Skip to main content
EURASIP Journal on Advances in Signal Processing
Download PDF

On the Solution of the Rational Matrix Equation

EURASIP Journal on Advances in Signal Processing volume 2007, Article number: 021850 (2007) Cite this article

Abstract

We study numerical methods for finding the maximal symmetric positive definite solution of the nonlinear matrix equation, where is symmetric positive definite and is nonsingular. Such equations arise for instance in the analysis of stationary Gaussian reciprocal processes over a finite interval. Its unique largest positive definite solution coincides with the unique positive definite solution of a related discrete-time algebraic Riccati equation (DARE). We discuss how to use the butterfly algorithm to solve the DARE. This approach is compared to several fixed-point and doubling-type iterative methods suggested in the literature.

References

  1. 1.

    Levy BC, Frezza R, Krener AJ: Modeling and estimation of discrete-time Gaussian reciprocal processes. IEEE Transactions on Automatic Control 1990,35(9):1013-1023. 10.1109/9.58529

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ferrante A, Levy BC:Hermitian solutions of the equation:. Linear Algebra and Its Applications 1996, 247: 359–373.

    MathSciNet  Article  Google Scholar 

  3. 3.

    Guo C-H, Lancaster P: Iterative solution of two matrix equations. Mathematics of Computation 1999,68(228):1589-1603. 10.1090/S0025-5718-99-01122-9

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ivanov IG, Hasanov VI, Uhlig F:Improved methods and starting values to solve the matrix equations: :iteratively. Mathematics of Computation 2005,74(249):263-278.

    MathSciNet  Article  Google Scholar 

  5. 5.

    Lin W-W, Xu S-F: Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations. SIAM Journal on Matrix Analysis and Applications 2006,28(1):26-39. 10.1137/040617650

    MathSciNet  Article  Google Scholar 

  6. 6.

    Meini B:Efficient computation of the extreme solutions of: and. Mathematics of Computation 2002,71(239):1189-1204.

    MathSciNet  Article  Google Scholar 

  7. 7.

    Reurings M: Symmetric matrix equations, Ph.D. thesis. Vrije Universiteit, Amsterdam, The Netherlands; 2003.

    Google Scholar 

  8. 8.

    Lancaster P, Tismenetsky M: The Theory of Matrices. 2nd edition. Academic Press, Orlando, Fla, USA; 1985.

    MATH  Google Scholar 

  9. 9.

    Lancaster P, Rodman L: Algebraic Riccati Equations. Oxford University Press, Oxford, UK; 1995.

    MATH  Google Scholar 

  10. 10.

    Ahlbrandt CD, Peterson AC: Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1998.

    MATH  Google Scholar 

  11. 11.

    Anderson BDO, Moore JB: Optimal Filtering. Prentice-Hall, Englewood Cliffs, NJ, USA; 1979.

    MATH  Google Scholar 

  12. 12.

    Anderson BDO, Vongpanitlerd B: Network Analysis and Synthesis. A Modern Systems Approach. Prentice-Hall, Englewood Cliffs, NJ, USA; 1972.

    Google Scholar 

  13. 13.

    Zhou K, Doyle JC, Glover K: Robust and Optimal Control. Prentice-Hall, Upper Saddle River, NJ, USA; 1995.

    MATH  Google Scholar 

  14. 14.

    Hewer GA: An iterative technique for the computation of the steady state gains for the discrete optimal regulator. IEEE Transactions on Automatic Control 1971,16(4):382-384. 10.1109/TAC.1971.1099755

    Article  Google Scholar 

  15. 15.

    Mehrmann VL: The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution, Lecture Notes in Control and Information Sciences. Volume 163. Springer, Heidelberg, Germany; 1991.

    Google Scholar 

  16. 16.

    Laub AJ: Algebraic aspects of generalized eigenvalue problems for solving Riccati equations. In Computational and Combinatorial Methods in Systems Theory. Edited by: Byrnes CI, Lindquist A. Elsevier/North-Holland, New York, NY, USA; 1986:213-227.

    Google Scholar 

  17. 17.

    Pappas T, Laub AJ, Sandell NR Jr.: On the numerical solution of the discrete-time algebraic Riccati equation. IEEE Transactions on Automatic Control 1980,25(4):631-641. 10.1109/TAC.1980.1102434

    MathSciNet  Article  Google Scholar 

  18. 18.

    Sima V: Algorithms for Linear-Quadratic Optimization, Pure and Applied Mathematics. Volume 200. Marcel Dekker, New York, NY, USA; 1996.

    MATH  Google Scholar 

  19. 19.

