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

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

doi:10.4230/DagSemProc.07461.7

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

Authors Dario A. Bini, Beatrice Meini, Federico Poloni

Part of: Volume: Dagstuhl Seminar Proceedings, Volume 7461
Part of: Series: Dagstuhl Seminar Proceedings (DagSemProc)
License: Creative Commons Attribution 4.0 International license
Publication Date: 2008-04-07

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Document Identifiers

DOI: 10.4230/DagSemProc.07461.7
URN: urn:nbn:de:0030-drops-13987

Subject Classification

Keywords

Algebraic Riccati Equation
Matrix Equation
Cyclic Reduction
Structured doubling algorithm

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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.

Cite As Get BibTex

Dario A. Bini, Beatrice Meini, and Federico Poloni. From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms. In Numerical Methods for Structured Markov Chains. Dagstuhl Seminar Proceedings, Volume 7461, pp. 1-28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008) https://doi.org/10.4230/DagSemProc.07461.7

Author Details

Dario A. Bini

Beatrice Meini

Federico Poloni

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