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An iterative algorithm for the least squares generalized reflexive solutions of the matrix equations $$AXB = E$$, $$CXD = F$$. (English) Zbl 1242.65085
Summary: The generalized coupled Sylvester systems play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. The iterative method is an important way to solve the generalized coupled Sylvester systems. In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem: $$||(\binom{AXB}{CXD}) - (\binom EF)|| = \min$$ over a generalized reflexive matrix $$X$$. For any initial generalized reflexive matrix $$X_1$$, by the iterative algorithm, the generalized reflexive solution $$X^\ast$$ can be obtained within finite iterative steps in the absence of round-off errors, and the unique least-norm generalized reflexive solution $$X^\ast$$ can also be derived when an appropriate initial iterative matrix is chosen. Furthermore, the unique optimal approximate solution $$\hat X$$ to a given matrix $$X_0$$ in Frobenius norm can be derived by finding the least-norm generalized reflexive solution $$\widetilde X^\ast$$ of a new corresponding minimum Frobenius norm residual problem: $$\min || (\binom{A\widetilde{X}B}{C\widetilde{X}D}) - (\binom{\widetilde{E}}{\widetilde{F}})$$ with $$\widetilde{E} = E - AX_0B, \widetilde{F} = F - CX_0D$$. Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.

##### MSC:
 65F30 Other matrix algorithms (MSC2010) 65F10 Iterative numerical methods for linear systems 15A24 Matrix equations and identities
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##### References:
 [1] F. Li, X. Hu, and L. Zhang, “The generalized reflexive solution for a class of matrix equations AX=B; XC=D,” Acta Mathematica Scientia Series B, vol. 28, no. 1, pp. 185-193, 2008. · Zbl 1150.15006 [2] J.-C. Zhang, X.-Y. Hu, and L. Zhang, “The (P,Q) generalized reflexive and anti-reflexive solutions of the matrix equation AX=B,” Applied Mathematics and Computation, vol. 209, no. 2, pp. 254-258, 2009. · Zbl 1168.15008 [3] B. Zhou, Z.-Y. Li, G.-R. Duan, and Y. Wang, “Weighted least squares solutions to general coupled Sylvester matrix equations,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 759-776, 2009. · Zbl 1161.65034 [4] A.-P. Liao and Y. Lei, “Least-squares solution with the minimum-norm for the matrix equation (AXB,GXH)=(C,D),” Computers & Mathematics with Applications, vol. 50, no. 3-4, pp. 539-549, 2005. · Zbl 1087.65040 [5] F. Ding, P. X. Liu, and J. Ding, “Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 41-50, 2008. · Zbl 1143.65035 [6] F. Ding and T. Chen, “On iterative solutions of general coupled matrix equations,” SIAM Journal on Control and Optimization, vol. 44, no. 6, pp. 2269-2284, 2006. · Zbl 1115.65035 [7] L. Xie, J. Ding, and F. Ding, “Gradient based iterative solutions for general linear matrix equations,” Computers & Mathematics with Applications, vol. 58, no. 7, pp. 1441-1448, 2009. · Zbl 1189.65083 [8] F. Ding and T. Chen, “Gradient based iterative algorithms for solving a class of matrix equations,” IEEE Transactions on Automatic Control, vol. 50, no. 8, pp. 1216-1221, 2005. · Zbl 1365.65083 [9] J. Ding, Y. Liu, and F. Ding, “Iterative solutions to matrix equations of the form AiXBi=Fi,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3500-3507, 2010. · Zbl 1197.15009 [10] F. Ding and T. Chen, “Iterative least-squares solutions of coupled Sylvester matrix equations,” Systems & Control Letters, vol. 54, no. 2, pp. 95-107, 2005. · Zbl 0746.15015 [11] F. Ding and T. Chen, “Hierarchical gradient-based identification of multivariable discrete-time systems,” Automatica, vol. 41, no. 2, pp. 315-325, 2005. · Zbl 1073.93012 [12] F. Ding and T. Chen, “Hierarchical least squares identification methods for multivariable systems,” IEEE Transactions on Automatic Control, vol. 50, no. 3, pp. 397-402, 2005. · Zbl 1365.93551 [13] F. Ding and T. Chen, “Hierarchical identification of lifted state-space models for general dual-rate systems,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 52, no. 6, pp. 1179-1187, 2005. · Zbl 1374.93342 [14] Y.-X. Peng, X.-Y. Hu, and L. Zhang, “An iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations,” Applied Mathematics and Computation, vol. 183, no. 2, pp. 1127-1137, 2006. · Zbl 1134.65032 [15] X. Sheng and G. Chen, “A finite iterative method for solving a pair of linear matrix equations (AXB,CXD)=(E,F),” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1350-1358, 2007. · Zbl 1133.65026 [16] Z.-H. Peng, X.-Y. Hu, and L. Zhang, “An efficient algorithm for the least-squares reflexive solution of the matrix equation A1XB1=C1, A2XB2=C2,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 988-999, 2006. · Zbl 1115.65048 [17] M. Dehghan and M. Hajarian, “An iterative algorithm for solving a pair of matrix equations AYB=E, CYD=F over generalized centro-symmetric matrices,” Computers & Mathematics with Applications, vol. 56, no. 12, pp. 3246-3260, 2008. · Zbl 1165.15301 [18] J. Cai and G. Chen, “An iterative algorithm for the least squares bisymmetric solutions of the matrix equations A1XB1=C1; A2XB2=C2,” Mathematical and Computer Modelling, vol. 50, no. 7-8, pp. 1237-1244, 2009. · Zbl 1190.65061 [19] A. Kılı\ccman and Z. A. A. A. Zhour, “Vector least-squares solutions for coupled singular matrix equations,” Journal of Computational and Applied Mathematics, vol. 206, no. 2, pp. 1051-1069, 2007. · Zbl 1137.65070 [20] M. Dehghan and M. Hajarian, “An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices,” Applied Mathematical Modelling, vol. 34, no. 3, pp. 639-654, 2010. · Zbl 1185.65054 [21] A.