### Theory of decomposition and bulge-chasing algorithms for the generalized eigenvalue problem

Watkins DS, Elsner L (1994)
SIAM Journal on matrix analysis and applications 15(3): 943-967.

Zeitschriftenaufsatz | Veröffentlicht | Englisch

Autor*in
Watkins, David S.; Elsner, LudwigUniBi
Einrichtung
Abstract / Bemerkung
A generic GZ algorithm for the generalized eigenvalue problem Ax = lambdaBx is presented. This is actually a large class of algorithms that includes multiple-step QZ and LZ algorithms, as well as QZ-LZ hybrids, as special cases. First the convergence properties of the GZ algorithm are discussed, then a study of implementations is undertaken. The notion of an elimination rule is introduced as a device for studying the QZ, LZ and other algorithms simultaneously. To each elimination rule there corresponds an explicit GZ algorithm. Through a careful study of the steps involved in executing the explicit algorithm, it is discovered how to implement the algorithm implicitly by bulge chasing. The approach taken here was introduced by Miminis and Paige in the context of the QR algorithm for the ordinary eigenvalue problem. It is more involved than the standard approach, but it yields a much clearer picture of the relationship between the implicit and explicit versions of the algorithm. Furthermore, it is more general than the standard approach, as it does not require the use of a theorem of ''Implicit-Q'' type. Finally a generalization of the implicit GZ algorithm, the generic bulge-chasing algorithm, is introduced. It is proved that the generic bulge-chasing algorithm implicitly performs iterations of the generic GZ algorithm. Thus the convergence theorems that are proved for the generic GZ algorithm hold for the generic bulge-chasing algorithm as well.
Stichworte
Generalized eigenvalue problem; GZ algorithm; QZ algorithm; Chasing the bulge
Erscheinungsjahr
1994
Zeitschriftentitel
SIAM Journal on matrix analysis and applications
Band
15
Ausgabe
3
Seite(n)
943-967
ISSN
0895-4798
eISSN
1095-7162
Page URI
https://pub.uni-bielefeld.de/record/1776231

### Zitieren

Watkins DS, Elsner L. Theory of decomposition and bulge-chasing algorithms for the generalized eigenvalue problem. SIAM Journal on matrix analysis and applications. 1994;15(3):943-967.
Watkins, D. S., & Elsner, L. (1994). Theory of decomposition and bulge-chasing algorithms for the generalized eigenvalue problem. SIAM Journal on matrix analysis and applications, 15(3), 943-967. doi:10.1137/S089547989122377X
Watkins, D. S., and Elsner, L. (1994). Theory of decomposition and bulge-chasing algorithms for the generalized eigenvalue problem. SIAM Journal on matrix analysis and applications 15, 943-967.
Watkins, D.S., & Elsner, L., 1994. Theory of decomposition and bulge-chasing algorithms for the generalized eigenvalue problem. SIAM Journal on matrix analysis and applications, 15(3), p 943-967.
D.S. Watkins and L. Elsner, “Theory of decomposition and bulge-chasing algorithms for the generalized eigenvalue problem”, SIAM Journal on matrix analysis and applications, vol. 15, 1994, pp. 943-967.
Watkins, D.S., Elsner, L.: Theory of decomposition and bulge-chasing algorithms for the generalized eigenvalue problem. SIAM Journal on matrix analysis and applications. 15, 943-967 (1994).
Watkins, David S., and Elsner, Ludwig. “Theory of decomposition and bulge-chasing algorithms for the generalized eigenvalue problem”. SIAM Journal on matrix analysis and applications 15.3 (1994): 943-967.
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Access Level
Open Access