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

100%
El Guennouni A.
Applicationes Mathematicae
|
1999
|
tom 26
|
nr 4
477-488
EN
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.
2
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Orthogonal polynomials and the Lanczos method

100%
Brezinski C., Sadok H., Redivo Zaglia M.
Banach Center Publications
|
1994
|
tom 29
|
nr 1
19-33
EN
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.
3
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Avoiding look-ahead in the Lanczos method and Padé approximation

100%
Ayachour E. H.
Applicationes Mathematicae
|
1999
|
tom 26
|
nr 1
33-62
EN
In the non-normal case, it is possible to use various look-ahead strategies for computing the elements of a family of regular orthogonal polynomials. These strategies consist in jumping over non-existing and singular orthogonal polynomials by solving triangular linear systems. We show how to avoid them by using a new method called ALA (Avoiding Look-Ahead), for which we give three principal implementations. The application of ALA to Padé approximation, extrapolation methods and Lanczos method for solving systems of linear equations is discussed.
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