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On the Solution of the Rational Matrix Equation
EURASIP Journal on Advances in Signal Processing volume 2007, Article number: 021850 (2007)
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.
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
- 2.
Ferrante A, Levy BC:Hermitian solutions of the equation:
. Linear Algebra and Its Applications 1996, 247: 359–373.
- 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
- 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.
- 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
- 6.
Meini B:Efficient computation of the extreme solutions of:
and
. Mathematics of Computation 2002,71(239):1189-1204.
- 7.
Reurings M: Symmetric matrix equations, Ph.D. thesis. Vrije Universiteit, Amsterdam, The Netherlands; 2003.
- 8.
Lancaster P, Tismenetsky M: The Theory of Matrices. 2nd edition. Academic Press, Orlando, Fla, USA; 1985.
- 9.
Lancaster P, Rodman L: Algebraic Riccati Equations. Oxford University Press, Oxford, UK; 1995.
- 10.
Ahlbrandt CD, Peterson AC: Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1998.
- 11.
Anderson BDO, Moore JB: Optimal Filtering. Prentice-Hall, Englewood Cliffs, NJ, USA; 1979.
- 12.
Anderson BDO, Vongpanitlerd B: Network Analysis and Synthesis. A Modern Systems Approach. Prentice-Hall, Englewood Cliffs, NJ, USA; 1972.
- 13.
Zhou K, Doyle JC, Glover K: Robust and Optimal Control. Prentice-Hall, Upper Saddle River, NJ, USA; 1995.
- 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
- 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.
- 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.
- 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
- 18.
Sima V: Algorithms for Linear-Quadratic Optimization, Pure and Applied Mathematics. Volume 200. Marcel Dekker, New York, NY, USA; 1996.
- 19.
Golub GH, van Loan CF: Matrix Computations. 3rd edition. Johns Hopkins University Press, Baltimore, Md, USA; 1996.
- 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
- 21.
Benner P: Contributions to the Numerical Solution of Algebraic Riccati Equations and Related Eigenvalue Problems. Logos, Berlin, Germany; 1997.
- 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
- 23.
Malyshev AN: Parallel algorithm for solving some spectral problems of linear algebra. Linear Algebra and Its Applications 1993,188-189(1):489-520.
- 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.
- 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.
- 26.
Faßbender H: Symplectic Methods for the Symplectic Eigenproblem. Kluwer Academic/Plenum Publishers, New York, NY, USA; 2000.
- 27.
Antoulas AC: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia, Pa, USA; 2005.
- 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
- 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.
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
- 31.
Horn R, Johnson CR: Topics in Matrix Analysis. Cambridge University Press, Cambridge, UK; 1994.
- 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
- 33.
Barraud AY:A numerical algorithm to solve:
. IEEE Transactions on Automatic Control 1977,22(5):883-885. 10.1109/TAC.1977.1101604
- 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,
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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
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Keywords
- Information Technology
- Iterative Method
- Quantum Information
- Matrix Equation
- Riccati Equation