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Documents authored by Poloni, Federico

Document
DOI: 10.4230/DagSemProc.07461.7
From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms

Authors: Dario A. Bini, Beatrice Meini, and Federico Poloni

Published in: Dagstuhl Seminar Proceedings, Volume 7461, Numerical Methods for Structured Markov Chains (2008)

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

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)

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@InProceedings{bini_et_al:DagSemProc.07461.7,
  author =	{Bini, Dario A. and Meini, Beatrice and Poloni, Federico},
  title =	{{From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms}},
  booktitle =	{Numerical Methods for Structured Markov Chains},
  pages =	{1--28},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{7461},
  editor =	{Dario Bini and Beatrice Meini and Vaidyanathan Ramaswami and Marie-Ange Remiche and Peter Taylor},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07461.7},
  URN =		{urn:nbn:de:0030-drops-13987},
  doi =		{10.4230/DagSemProc.07461.7},
  annote =	{Keywords: Algebraic Riccati Equation, Matrix Equation, Cyclic Reduction, Structured doubling algorithm}
}
Document
DOI: 10.4230/DagSemProc.07461.11
Nonsymmetric algebraic Riccati equations associated with an M-matrix: recent advances and algorithms

Authors: Dario A. Bini, Bruno Iannazzo, Beatrice Meini, and Federico Poloni

Published in: Dagstuhl Seminar Proceedings, Volume 7461, Numerical Methods for Structured Markov Chains (2008)

Abstract
We survey on theoretical properties and algorithms concerning the problem of solving a nonsymmetric algebraic Riccati equation, and we report on some known methods and new algorithmic advances. In particular, some results on the number of positive solutions are proved and a careful convergence analysis of Newton's iteration is carried out in the cases of interest where some singularity conditions are encountered. From this analysis we determine initial approximations which still guarantee the quadratic convergence.
Cite as

Dario A. Bini, Bruno Iannazzo, Beatrice Meini, and Federico Poloni. Nonsymmetric algebraic Riccati equations associated with an M-matrix: recent advances and algorithms. In Numerical Methods for Structured Markov Chains. Dagstuhl Seminar Proceedings, Volume 7461, pp. 1-31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{bini_et_al:DagSemProc.07461.11,
  author =	{Bini, Dario A. and Iannazzo, Bruno and Meini, Beatrice and Poloni, Federico},
  title =	{{Nonsymmetric algebraic Riccati equations associated with an M-matrix: recent advances and algorithms}},
  booktitle =	{Numerical Methods for Structured Markov Chains},
  pages =	{1--31},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{7461},
  editor =	{Dario Bini and Beatrice Meini and Vaidyanathan Ramaswami and Marie-Ange Remiche and Peter Taylor},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07461.11},
  URN =		{urn:nbn:de:0030-drops-13958},
  doi =		{10.4230/DagSemProc.07461.11},
  annote =	{Keywords: Nonsymmetric algebraic Riccati equations, matrix equation, M-matrices, Newton method, quadratically convergent algorithms, cyclic reduction, doubling}
}
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