On the Computation of Derivatives within LD Factorization of Parametrized Matrices
The paper presents a new method for calculating the values of derivatives in the LD factorization of parametrized matrices, based on the direct procedure for the modified weighted Gram-Schmidt orthogonalization.
The need for calculating the values of derivatives in matrix orthogonal transformations arises in the theory of perturbations and control, in differential geometry, in solving problems such as the Lyapunov exponential calculation, the problems of automatic differentiation, the calculation of the numerical solution of the matrix differential Riccati equation, the calculation of high-order derivatives in the optimal input design. In the theory of parameter identification of mathematical models of discrete linear stochastic systems, such problems are solved by developing numerically effective algorithms for finding the solution of the matrix difference Riccati sensitivity equation.
In this paper, we have posed and solved a new problem of calculating the values of derivatives. Lemma 1 represents the main theoretical result. The practical result is the computational algorithm 2. The software implementation of the algorithm allows us to calculate the values of derivatives of the parametrized matrices that are the result of a direct procedure of the LD factorization quickly and with high accuracy. It is not necessary to calculate the values of derivatives of the matrix of weighted orthogonal transformation. The algorithm has a simple structure and does not contain complex operations of symbolic or numerical differentiation. Only one inversion of the triangular matrix and simple matrix operations of addition and multiplication are required.
Two numerical examples are considered that show the operability and numerical efficiency of the proposed algorithm 2.
The results obtained in this paper will be used to construct new classes of adaptive LD filters in the area of parameter identification of mathematical models of discrete linear stochastic systems.
About the Authors
Yulia V. Tsyganova, Dr. Sci. (Phys.–Math.), Assoc. Prof., Ulyanovsk State University, 42, Leo Tolstoy st., Ulyanovsk, 432017, Russian Federation, e-mail: email@example.com
Andrey V. Tsyganov, Cand. Sci. (Phys.–Math.), Assoc. Prof., Ulyanovsk State Pedagogical University named after I. N. Ulyanov, 4, 100th anniversary of V. I. Lenin’s birth Sq., Ulyanovsk, 432071, Russian Federation, e-mail: firstname.lastname@example.org
1. Semushin I.V., Tsyganova Yu.V., Kulikova M.V. et al. Adaptive Systems of Filtering, Control, and Fault Detection. Collective monograph. Ulyanovsk: USU Publishers, 2011. 298 p. ISBN 978-5-88866-399-8 (in Russian)
2. Tsyganova Yu. V. Computing the gradient of the auxiliary quality functional in the parametric identification problem for stochastic systems. Automation and Remote Control, 2011, vol. 72, no 9, pp. 1925–1940. https://doi.org/10.1134/S0005117911090141
3. Tsyganova Yu.V., Kulikova M.V. On efficient parametric identification methods forlinear discrete stochastic systems. Automation and Remote Control, 2012, vol. 73, no 6, pp. 962–975. https://doi.org/10.1134/S0005117912060033
4. Shary S. P. Course of computational methods. Electronic textbook. Institute of Computational Technologies SB RAS, 2012. 315 p. (in Russian)
5. Bierman G. J. Factorization Methods For Discrete Sequential Estimation. NewYork : Academic Press, 1977. 256 p. ISBN 0-12-097350-2
6. Bierman G. J., Belzer M. R., Vandergraft J. S., Porter D. V. Maximum likelihood estimation using square root information filters. IEEE Trans. on Automatic Control, 1990, Dec., vol. 35, no 12, pp. 1293-1298. https://doi.org/10.1109/9.61004
7. Bjorck A. Solving least squares problems by orthogonalization. BIT, 1967, vol. 7,pp. 1–21. https://doi.org/10.1007/BF01934122
8. Dieci L., Russell R. D., Van Vleck E. S. On the Computation of Lyapunov Exponents for Continuous Dynamical Systems SIAM J. Numer. Anal., 1997, vol. 34, no 1, pp. 402-423. https://doi.org/10.1137/S0036142993247311
9. Dieci L., Eirola T. Applications of Smooth Orthogonal Factorizations of Matrices. Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, Springer New York, 2000, vol. 119 of the IMA Volumes in Mathematics and its Applications, pp. 141-162. https://doi.org/10.1007/978-1-4612-1208-9
10. An extended collection of matrix derivative results for forward and reverse mode algorithmic differentiation : Report : 08/01 Executor : M. Giles, Oxford University Computing Laboratory, Parks Road, Oxford, U.K., 2008, January, 23 p.
11. Gupta N. K., Mehra R. K. Computational aspects of maximum likelihood estimation and reduction in sensitivity function calculations. IEEE Trans. on Automatic Control, 1974, no AC-19, pp. 774-783. hhttps://doi.org/10.1109/TAC.1974.1100714
12. Jordan T. L. Experiments on error growth associated with some linearleast squares procedures. Math. Comp., 1968, vol. 22, no 1, pp. 579-588. https://doi.org/10.1090/S0025-5718-1968-0229373-X
13. Jover J. M., Kailath T. A Parallel Architecture for Kalman Filter Measurement Update and Parameter Estimation. Automatica, 1986, vol. 22, no. 1, pp. 43–57. https://doi.org/10.1016/0005-1098(86)90104-4
14. Kulikova M. V. Maximum likelihood estimation via the extended covariance and combined square-root filters. Mathematics and Computers in Simulation, 2009,vol. 79, no. 5, pp. 1641–1657. https://doi.org/10.1016/j.matcom.2008.08.004
15. Kulikova M. V. Likelihood gradient evaluation using square-root covariance filters. IEEE Trans. on Automatic Control, 2009, Mar., vol. 54, no 3, pp. 646-651. https://doi.org/10.1109/TAC.2008.2010989
16. Kulikova M. V., Tsyganova J. V. Constructing numerically stable Kalman filter-based algorithms for gradient-based adaptive filtering. Int. J. Adapt. Control Signal Process, 2015, Nov., vol. 29, no 11, pp. 1411-1426. https://doi.org/10.1002/acs.2552
17. Kunkel P., Mehrmann V. Smooth factorizations of matrix valued functions and their derivatives. Numerische Mathematik, 1991, Dec., vol. 60, no 1, pp. 115-131. https://doi.org/10.1007/BF01385717
18. Thornton C. L. Triangular Covariance Factorizations for Kalman Filtering : Ph.D.thesis. School of Engineering, University of California at Los Angeles, 1976.
19. Tsyganova J. V., Kulikova M. V. State sensitivity evaluation within UD basedarray covariance filters. IEEE Trans. on Automatic Control, 2013, Nov., vol. 58,no 11, pp. 2944-2950. https://doi.org/10.1109/TAC.2013.2259093
20. Walter S. F. Structured Higher-Order Algorithmic Differentiation in the Forwardand Reverse Mode with Application in Optimum Experimental Design : Ph.D.thesis, Humboldt-Universitat zu Berlin, 2011. 221 p.