## Gradient based and least squares based iterative algorithms for matrix equations $$AXB + CX^{T}D = F$$.(English)Zbl 1210.65097

A gradient based iterative algorithm and a least squares based iterative algorithm are developed and presented for the solution of the matrix equation $$AXB + CX^{T}D = F$$. The hierarchical identification principle is applied to the matrix equation in order to decompose the system under consideration into two subsystems and to derive the iterative algorithms by extending the iterative methods for solving $$Ax = b$$ and $$AXB = F$$. Further analysis shows that when the matrix equation has a unique solution, under the sense of least squares, the iterative solution converges to the exact solution for any initial values. A numerical example is used to verify the proposed methods.

### MSC:

 65F30 Other matrix algorithms (MSC2010) 65F10 Iterative numerical methods for linear systems 15A24 Matrix equations and identities
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### References:

 [1] Ding, F.; Chen, T., Gradient based iterative algorithms for solving a class of matrix equations, IEEE Transactions on Automatic Control, 50, 8, 1216-1221 (2005) · Zbl 1365.65083 [2] Ding, F.; Chen, T., Iterative least squares solutions of coupled Sylvester matrix equations, Systems and Control Letters, 54, 2, 95-107 (2005) · Zbl 1129.65306 [3] Ding, F.; Chen, T., On iterative solutions of general coupled matrix equations, SIAM Journal on Control and Optimization, 44, 6, 2269-2284 (2006) · Zbl 1115.65035 [4] Ding, F.; Liu, P. X.; Ding, J., Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Applied Mathematics and Computation, 197, 1, 41-50 (2008) · Zbl 1143.65035 [5] Shi, Y.; Ding, F.; Chen, T., 2-Norm based recursive design of transmultiplexers with designable filter length, Circuits, Systems, and Signal Processing, 25, 4, 447-462 (2006) · Zbl 1130.94312 [6] Wang, D. Q.; Ding, F., Input-output data filtering based recursive least squares identification for CARARMA systems, Digital Signal Processing, 20, 4, 991-999 (2010) [7] Liu, Y. J.; Yu, L.; Ding, F., Multi-innovation extended stochastic gradient algorithm and its performance analysis, Circuits, Systems and Signal Processing, 29, 4, 649-667 (2010) · Zbl 1196.94026 [8] Ding, F.; X Liu, P.; Liu, G., Multi-innovation least squares identification for linear and pseudo-linear regression models, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 40, 3, 767-778 (2010) [9] Ding, F.; Shi, Y.; Chen, T., Performance analysis of estimation algorithms of non-stationary ARMA processes, IEEE Transactions on Signal Processing, 54, 3, 1041-1053 (2006) · Zbl 1373.94569 [10] Ding, F.; Liu, P. X.; Liu, G., Auxiliary model based multi-innovation extended stochastic gradient parameter estimation with colored measurement noises, Signal Processing, 89, 10, 1883-1890 (2009) · Zbl 1178.94137 [11] Ding, F.; Chen, T., Performance analysis of multi-innovation gradient type identification methods, Automatica, 43, 1, 1-14 (2007) · Zbl 1140.93488 [12] Ding, F., Several multi-innovation identification methods, Digital Signal Processing, 20, 4, 1027-1039 (2010) [13] Wang, D. Q.; Ding, F., Performance analysis of the auxiliary models based multi-innovation stochastic gradient estimation algorithm for output error systems, Digital Signal Processing, 20, 3, 750-762 (2010) [14] Dehghan, M.; Hajarian, M., An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation, Applied Mathematics and Computation, 202, 2, 571-588 (2008) · Zbl 1154.65023 [15] Kilicman, A.; Al Zhour, Z., Vector least-squares solutions for coupled singular matrix equations, Journal of Computational and Applied Mathematics, 206, 2, 1051-1069 (2007) · Zbl 1132.65034 [16] Ding, F., Transformations between some special matrices, Computers and Mathematics with Applications, 59, 8, 2676-2695 (2010) · Zbl 1193.15028 [17] Tian, Z. L.; Gu, C. Q., A numerical algorithm for Lyapunov equations, Applied Mathematics and Computation, 202, 1, 44-53 (2008) · Zbl 1154.65027 [18] Ding, F.; Liu, P. X.; Liu, G., Gradient based and least-squares based iterative identification methods for OE and OEMA systems, Digital Signal Processing, 20, 3, 664-677 (2010) [19] Liu, Y. J.; Wang, D. Q.; Ding, F., Least-squares based iterative algorithms for identifying Box-Jenkins models with finite measurement data, Digital Signal Processing, 20, 5, 1458-1467 (2010) [20] Wang, D. Q.; Ding, F., Gradient-based iterative parameter estimation for Box-Jenkins systems, Computers and Mathematics with Applications (2010) · Zbl 1201.94046 [21] Peng, Z. Y., An iterative method for the least squares symmetric solution of the linear matrix