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The unitary completion and QR iterations for a class of structured matrices
Author(s):
D.
A.
Bini;
Y.
Eidelman;
L.
Gemignani;
I.
Gohberg.
Journal:
Math. Comp.
77
(2008),
353-378.
MSC (2000):
Primary 15A18, 15A57, 65F15
Posted:
June 22, 2007
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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.
References:
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On the shifted  iteration applied to companion matrices.
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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:
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
Posted:
June 22, 2007
Additional Notes:
This work was partially supported by MIUR grant number 2004015437
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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