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URN: urn:nbn:de:0030-drops-13987
URL: https://drops.dagstuhl.de/opus/volltexte/2008/1398/
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Bini, Dario A. ;
Meini, Beatrice ;
Poloni, Federico
From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms
Abstract
The problem of reducing an algebraic Riccati equation $XCX-AX-XD+B=0$ to a unilateral quadratic matrix equation (UQME) of the
kind $PX^2+QX+R$ is analyzed. New reductions are introduced
which enable one to prove some theoretical and computational properties.
In particular we show that the structure preserving doubling algorithm
of B.D.O. Anderson [Internat. J. Control, 1978] is nothing else but the
cyclic reduction algorithm applied to a suitable UQME. A new algorithm
obtained by complementing our reductions with the shrink-and-shift tech-
nique of Ramaswami is presented. Finally, faster algorithms which require
some non-singularity conditions, are designed. The non-singularity re-
striction is relaxed by introducing a suitable similarity transformation of
the Hamiltonian.
BibTeX - Entry
@InProceedings{bini_et_al:DSP:2008:1398,
author = {Dario A. Bini and Beatrice Meini and Federico Poloni},
title = {From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms},
booktitle = {Numerical Methods for Structured Markov Chains},
year = {2008},
editor = {Dario Bini and Beatrice Meini and Vaidyanathan Ramaswami and Marie-Ange Remiche and Peter Taylor},
number = {07461},
series = {Dagstuhl Seminar Proceedings},
ISSN = {1862-4405},
publisher = {Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2008/1398},
annote = {Keywords: Algebraic Riccati Equation, Matrix Equation, Cyclic Reduction, Structured doubling algorithm}
}
Keywords: |
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Algebraic Riccati Equation, Matrix Equation, Cyclic Reduction, Structured doubling algorithm |
Collection: |
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07461 - Numerical Methods for Structured Markov Chains |
Issue Date: |
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2008 |
Date of publication: |
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07.04.2008 |