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

Dagstuhl Seminar Proceedings

Volumn 7461, Issue , 2008, Pages

  Bini, Dario A ^a   Meini, Beatrice ^a   Poloni, Federico ^b  

^a UNIVERSITY OF PISA   (Italy)

^b SCUOLA NORMALE SUPERIORE   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

MATRIX ALGEBRA;

% REDUCTIONS; ALGEBRAIC RICCATI EQUATIONS; ANDERSONS; CYCLIC REDUCTION; DOUBLINGS; MATRIX EQUATIONS; NONSINGULARITY; REDUCTION ALGORITHMS; SHIFT TECHNIQUE; STRUCTURE-PRESERVING;

RICCATI EQUATIONS;

EID: 84964984830     PISSN: 18624405     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (27)

1. B. D. O. Anderson. Second-order convergent algorithms for the steady-state Riccati equation. Internat. J. Control, 28(2):295-306, 1978.
2. N. G. Bean, M. M. O'Reilly, and P. G. Taylor. Algorithms for return probabilities for stochastic fluid flows. Stochastic Models, 21(1):149-184, 2005.
3. D. Bini and B. Meini. On the solution of a nonlinear matrix equation arising in queueing problems. SIAM J. Matrix Anal. Appl., 17(4):906-926, 1996.
4. D. Bini, B. Meini, and V. Ramaswami. A probabilistic interpretation of cyclic reduction and its relationships with logarithmic reduction. Calcolo, 2008. To appear.
5. D. A. Bini, L. Gemignani, and B. Meini. Computations with infinite Toeplitz matrices and polynomials. Linear Algebra Appl., 343/344:21-61, 2002. Special issue on structured and infinite systems of linear equations.
6. D. A. Bini, B. Iannazzo, G. Latouche, and B. Meini. On the solution of algebraic Riccati equations arising in fluid queues. Linear Algebra Appl., 413(2-3):474-494, 2006.
7. D. A. Bini, B. Iannazzo, B. Meini, and F. Poloni. Nonsymmetric algebraic Riccati equations associated with an M-matrix: recent advances and algorithms. In Dagstuhl Seminar Proceedings, "Numerical Methods for Structured Markov Chains", 07461, 2007.
8. D. A. Bini, G. Latouche, and B. Meini. Numerical methods for structured Markov chains. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2005.
9. B. L. Buzbee, G. H. Golub, and C. W. Nielson. On direct methods for solving Poisson's equations. SIAM J. Numer. Anal., 7:627-656, 1970.
10. G. H. Golub and C. F. Van Loan. Matrix computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, third edition, 1996.
11. C.-H. Guo. Nonsymmetric algebraic Riccati equations and Wiener-Hopf factorization for M-matrices. SIAM J. Matrix Anal. Appl., 23(1):225-242 (electronic), 2001.
12. C.-H. Guo. Eficient methods for solving a nonsymmetric algebraic Riccati equation arising in stochastic fluid models. J. Comput. Appl. Math., 192(2):353-373, 2006.
13. C.-H. Guo. A new class of nonsymmetric algebraic Riccati equations. Linear Algebra Appl., 426(2-3):636-649, 2007.
14. C.-H. Guo and N. J. Higham. Iterative solution of a nonsymmetric algebraic Riccati equation. Numer. Linear Algebra Appl., 12(2-3):191-200, 2005.
15. C.-H. Guo, B. Iannazzo, and B. Meini. On the doubling algorithm for a (shifted) nonsymmetric algebraic Riccati equation. Technical report, Dipartimento di Matematica, Università di Pisa, Pisa, Italy, May 2006. To appear in SIMAX.
16. C.-H. Guo and A. J. Laub. On the iterative solution of a class of nonsymmetric algebraic Riccati equations. SIAM J. Matrix Anal. Appl., 22(2):376-391 (electronic), 2000.
17. C.-H. Guo and W.-W. Lin. Convergence rate of the cyclic reduction algorithm for null recurrent quasi-birth-death problems. Technical report, Personal communication, 2007.
18. X.-X. Guo, W.-W. Lin, and S.-F. Xu. A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation. Numer. Math., 103(3):393-412, 2006.
19. R. W. Hockney. A fast direct solution of Poisson's equation using Fourier analysis. J. Assoc. Comput. Mach., 12:95-113, 1965.
20. R. A. Horn and C. R. Johnson. Topics in matrix analysis. Cambridge University Press, Cambridge, 1994. Corrected reprint of the 1991 original.
21. B. Iannazzo and D. Bini. A cyclic reduction method for solving algebraic Riccati equations. Technical report, Dipartimento di Matematica, Università di Pisa, 2003.
22. J. Juang and W.-W. Lin. Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices. SIAM J. Matrix Anal. Appl., 20(1):228-243 (electronic), 1999.
23. P. Lancaster and L. Rodman. Algebraic Riccati equations. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, 1995.
24. G. Latouche and V. Ramaswami. A logarithmic reduction algorithm for quasi-birth-death processes. J. Appl. Probab., 30(3):650-674, 1993.
25. W.-W. Lin and S.-F. Xu. Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations. SIAM J. Matrix Anal. Appl., 28(1):26-39 (electronic), 2006
26. V. L. Mehrmann. The autonomous linear quadratic control problem, volume 163 of Lecture Notes in Control and Information Sciences. Springer-Verlag, Berlin, 1991. Theory and numerical solution.
27. V. Ramaswami. Matrix analytic methods for stochastic fluid flows. In Proceedings of the 16th International Teletrafic Congress, pages 19-30. Elsevier Science, Edinburg, 1999.