Weighted least squares solutions to general coupled Sylvester matrix equations. (English) Zbl 1161.65034

The paper is devoted to the weighted least squares solutions problem for general coupled Sylvester matrix equations. By adopting the gradient search principle in optimization theory, the authors provide a general gradient based iterative algorithm to solve this problem. The method used in the paper is quite different from the well known one. Especially, the authors introduce a necessary and sufficient condition to guarantee the convergence of the proposed algorithm. Based on this necessary and sufficient condition, a sufficient but wieldy condition can be induced. In deriving some of the results, they discover that a related result in the literature is in fact incorrect. Moreover, they suggest a method to choose the optimal step size in the algorithm, such that the proposed iteration converges fastest. They also provide several numerical examples to show the effectiveness of the proposed approach.
The paper is organized as follows. The problem formulation and some preliminary results are given in the beginning. The main results of the paper are shown next. Then, numerical examples are provided to illustrate the effectiveness of the proposed algorithm. Some concluding remarks are finally given.
The paper will be useful for all students, specialists and researchers, working in the area of weighted least squares solutions for coupled Sylvester matrix equations. It also is suitable for practitioners who wish to brush up these fundamental concepts.


65F30 Other matrix algorithms (MSC2010)
15A24 Matrix equations and identities
65F20 Numerical solutions to overdetermined systems, pseudoinverses


Full Text: DOI


[1] Wang, D., Some topics on weighted Moore-Penrose inverse, weighted least squares and weighted regularized Tikhonov problems, Applied mathematics and computation, 157, 243-267, (2004) · Zbl 1064.65032
[2] Kılıcman, Adem; Abdel Aziz Al Zhour, Zeyad, Vector least-squares solutions for coupled singular matrix equations, Journal of computational and applied mathematics, 206, 1051-1069, (2007) · Zbl 1132.65034
[3] Zhou, B.; Duan, G.R., An explicit solution to the matrix equation \(A X - X F = B Y\), Linear algebra and its applications, 402, 1, 345-366, (2005)
[4] Zhou, B.; Duan, G.R., A new solution to the generalized Sylvester matrix equation \(A V - E V F = B W\), Systems & control letters, 55, 3, 193-198, (2006)
[5] Duan, G.R.; Zhou, B., Solution to the second-order Sylvester matrix equation \(M V F^2 + D V F + K V = B W\), IEEE transactions on automatic control, 51, 5, 805-809, (2006)
[6] Zhou, B.; Duan, G.R., On the generalized Sylvester mapping and matrix equations, Systems & control letters, 57, 3, 200-208, (2008) · Zbl 1129.93018
[7] Zhou, B.; Duan, G.R., Solutions to generalized Sylvester matrix equation by Schur decomposition, International journal of systems science, 38, 5, 369-375, (2007) · Zbl 1126.65034
[8] Peng, Z.H.; Hu, X.Y.; Zhang, L., The bisymmetric solutions of the matrix equation \(A_1 X_1 B_1 + A_2 X_2 B_2 + \cdots + A_l X_l B_l = C\), Linear algebra and its applications, 426, 583-595, (2007)
[9] T. Stykel, Solving projected generalized Lyapunov equations using SLICOT, in: 2006 IEEE International Symposium on Computer-Aided Control Systems Design, pp. 14-18
[10] Chen, T.; Francis, B.A., Optimal sampled-date control systems, (1995), Springer Londom
[11] Costa, O.L.V.; Fragoso, M.D., Stability results for discrete-time linear systems with Markovian jumping parameters, Journal of mathematical analysis and applications, 179, 1, 154-178, (1993) · Zbl 0790.93108
[12] Borno, I.; Gajic, Z., Parallel algorithm for solving coupled algebraic Lyapunov equations of discrete-time jump linear systems, Computers and mathematics with applications, 30, 7, 1-4, (1995) · Zbl 0837.93075
[13] Wang, Q.; Lam, J.; Wei, Y.; Chen, T., Iterative solutions of coupled discrete Markovian jump Lyapunov equations, Computers and mathematics with applications, 55, 4, 843-850, (2008) · Zbl 1139.60334
[14] Chen, J.L.; Chen, X.H., Special matrices, (2002), Tsinghua University Press, (in Chinese)
[15] Kelley, C.T., Iterative methods for linear and nonlinear equations, (1995), SIAM Philadelphia · Zbl 0832.65046
[16] Zhou, K.; Doyle, J.; Glover, K., Robust and optimal control, (1996), Prentice-Hell
[17] Mouroutsos, G.; Sparis, P.D., Taylor series approach to system identification, analysis and optimal control, Journal of franklin institute pergamon, 319, 3, 359-371, (1985) · Zbl 0561.93018
[18] Huang, G.X.; Yin, F.; Guo, K., An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation \(A X B = C\), Journal of computational and applied mathematics, 212, 2, 231-244, (2008) · Zbl 1146.65036
[19] Peng, X.Y.; Hu, X.Y.; Zhang, L., The reflexive and anti-reflexive solutions of the matrix equation \(A^{\operatorname{H}} X B = C\), Journal of computational and applied mathematics, 200, 2, 749-760, (2007) · Zbl 1115.15014
[20] Thiran, J.P.; Matelart, M.; Le Bailly, B., On the generalized ADI method for the matrix equation \(X - A X B = C\), Journal of computational and applied mathematics, 156, 2, 285-302, (2003) · Zbl 1032.65041
[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.; 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
[23] 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
[24] 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
[25] Sheng, X.P.; Chen, G.L., A finite iterative method for solving a pair of linear matrix equations \((A X B, C X D) = (E, F)\), Applied mathematics and computation, 189, 1350-1358, (2007) · Zbl 1133.65026
[26] Peng, Z.Y., An iterative method for the least squares symmetric solution of the linear matrix equation \(A X B = C\), Applied mathematics and computation, 170, 711-723, (2005) · Zbl 1081.65039
[27] Peng, Z.H.; Hu, X.Y.; Zhang, L., An effective algorithm for the least-squares reflexive solution of the matrix equation \(A_1 X B_1 = C_1, A_2 X B_2 = C_2\), Applied mathematics and computation, 181, 988-999, (2006) · Zbl 1115.65048
[28] Qiu, Y.Y.; Zhang, Z.Y.; Lu, J.F., Matrix iterative solutions to the least squares problem of \(B X A^{\operatorname{T}} = F\) with some linear constraints, Applied mathematics and computation, 185, 284-300, (2007)
[29] De Pierro, A.R.; Wei, M., Some new properties of the equality constrained and weighted least squares problem, Linear algebra and its applications, 320, 145-165, (2000) · Zbl 0985.65030
[30] Wang, G.R.; Zheng, B., The weighted generalized inverses of a partitioned matrix, Linear algebra and its applications, 320, 145-165, (2000)
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