-G. Wu, G. Feng, G.-R. Duan, and W.-J. Wu, “Iterative solutions to coupled Sylvester-conjugate matrix equations,” Computers & Mathematics with Applications, vol. 60, no. 1, pp. 54-66, 2010. · Zbl 1198.65083 [22] A.-G. Wu, B. Li, Y. Zhang, and G.-R. Duan, “Finite iterative solutions to coupled Sylvester-conjugate matrix equations,” Applied Mathematical Modelling, vol. 35, no. 3, pp. 1065-1080, 2011. · Zbl 1211.15024 [23] A.-G. Wu, G. Feng, G.-R. Duan, and W.-J. Wu, “Finite iterative solutions to a class of complex matrix equations with conjugate and transpose of the unknowns,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1463-1478, 2010. · Zbl 1205.15027 [24] I. Jonsson and B. Kågström, “Recursive blocked algorithm for solving triangular systems. I. One-sided and coupled Sylvester-type matrix equations,” ACM Transactions on Mathematical Software, vol. 28, no. 4, pp. 392-415, 2002. · Zbl 1072.65061 [25] I. Jonsson and B. Kågström, “Recursive blocked algorithm for solving triangular systems. II. Two-sided and generalized Sylvester and Lyapunov matrix equations,” ACM Transactions on Mathematical Software, vol. 28, no. 4, pp. 416-435, 2002. · Zbl 1072.65062 [26] G. X. Huang, N. Wu, F. Yin, Z. L. Zhou, and K. Guo, “Finite iterative algorithms for solving generalized coupled Sylvester systems-part I: one-sided and generalized coupled Sylvester matrix equations over generalized reflexive solutions,” Applied Mathematical Modelling, vol. 36, no. 4, pp. 1589-1603, 2012. · Zbl 1243.65045 [27] F. Yin, G. X. Huang, and D. Q. Chen, “Finite iterative algorithms for solving generalized coupled Sylvester systems-Part II: two-sided and generalized coupled Sylvester matrix equations over reflexive solutions,” Applied Mathematical Modelling, vol. 36, no. 4, pp. 1604-1614, 2012. · Zbl 1243.65046 [28] M. Dehghan and M. Hajarian, “The general coupled matrix equations over generalized bisymmetric matrices,” Linear Algebra and Its Applications, vol. 432, no. 6, pp. 1531-1552, 2010. · Zbl 1187.65042 [29] M. Dehghan and M. Hajarian, “Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations,” Applied Mathematical Modelling, vol. 35, no. 7, pp. 3285-3300, 2011. · Zbl 1227.65037 [30] G.-X. Huang, F. Yin, and K. Guo, “An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=C,” Journal of Computational and Applied Mathematics, vol. 212, no. 2, pp. 231-244, 2008. · Zbl 1146.65036 [31] Z.-Y. Peng, “New matrix iterative methods for constraint solutions of the matrix equation AXB=C,” Journal of Computational and Applied Mathematics, vol. 235, no. 3, pp. 726-735, 2010. · Zbl 1206.65145 [32] Z.-Y. Peng and X.-Y. Hu, “The reflexive and anti-reflexive solutions of the matrix equation AX=B,” Linear Algebra and Its Applications, vol. 375, pp. 147-155, 2003. · Zbl 1050.15016 [33] Z.-h. Peng, X.-Y. Hu, and L. Zhang, “An efficient algorithm for the least-squares reflexive solution of the matrix equation A1XB1=C1, A2XB2=C2,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 988-999, 2006. · Zbl 1115.65048 [34] X. Sheng and G. Chen, “An iterative method for the symmetric and skew symmetric solutions of a linear matrix equation AXB+CYD=E,” Journal of Computational and Applied Mathematics, vol. 233, no. 11, pp. 3030-3040, 2010. · Zbl 1190.65071 [35] Q.-W. Wang, J.-H. Sun, and S.-Z. Li, “Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra,” Linear Algebra and Its Applications, vol. 353, pp. 169-182, 2002. · Zbl 1004.15017 [36] Q.-W. Wang, “A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity,” Linear Algebra and Its Applications, vol. 384, pp. 43-54, 2004. · Zbl 1058.15015 [37] Q.-W. Wang, “Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations,” Computers & Mathematics with Applications, vol. 49, no. 5-6, pp. 641-650, 2005. · Zbl 1138.15003 [38] A.-G. Wu, G.-R. Duan, and Y. Xue, “Kronecker maps and Sylvester-polynomial matrix equations,” IEEE Transactions on Automatic Control, vol. 52, no. 5, pp. 905-910, 2007. · Zbl 1366.93190 [39] A.-G. Wu, G. Feng, G.-R. Duan, and W.-J. Wu, “Closed-form solutions to Sylvester-conjugate matrix equations,” Computers & Mathematics with Applications, vol. 60, no. 1, pp. 95-111, 2010. · Zbl 1198.15013 [40] Y. Yuan and H. Dai, “Generalized reflexive solutions of the matrix equation AXB=D and an associated optimal approximation problem,” Computers & Mathematics with Applications, vol. 56, no. 6, pp. 1643-1649, 2008. · Zbl 1155.15301 [41] A. Antoniou and W.-S. Lu, Practical Optimization: Algorithm and Engineering Applications, Springer, New York, NY, USA, 2007. · Zbl 1128.90001
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