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Gradient based iterative solutions for general linear matrix equations. (English) Zbl 1189.65083
Summary: We present a gradient based iterative algorithm for solving general linear matrix equations by extending the Jacobi iteration and by applying the hierarchical identification principle. Convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices. Two numerical examples are provided to show that the proposed algorithm is effective.

65F30Other matrix algorithms
15A24Matrix equations and identities
Full Text: DOI
[1] Golub, G. H.; Van Loan, C. F.: Matrix computations, (1996) · Zbl 0865.65009
[2] Ding, F.; Chen, T.: Gradient based iterative algorithms for solving a class of matrix equations, IEEE transactions on automatic control 50, No. 8, 1216-1221 (2005)
[3] 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, No. 1, 41-50 (2008) · Zbl 1143.65035 · doi:10.1016/j.amc.2007.07.040
[4] Tian, Z. L.; Gu, C. Q.: A numerical algorithm for Lyapunov equations, Applied mathematics and computation 202, No. 1, 44-53 (2008) · Zbl 1154.65027 · doi:10.1016/j.amc.2007.12.057
[5] Kilicman, A.; Zhour, Z. Al: Vector least-squares solutions for coupled singular matrix equations, Journal of computational and applied mathematics 206, No. 2, 1051-1069 (2007) · Zbl 1132.65034 · doi:10.1016/j.cam.2006.09.009
[6] Ding, F.; Qiu, L.; Chen, T.: Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems, Automatica 45, No. 2, 324-332 (2009) · Zbl 1158.93365 · doi:10.1016/j.automatica.2008.08.007
[7] Ding, F.; Chen, T.: Performance analysis of multi-innovation gradient type identification methods, Automatica 43, No. 1, 1-14 (2007) · Zbl 1140.93488 · doi:10.1016/j.automatica.2006.07.024
[8] Ding, F.; Liu, P. X.; Liu, G.: Auxiliary model based multi-innovation extended stochastic gradient parameter estimation with colored measurement noises, Signal processing 89, No. 10, 1883-1890 (2009) · Zbl 1178.94137 · doi:10.1016/j.sigpro.2009.03.020
[9] 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, No. 4, 966-975 (2008)
[10] Ding, F.; Yang, H. Z.; Liu, F.: Performance analysis of stochastic gradient algorithms under weak conditions, Science in China series F--information sciences 51, No. 9, 1269-1280 (2008) · Zbl 1145.93050 · doi:10.1007/s11432-008-0117-y
[11] 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, No. 2, 571-588 (2008) · Zbl 1154.65023 · doi:10.1016/j.amc.2008.02.035
[12] Mukaidani, H.; Yamamoto, S.; Yamamoto, T.: A numerical algorithm for finding solution of cross-coupled algebraic Riccati equations, IEICE transactions on fundamentals of electronics communications and computer sciences 91A, No. 2, 682-685 (2008)
[13] Zhou, B.; Duan, G. R.: Solutions to generalized Sylvester matrix equation by Schur decomposition, International journal of systems science 38, No. 5, 369-375 (2007) · Zbl 1126.65034 · doi:10.1080/00207720601160215
[14] Zhou, B.; Duan, G. R.: On the generalized Sylvester mapping and matrix equations, Systems control letters 57, No. 3, 200-208 (2008) · Zbl 1129.93018 · doi:10.1016/j.sysconle.2007.08.010
[15] Ding, F.; Chen, T.: Iterative least squares solutions of coupled Sylvester matrix equations, Systems control letters 54, No. 2, 95-107 (2005) · Zbl 1129.65306 · doi:10.1016/j.sysconle.2004.06.008
[16] Ding, F.; Chen, T.: On iterative solutions of general coupled matrix equations, SIAM journal on control and optimization 44, No. 6, 2269-2284 (2005) · Zbl 1115.65035 · doi:10.1137/S0363012904441350
[17] Zhour, Z. Al; Kilicman, A.: Some new connections between matrix products for partitioned and non-partitioned matrices, Computers mathematics with applications 54, No. 6, 763-784 (2007) · Zbl 1146.15014 · doi:10.1016/j.camwa.2006.12.045
[18] Zhou, B.; Duan, G. R.: An explicit solution to the matrix equation AX-XF=BY, Linear algebra and its applications 402, No. 1--3, 345-366 (2005) · Zbl 1076.15016 · doi:10.1016/j.laa.2005.01.018
[19] Zhou, B.; Duan, G. R.: A new solution to the generalized Sylvester matrix equation AV-EVF=BW, Systems control letters 55, No. 3, 193-198 (2006) · Zbl 1129.15300 · doi:10.1016/j.sysconle.2005.07.002
[20] Zhou, B.; Duan, G. R.: Parametric solutions to the generalized Sylvester matrix equation AX-XF=BY and the regulator equation AX-XF=BY+R, Asian journal of control 9, No. 4, 475-483 (2007)
[21] Ding, F.; Chen, T.: Hierarchical gradient based identification of multivariable discrete-time systems, Automatica 41, No. 2, 315-325 (2005) · Zbl 1073.93012 · doi:10.1016/j.automatica.2004.10.010
[22] Ding, F.; Chen, T.: Hierarchical least squares identification methods for multivariable systems, IEEE transactions on automatic control 50, No. 3, 397-402 (2005)