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  • Research Article
  • Open Access

On the Solution of the Rational Matrix Equation

EURASIP Journal on Advances in Signal Processing20072007:021850

  • Received: 30 September 2006
  • Accepted: 22 February 2007
  • Published:


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.


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

Authors’ Affiliations

Fakultät für Mathematik, Technische Universität Chemnitz, Chemnitz, 09107, Germany
Institut Computational Mathematics, Technische Universität Braunschweig, Braunschweig, 38106, Germany


  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. 10.1109/9.58529MathSciNetView ArticleGoogle Scholar
  2. Ferrante A, Levy BC:Hermitian solutions of the equation: . Linear Algebra and Its Applications 1996, 247: 359–373.MathSciNetView ArticleGoogle Scholar
  3. Guo C-H, Lancaster P: Iterative solution of two matrix equations. Mathematics of Computation 1999,68(228):1589-1603. 10.1090/S0025-5718-99-01122-9MathSciNetView ArticleGoogle Scholar
  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.MathSciNetView ArticleGoogle Scholar
  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. 10.1137/040617650MathSciNetView ArticleGoogle Scholar
  6. Meini B:Efficient computation of the extreme solutions of: and . Mathematics of Computation 2002,71(239):1189-1204.MathSciNetView ArticleGoogle Scholar
  7. Reurings M: Symmetric matrix equations, Ph.D. thesis. Vrije Universiteit, Amsterdam, The Netherlands; 2003.Google Scholar
  8. Lancaster P, Tismenetsky M: The Theory of Matrices. 2nd edition. Academic Press, Orlando, Fla, USA; 1985.MATHGoogle Scholar
  9. Lancaster P, Rodman L: Algebraic Riccati Equations. Oxford University Press, Oxford, UK; 1995.MATHGoogle Scholar
  10. Ahlbrandt CD, Peterson AC: Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1998.MATHGoogle Scholar
  11. Anderson BDO, Moore JB: Optimal Filtering. Prentice-Hall, Englewood Cliffs, NJ, USA; 1979.MATHGoogle Scholar
  12. Anderson BDO, Vongpanitlerd B: Network Analysis and Synthesis. A Modern Systems Approach. Prentice-Hall, Englewood Cliffs, NJ, USA; 1972.Google Scholar
  13. Zhou K, Doyle JC, Glover K: Robust and Optimal Control. Prentice-Hall, Upper Saddle River, NJ, USA; 1995.MATHGoogle Scholar
  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.1971.1099755View ArticleGoogle Scholar
  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. 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. 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.1980.1102434MathSciNetView ArticleGoogle Scholar
  18. Sima V: Algorithms for Linear-Quadratic Optimization, Pure and Applied Mathematics. Volume 200. Marcel Dekker, New York, NY, USA; 1996.MATHGoogle Scholar
  19. Golub GH, van Loan CF: Matrix Computations. 3rd edition. Johns Hopkins University Press, Baltimore, Md, USA; 1996.MATHGoogle Scholar
  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. 10.1080/00207170410001714988MathSciNetView ArticleGoogle Scholar
  21. Benner P: Contributions to the Numerical Solution of Algebraic Riccati Equations and Related Eigenvalue Problems. Logos, Berlin, Germany; 1997.MATHGoogle Scholar
  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/s002110050264MathSciNetView ArticleGoogle Scholar
  23. Malyshev AN: Parallel algorithm for solving some spectral problems of linear algebra. Linear Algebra and Its Applications 1993,188-189(1):489-520.MathSciNetView ArticleGoogle Scholar
  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.MathSciNetView ArticleGoogle Scholar
  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.MathSciNetView ArticleGoogle Scholar
  26. Faßbender H: Symplectic Methods for the Symplectic Eigenproblem. Kluwer Academic/Plenum Publishers, New York, NY, USA; 2000.MATHGoogle Scholar
  27. Antoulas AC: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia, Pa, USA; 2005.View ArticleGoogle Scholar
  28. Anderson BDO: Second-order convergent algorithms for the steady-state Riccati equation. International Journal of Control 1978,28(2):295-306. 10.1080/00207177808922455MathSciNetView ArticleGoogle Scholar
  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, preprint, 2006, Fakultät für Mathematik, TU Chemnitz, Germany,
  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/S089547989122377XMathSciNetView ArticleGoogle Scholar
  31. Horn R, Johnson CR: Topics in Matrix Analysis. Cambridge University Press, Cambridge, UK; 1994.MATHGoogle Scholar
  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. 10.1145/146847.146929MathSciNetView ArticleGoogle Scholar
  33. Barraud AY:A numerical algorithm to solve: . IEEE Transactions on Automatic Control 1977,22(5):883-885. 10.1109/TAC.1977.1101604MathSciNetView ArticleGoogle Scholar
  34. Slowik M, Benner P, Sima V: Evaluation of the Linear Matrix Equation Solvers in SLICOT. SLICOT Working Note, 2004—1, 2004, SLICOT Working Note, 2004—1, 2004,Google Scholar


© P. Benner and H. Faßbender 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.