Numerische Mathematik

, Volume 116, Issue 4, pp 553–578 | Cite as

Transforming algebraic Riccati equations into unilateral quadratic matrix equations

Article

Abstract

The problem of reducing an algebraic Riccati equation XCXAXXD + B = 0 to a unilateral quadratic matrix equation (UQME) of the kind PX2 + QX + R = 0 is analyzed. New transformations are introduced which enable one to prove some theoretical and computational properties. In particular we show that the structure preserving doubling algorithm (SDA) of Anderson (Int J Control 28(2):295–306, 1978) is in fact the cyclic reduction algorithm of Hockney (J Assoc Comput Mach 12:95–113, 1965) and Buzbee et al. (SIAM J Numer Anal 7:627–656, 1970), applied to a suitable UQME. A new algorithm obtained by complementing our transformations with the shrink-and-shift technique of Ramaswami is presented. The new algorithm is accurate and much faster than SDA when applied to some examples concerning fluid queue models.

Mathematics Subject Classification (2000)

15A24 65F30 65H10 

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References

  1. 1.
    Anderson B.D.O.: Second-order convergent algorithms for the steady-state Riccati equation. Int. J. Control 28(2), 295–306 (1978)MATHCrossRefGoogle Scholar
  2. 2.
    Bean N.G., O’Reilly M.M., Taylor P.G.: Algorithms for return probabilities for stochastic fluid flows. Stoch. Models 21(1), 149–184 (2005)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bini D., Meini B.: On the solution of a nonlinear matrix equation arising in queueing problems. SIAM J. Matrix Anal. Appl. 17(4), 906–926 (1996)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bini D., Meini B.: The cyclic reduction algorithm: from Poisson equation to stochastic processes and beyond. Numer. Algorithm 51(1), 23–60 (2009)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bini D., Meini B., Ramaswami V.: A probabilistic interpretation of cyclic reduction and its relationships with logarithmic reduction. Calcolo 45, 207–216 (2008)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bini, D.A., Gemignani, L., Meini, B.: 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)Google Scholar
  7. 7.
    Bini D.A., Iannazzo B., Latouche G., Meini B.: On the solution of algebraic Riccati equations arising in fluid queues. Linear Algebra Appl. 413(2-3), 474–494 (2006)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bini, D.A., Iannazzo, B., Meini, B., Poloni, F.: Nonsymmetric algebraic Riccati equations associated with an M-matrix: recent advances and algorithms. In: Olshevsky, V., Tyrtyshnikov, E. (eds) Matrix Methods: Theory, Algorithms and Applications, pp. 176–209, World Scientific Publishing, April 2010Google Scholar
  9. 9.
    Bini D.A., Latouche G., Meini B.: Numerical Methods for Structured Markov Chains. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford Science Publications, New York (2005)CrossRefGoogle Scholar
  10. 10.
    Buzbee B.L., Golub G.H., Nielson C.W.: On direct methods for solving Poisson’s equations. SIAM J. Numer. Anal. 7, 627–656 (1970)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Chiang C.-Y., Chu E.K.-W., Guo C.-H., Huang T.-M., Lin W.-W., Xu S.-F.: Convergence analysis of the doubling algorithm for several nonlinear matrix equations in the critical case. SIAM J. Matrix Anal. Appl. 31(2), 227–247 (2009)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Golub G.H., Van Loan C.F.: Matrix computations. Johns Hopkins Studies in the Mathematical Sciences, 3rd edn. Johns Hopkins University Press, Baltimore (1996)Google Scholar
  13. 13.
    Guo C.-H.: Nonsymmetric algebraic Riccati equations and Wiener-Hopf factorization for M-matrices. SIAM J. Matrix Anal. Appl. 23(1), 225–242 (2001)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Guo C.-H.: Efficient methods for solving a nonsymmetric algebraic Riccati equation arising in stochastic fluid models. J. Comput. Appl. Math. 192(2), 353–373 (2006)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Guo C.-H.: A new class of nonsymmetric algebraic Riccati equations. Linear Algebra Appl. 426(2–3), 636–649 (2007)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Guo C.-H., Higham N.J.: Iterative solution of a nonsymmetric algebraic Riccati equation. SIAM J. Matrix Anal. Appl. 29(2), 396–412 (2007)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Guo C.-H., Iannazzo B., Meini B.: On the doubling algorithm for a (shifted) nonsymmetric algebraic Riccati equation. SIAM J. Matrix Anal. Appl. 29(4), 1083–1100 (2007)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Guo C.-H., Laub A.J.: On the iterative solution of a class of nonsymmetric algebraic Riccati equations. SIAM J. Matrix Anal. Appl. 22(2), 376–391 (2000)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Guo X.-X., Lin W.-W., Xu S.-F.: A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation. Numer. Math. 103(3), 393–412 (2006)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Hockney R.W.: A fast direct solution of Poisson’s equation using Fourier analysis. J. Assoc. Comput. Mach. 12, 95–113 (1965)MATHMathSciNetGoogle Scholar
  21. 21.
    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1994) (Corrected reprint of the 1991 original)Google Scholar
  22. 22.
    Iannazzo, B., Bini, D.: A Cyclic Reduction Method for Solving Algebraic Riccati Equations. Technical report. Dipartimento di Matematica, Università di Pisa (2003)Google Scholar
  23. 23.
    Juang J., Lin W.-W.: Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices. SIAM J. Matrix Anal. Appl. 20(1), 228–243 (1999)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Lancaster P., Rodman L.: Existence and uniqueness theorems for the algebraic Riccati equation. Int. J. Control 32(2), 285–309 (1980)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Lancaster P., Rodman L.: Algebraic Riccati equations. Oxford Science Publications/The Clarendon Press Oxford University Press, New York (1995)MATHGoogle Scholar
  26. 26.
    Latouche G., Ramaswami V.: A logarithmic reduction algorithm for quasi-birth-death processes. J. Appl. Probab. 30(3), 650–674 (1993)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Lin W.-W., Xu S.-F.: Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations. SIAM J. Matrix Anal. Appl. 28(1), 26–39 (2006)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Mehrmann V.L.: The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution, vol. 163 of Lecture Notes in Control and Information Sciences. Springer, Berlin (1991)Google Scholar
  29. 29.
    Ramaswami, V.: Matrix analytic methods for stochastic fluid flows. In: Proceedings of the 16th International Teletraffic Congress, pp. 19–30. Elsevier Science, Edinburg (1999)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Dario A. Bini
    • 1
  • Beatrice Meini
    • 1
  • Federico Poloni
    • 2
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.Scuola Normale SuperiorePisaItaly

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