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Iterative solutions to matrix equations of the form \(A_{i}XB_{i}=F_{i}\). (English) Zbl 1197.15009

Summary: This paper is concerned with the numerical solutions to the linear matrix equations \(A_{1}XB_{1}=F_{1}\) and \(A_{2}XB_{2}=F_{2}\); two iterative algorithms are presented to obtain the solutions. For any initial value, it is proved that the iterative solutions obtained by the proposed algorithms converge to their true values. Finally, simulation examples are given to verify the proposed convergence theorems.

MSC:

15A24 Matrix equations and identities
65F30 Other matrix algorithms (MSC2010)
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References:

[1] Zheng, B.; Ye, L.; Cvetkovic-Ilic, D. S., The congruence class of the solutions of some matrix equations, Computers & Mathematics with Applications, 57, 4, 540-549 (2009) · Zbl 1165.15303
[2] Xie, L.; Ding, J.; Ding, F., Gradient based iterative solutions for general linear matrix equations, Computers & Mathematics with Applications, 58, 7, 1441-1448 (2009) · Zbl 1189.65083
[3] Zhou, B.; Li, Z. Y.; Duan, G. R.; Wang, Y., Weighted least squares solutions to general coupled Sylvester matrix equations, Journal of Computational and Applied Mathematics, 224, 2, 759-776 (2009) · Zbl 1161.65034
[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] 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
[6] Dehghan, M.; Hajarian, M., Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation \(A_1 X_1 B_1 + A_2 X_2 B_2 = C\), Mathematical and Computer Modelling, 49, 9-10, 1937-1959 (2009) · Zbl 1171.15310
[7] Qiu, J. Q.; Jiang, W. H.; Shi, Y.; Xi, D. M., Positive solutions for nonlinear \(n\) th-order \(m\)-point boundary value problem with the first derivative, International Journal of Innovative Computing, Information and Control, 5, 8, 2405-2414 (2009)
[8] Zhong, X. Z.; Zhang, T.; Shi, Y., Oscillation and nonoscillation of neutral difference equation with positive and negative coefficients, International Journal of Innovative Computing, Information and Control, 5, 5, 1329-1342 (2009)
[9] Yu, C., Existence and uniqueness of the solution for FM-BEM based on GMRES(m) algorithm, ICIC Express Letters, 2, 1, 89-93 (2008)
[10] Liu, J.; Yu, C.; Chen, Y.; Li, Xia, Computational formulations for the fundamental solution and kernel functions of elasto-plastic FM-BEM in spherical coordinate system, ICIC Express Letters, 2, 2, 207-212 (2008)
[11] Navarra, A.; Odell, P. L.; Young, D. M., A representation of the general common solution to the matrix equations \(A_1 X B_1 = C_1\) and \(A_2 X B_2 = C_2\) with applications, Computers & Mathematics with Applications, 41, 7-8, 929-935 (2001) · Zbl 0983.15016
[12] Liao, A.; Lei, Y., Least-squares solution with the minimum-norm for the matrix equation \((A X B, G X H) = (C, D)\), Computers & Mathematics with Applications, 50, 3-4, 539-549 (2005) · Zbl 1087.65040
[13] Liu, Y. H., Ranks of least squares solutions of the matrix equation \(A X B = C\), Computers & Mathematics with Applications, 55, 6, 1270-1278 (2008) · Zbl 1157.15014
[14] Yuan, S.; Liao, A.; Lei, Y., Least squares Hermitian solution of the matrix equation \((A X B, C X D) = (E, F)\) with the least norm over the skew field of quaternions, Mathematical and Computer Modelling, 48, 1-2, 91-100 (2008) · Zbl 1145.15303
[15] Wang, D. Q.; Ding, F., Extended stochastic gradient identification algorithms for Hammerstein-Wiener ARMAX systems, Computers & Mathematics with Applications, 56, 12, 3157-3164 (2008) · Zbl 1165.65308
[16] Ding, J.; Ding, F., The residual based extended least squares identification method for dual-rate systems, Computers & Mathematics with Applications, 56, 6, 1479-1487 (2008) · Zbl 1155.93435
[17] Han, L. L.; Ding, F., Multi-innovation stochastic gradient algorithms for multi-input multi-output systems, Digital Signal Processing, 19, 4, 545-554 (2009)
[18] Dehghan, M.; Hajarian, M., An iterative algorithm for solving a pair of matrix equations \(A Y B = E, C Y D = F\) over generalized centro-symmetric matrices, Computers & Mathematics with Applications, 56, 12, 3246-3260 (2008) · Zbl 1165.15301
[19] Cai, J.; Chen, G., An iterative algorithm for the least squares bisymmetric solutions of the matrix equations \(A_1 X B_1 = C_1\) and \(A_2 X B_2 = C_2\), Mathematical and Computer Modelling, 50, 7-8, 1237-1244 (2009) · Zbl 1190.65061
[20] Ding, F.; Chen, T., Iterative least squares solutions of coupled Sylvester matrix equations, Systems & Control Letters, 54, 2, 95-107 (2005) · Zbl 1129.65306
[21] 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
[22] 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
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