DagSemProc.07461.7.pdf
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From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms
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Dagstuhl Seminar Proceedings, Volume 7461
Series: Dagstuhl Seminar Proceedings (DagSemProc) - License:
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- Publication Date: 2008-04-07
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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
https://doi.org/10.4230/DagSemProc.07461.7
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.
Keywords
- Algebraic Riccati Equation
- Matrix Equation
- Cyclic Reduction
- Structured doubling algorithm
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