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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



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Dario A. Bini
Beatrice Meini
Federico Poloni

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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

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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