ and its variants, the normalized and unnormalized BIORES algorithm and the composite step biconjugate algorithm. We prove that all these algorithms can be derived from a unified framework; in fact, we give a formalism for finding all the recurrence relationships used in these algorithms, which shows that the three strategies use the same techniques.},

author = {El Guennouni, A.},

journal = {Applicationes Mathematicae},

keywords = {deficient polynomials; orthogonal polynomials; Lanczos method; biconjugate algorithm},

language = {eng},

number = {4},

pages = {477-488},

title = {A unified approach to some strategies for the treatment of breakdown in Lanczos-type algorithms},

url = {http://eudml.org/doc/219253},

volume = {26},

year = {1999},

}

TY - JOUR

AU - El Guennouni, A.

TI - A unified approach to some strategies for the treatment of breakdown in Lanczos-type algorithms

JO - Applicationes Mathematicae

PY - 1999

VL - 26

IS - 4

SP - 477

EP - 488

AB - The Lanczos method for solving systems of linear equations is implemented by using some recurrence relationships between polynomials of a family of formal orthogonal polynomials or between those of two adjacent families of formal orthogonal polynomials. A division by zero can occur in these relations, thus producing a breakdown in the algorithm which has to be stopped. In this paper, three strategies to avoid this drawback are discussed: the MRZ and its variants, the normalized and unnormalized BIORES algorithm and the composite step biconjugate algorithm. We prove that all these algorithms can be derived from a unified framework; in fact, we give a formalism for finding all the recurrence relationships used in these algorithms, which shows that the three strategies use the same techniques.

LA - eng

KW - deficient polynomials; orthogonal polynomials; Lanczos method; biconjugate algorithm

UR - http://eudml.org/doc/219253

ER -

## References

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- [2] C. Brezinski, M. Redivo Zaglia and H. Sadok, Avoiding breakdown and near-breakdown in Lanczos type algorithms, Numer. Algorithms 1 (1991), 261-284.
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- [8] A. Draux, Formal orthogonal polynomials revisited. Applications, Numer. Algorithms 11 (1996), 143-158.
- [9] R. Fletcher, Conjugate gradient methods for indefinite systems, in: Numerical Analysis (Dundee, 1975), G. A. Watson (ed.), Lecture Notes in Math. 506, Springer, Berlin, 1976, 73-89.
- [10] M. H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms. I, SIAM J. Matrix Anal. 13 (1992), 594-639. Zbl0760.65039
- [11] C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Standards 45 (1950), 255-282.
- [12] C. Lanczos, Solution of systems of linear equations by minimized iterations, ibid. 49 (1952), 33-53.

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