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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The unitary completion and QR iterations for a class of structured matrices


Authors: D. A. Bini, Y. Eidelman, L. Gemignani and I. Gohberg
Journal: Math. Comp. 77 (2008), 353-378
MSC (2000): Primary 15A18, 15A57, 65F15
Published electronically: June 22, 2007
MathSciNet review: 2353957
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Abstract: We consider the problem of completion of a matrix with a specified lower triangular part to a unitary matrix. In this paper we obtain the necessary and sufficient conditions of existence of a unitary completion without any additional constraints and give a general formula for this completion. The paper is mainly focused on matrices with the specified lower triangular part of a special form. For such a specified part the unitary completion is a structured matrix, and we derive in this paper the formulas for its structure. Next we apply the unitary completion method to the solution of the eigenvalue problem for a class of structured matrices via structured QR iterations.


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Additional Information

D. A. Bini
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
Email: bini@dm.unipi.it

Y. Eidelman
Affiliation: School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Ramat-Aviv 69978, Israel
Email: eideyu@post.tau.ac.il

L. Gemignani
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
Email: gemignan@dm.unipi.it

I. Gohberg
Affiliation: School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Ramat-Aviv 69978, Israel
Email: gohberg@post.tau.ac.il

DOI: http://dx.doi.org/10.1090/S0025-5718-07-02004-2
PII: S 0025-5718(07)02004-2
Keywords: Unitary completion, Hessenberg matrices, rank-one perturbations, unitary matrices, companion matrices, quasiseparable matrices, $QR$ iteration, eigenvalue computation, complexity
Received by editor(s): September 4, 2005
Received by editor(s) in revised form: September 20, 2006
Published electronically: June 22, 2007
Additional Notes: This work was partially supported by MIUR grant number 2004015437
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.