equation $$AXB =C$$, Applied Mathematics and Computation, 170, 1, 711-723 (2005) · Zbl 1081.65039 [22] Evans, D. J.; Martins, M. M., AOR method for $$AX − XB =C$$, International Journal of Computer Mathematics, 52, 1-2, 75-82 (1994) [23] Berlin, U. B.; Chemnitz, P. B., Factorized solution of Lyapunov equations based on hierarchical matrix arithmetic, Computing, 78, 3, 211-234 (2006) · Zbl 1111.65039 [24] Benner, P.; Quintana-Ortí, E. S.; Quintana-Ortí, G., Solving stable Sylvester equations via rational iterative schemes, Journal of Scientific Computing, 28, 1, 51-83 (2006) · Zbl 1098.65041 [25] Ding, F.; Chen, T., Hierarchical gradient-based identification of multivariable discrete-time systems, Automatica, 41, 2, 315-325 (2005) · Zbl 1073.93012 [26] Ding, F.; Chen, T., Hierarchical least squares identification methods for multivariable systems, IEEE Transactions on Automatic Control, 50, 3, 397-402 (2005) · Zbl 1365.93551 [27] Han, H. Q.; Xie, L.; Ding, F.; Liu, X. G., Hierarchical least squares based iterative identification for multivariable systems with moving average noises, Mathematical and Computer Modelling, 51, 9-10, 1213-1220 (2010) · Zbl 1198.93216 [28] Xie, L.; Ding, J.; Ding, F., Gradient based iterative solutions for general linear matrix equations, Computers and Mathematics with Applications, 58, 7, 1441-1448 (2009) · Zbl 1189.65083 [29] Ding, J.; Liu, Y. J.; Ding, F., Iterative solutions to matrix equations of form AiXBi=Fi, Computers and Mathematics with Applications, 59, 11, 3500-3507 (2010) · Zbl 1197.15009 [31] Ding, F., System Identification Theory and Methods + Matlab Simulations (2010), China Electric Power Press: China Electric Power Press Beijing, (in Chinese) [32] Ding, F.; Chen, H. B.; Li, M., Multi-innovation least squares identification methods based on the auxiliary model for MISO systems, Applied Mathematics and Computation, 187, 2, 658-668 (2007) · Zbl 1114.93101 [33] Han, L. L.; Ding, F., Multi-innovation stochastic gradient algorithms for multi-input multi-output systems, Digital Signal Processing, 19, 4, 545-554 (2009) [34] Ding, F.; Chen, T., Identification of Hammerstein nonlinear ARMAX systems, Automatica, 41, 9, 1479-1489 (2005) · Zbl 1086.93063 [35] Ding, F.; Shi, Y.; Chen, T., Auxiliary model based least-squares identification methods for Hammerstein output-error systems, Systems and Control Letters, 56, 5, 373-380 (2007) · Zbl 1130.93055 [36] Wang, D. Q.; Ding, F., Extended stochastic gradient identification algorithms for Hammerstein-Wiener ARMAX Systems, Computers and Mathematics with Applications, 56, 12, 3157-3164 (2008) · Zbl 1165.65308 [37] Wang, D. Q.; Chu, Y. Y.; Ding, F., Auxiliary model-based RELS and MI-ELS algorithms for Hammerstein OEMA systems, Computers and Mathematics with Applications, 59, 9, 3092-3098 (2010) · Zbl 1193.93170 [38] Wang, D. Q.; Chu, Y. Y.; Yang, G. W.; Ding, F., Auxiliary model-based recursive generalized least squares parameter estimation for Hammerstein OEAR systems, Mathematical and Computer Modelling, 52, 1-2, 309-317 (2010) · Zbl 1201.93134 [39] Ding, F.; Liu, X. P.; Shi, Y., Convergence analysis of estimation algorithms of dual-rate stochastic systems, Applied Mathematics and Computation, 176, 1, 245-261 (2006) · Zbl 1095.65056 [40] Ding, F.; Liu, P. X.; Yang, H. Z., Parameter identification and intersample output estimation for dual-rate systems, IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans, 38, 4, 966-975 (2008) [41] Ding, J.; Shi, Y.; Wang, H. G.; Ding, F., A modified stochastic gradient based parameter estimation algorithm for dual-rate sampled-data systems, Digital Signal Processing, 20, 4, 1238-1249 (2010) [42] Ding, F.; Qiu, L.; Chen, T., Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems, Automatica, 45, 2, 324-332 (2009) · Zbl 1158.93365 [43] Liu, Y. J.; Xie, L.; Ding, F.; systems, An auxiliary model based recursive least squares parameter estimation algorithm for non-uniformly sampled multirate, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 223, 4, 445-454 (2009) [44] Xie, L.; Yang, H. Z.; Ding, F., Modeling and identification for non-uniformly periodically sampled-data systems, IET Control Theory and Applications, 4, 5, 784-794 (2010) [45] Ding, F.; Liu, G.; Liu, X. P., Partially coupled stochastic gradient identification methods for non-uniformly sampled systems, IEEE Transaction on Automatic Control, 55, 8 (2010) · Zbl 1368.93121 [46] Ding, F.; Ding, J., Least squares parameter estimation with irregularly missing data, International Journal of Adaptive Control and Signal Processing, 24, 7, 540-553 (2010) · Zbl 1200.93130
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