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Zbl 0760.65039
Gutknecht, Martin H.
A completed theory of the unsymmetric Lanczos process and related algorithms. I. (English)
[J] SIAM J. Matrix Anal. Appl. 13, No.2, 594-639 (1992). ISSN 0895-4798; ISSN 1095-7162/e

This substantial paper provides a general theory of the unsymmetric Lanczos algorithm. The whole theory is here developed from the point of view of orthogonal polynomials and Padé approximants. A rather comprehensive introduction to this subject is therefore provided. The recurrence relations for formal orthogonal polynomials are then interpreted in a matrix setting, leading naturally to the Lanczos process.\par Using the connection to Padé approximants, the block structure theorem for those is used to overcome the difficulties associated with nongeneric break-down. Connections are also established to other algorithms from the same group as Lanczos's, such as BIORES or BIORTHORES, and bi-conjugate gradients.
[H.Matthies (Hamburg)]

MSC 2000:: *65F15 Eigenvalues (numerical linear algebra)
65F10 Iterative methods for linear systems
41A21 Pade approximation

Keywords: bi-orthogonalization; BO algorithm; continued fraction; quotient difference algorithm; qd algorithm; unsymmetric Lanczos algorithm; orthogonal polynomials; Padé approximants; recurrence relations; BIORES; BIORTHORES; bi-conjugate gradients

Cited in: Zbl 0809.65028 Zbl 0773.65018

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