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URN: urn:nbn:de:0030-drops-13987
URL: http://drops.dagstuhl.de/opus/volltexte/2008/1398/

Bini, Dario A. ; Meini, Beatrice ; Poloni, Federico

From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms

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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: Algebraic Riccati Equation, Matrix Equation, Cyclic Reduction, Structured doubling algorithm
Seminar: 07461 - Numerical Methods for Structured Markov Chains
Issue date: 2008
Date of publication: 07.04.2008


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