    Golub GH, van Loan CF: Matrix Computations. 3rd edition. Johns Hopkins University Press, Baltimore, Md, USA; 1996.

    MATH  Google Scholar 

  20. 20.

    Chu EK-W, Fan H-Y, Lin W-W, Wang C-S: Structure-preserving algorithms for periodic discrete-time algebraic Riccati equations. International Journal of Control 2004,77(8):767-788. 10.1080/00207170410001714988

    MathSciNet  Article  Google Scholar 

  21. 21.

    Benner P: Contributions to the Numerical Solution of Algebraic Riccati Equations and Related Eigenvalue Problems. Logos, Berlin, Germany; 1997.

    MATH  Google Scholar 

  22. 22.

    Bai Z, Demmel J, Gu M: An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems. Numerische Mathematik 1997,76(3):279-308. 10.1007/s002110050264

    MathSciNet  Article  Google Scholar 

  23. 23.

    Malyshev AN: Parallel algorithm for solving some spectral problems of linear algebra. Linear Algebra and Its Applications 1993,188-189(1):489-520.

    MathSciNet  Article  Google Scholar 

  24. 24.

    Benner P, Faßbender H:The symplectic eigenvalue problem, the butterfly form, the: :algorithm, and the Lanczos method. Linear Algebra and Its Applications 1998, 275-276: 19–47.

    MathSciNet  Article  Google Scholar 

  25. 25.

    Benner P, Faßbender H, Watkins DS: and :algorithms for the symplectic (butterfly) eigenproblem. Linear Algebra and Its Applications 1999,287(1–3):41-76.

    MathSciNet  Article  Google Scholar 

  26. 26.

    Faßbender H: Symplectic Methods for the Symplectic Eigenproblem. Kluwer Academic/Plenum Publishers, New York, NY, USA; 2000.

    MATH  Google Scholar 

  27. 27.

    Antoulas AC: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia, Pa, USA; 2005.

    Book  Google Scholar 

  28. 28.

    Anderson BDO: Second-order convergent algorithms for the steady-state Riccati equation. International Journal of Control 1978,28(2):295-306. 10.1080/00207177808922455

    MathSciNet  Article  Google Scholar 

  29. 29.

    Barrachina S, Benner P, Quintana-Ortí ES: Solution of discrete-time Riccati equations via structure-preserving doubling algorithms on a cluster of SMPs. preprint, 2006, Fakultät für Mathematik, TU Chemnitz, Germany, https://doi.org/www.tu-chemnitz.de/~benner/pub/bbq-sda.pdf preprint, 2006, Fakultät für Mathematik, TU Chemnitz, Germany,

  30. 30.

    Watkins DS, Elsner L: Theory of decomposition and bulge-chasing algorithms for the generalized eigenvalue problem. SIAM Journal on Matrix Analysis and Applications 1994,15(3):943-967. 10.1137/S089547989122377X

    MathSciNet  Article  Google Scholar 

  31. 31.

    Horn R, Johnson CR: Topics in Matrix Analysis. Cambridge University Press, Cambridge, UK; 1994.

    MATH  Google Scholar 

  32. 32.

    Gardiner JD, Laub AJ, Amato JJ, Moler CB:Solution of the Sylvester matrix equation:. ACM Transactions on Mathematical Software 1992,18(2):223-231. 10.1145/146847.146929

    MathSciNet  Article  Google Scholar 

  33. 33.

    Barraud AY:A numerical algorithm to solve:. IEEE Transactions on Automatic Control 1977,22(5):883-885. 10.1109/TAC.1977.1101604

    MathSciNet  Article  Google Scholar 

  34. 34.

    Slowik M, Benner P, Sima V: Evaluation of the Linear Matrix Equation Solvers in SLICOT. SLICOT Working Note, 2004—1, 2004, https://doi.org/www.icm.tu-bs.de/NICONET/ SLICOT Working Note, 2004—1, 2004,

    Google Scholar 

Download references

Author information

Affiliations

  1. Fakultät für Mathematik, Technische Universität Chemnitz, Chemnitz, 09107, Germany

    Peter Benner

  2. Institut Computational Mathematics, Technische Universität Braunschweig, Braunschweig, 38106, Germany

    Heike Faßbender

Authors
  1. Peter Benner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Heike Faßbender
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Benner.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://doi.org/creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Benner, P., Faßbender, H. On the Solution of the Rational Matrix Equation. EURASIP J. Adv. Signal Process. 2007, 021850 (2007). https://doi.org/10.1155/2007/21850

Download citation

Keywords

  • Information Technology
  • Iterative Method
  • Quantum Information
  • Matrix Equation
  • Riccati Equation