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An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices. (English) Zbl 1185.65054

Summary: We present a general family of iterative methods to solve linear equations, which includes the well-known Jacobi and Gauss-Seidel iterations as its special cases. The methods are extended to solve coupled Sylvester matrix equations. In our approach, we regard the unknown matrices to be solved as the system parameters to be identified, and propose a least-squares iterative algorithm by applying a hierarchical identification principle and by introducing the block-matrix inner product (the star product for short). We prove that the iterative solution consistently converges to the exact solution for any initial value. The algorithms proposed require less storage capacity than the existing numerical ones. Finally, the algorithms are tested on computer and the results verify the theoretical findings.

MSC:

65F10 Iterative numerical methods for linear systems
15A24 Matrix equations and identities
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[1] Wang, Q.W., Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. math. appl., 49, 641-650, (2005) · Zbl 1138.15003
[2] Wang, Q.W.; Sun, J.H.; Li, S.Z., Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra, Linear algebra appl., 353, 169-182, (2002) · Zbl 1004.15017
[3] Wang, Q.W.; Zhang, F., The reflexive re-nonnegative definite solution to a quaternion matrix equation, Electron. J. linear algebra, 17, 88-101, (2008) · Zbl 1147.15012
[4] Wang, Q.W.; Zhang, H.S.; Yu, S.W., On solutions to the quaternion matrix equation AXB+CY D=E, Electron. J. linear algebra, 17, 343-358, (2008) · Zbl 1154.15019
[5] Wang, Q.W.; Chang, H.X.; Ning, Q., The common solution to six quaternion matrix equations with applications, Appl. math. comput., 198, 209-226, (2008) · Zbl 1141.15016
[6] Wang, Q.W.; Li, C.K., Ranks and the least-norm of the general solution to a system of quaternion matrix equations, Linear algebra appl., 430, 1626-1640, (2009) · Zbl 1158.15010
[7] Zhou, B.; Duan, G.R.; Li, Z.Y., Gradient based iterative algorithm for solving coupled matrix equations, Syst. contr. lett., 58, 327-333, (2009) · Zbl 1159.93323
[8] Huang, G.X.; Yin, F.; Guo, K., An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=C, J. comput. appl. math., 212, 231-244, (2008) · Zbl 1146.65036
[9] Ding, F.; Liu, P.X.; Ding, J., Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Appl. math. comput., 197, 41-50, (2008) · Zbl 1143.65035
[10] Wang, Q.W., A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity, Linear algebra appl., 384, 43-54, (2004) · Zbl 1058.15015
[11] Piao, F.; Zhang, Q.; Wang, Z., The solution to matrix equation \(\mathit{AX} + X^T C = B\), J. franklin inst., 344, 1056-1062, (2007) · Zbl 1171.15015
[12] Starke, G.; Niethammer, W., SOR for AX-XB=C, Linear algebra appl., 154, 355-375, (1991) · Zbl 0736.65031
[13] Kitagawa, G., An algorithm for solving the matrix equation \(X = \mathit{FXF}^T + S\), Int. J. contr., 25, 745-753, (1977) · Zbl 0364.65025
[14] Kleinman, D.L.; Rao, P.K., Extensions to the bartels-stewart algorithm for linear matrix equation, IEEE trans. autom. contr., 23, 85-87, (1978) · Zbl 0369.65005
[15] Golub, G.H.; Nash, S.; Van Loan, C., A Hessenberg-Schur method for the problem AX+XB=C, IEEE trans. automat. contr., 24, 909-913, (1979) · Zbl 0421.65022
[16] Ding, F.; Chen, T., Gradient based iterative algorithms for solving a class of matrix equations, IEEE trans. autom. contr., 50, 1216-1221, (2005) · Zbl 1365.65083
[17] Ding, F.; Chen, T., Iterative least squares solutions of coupled Sylvester matrix equations, Syst. contr. lett., 54, 95-107, (2005) · Zbl 1129.65306
[18] Ding, F.; Chen, T., On iterative solutions of general coupled matrix equations, SIAM J. contr. optim., 44, 2269-2284, (2006) · Zbl 1115.65035
[19] Hu, D.Y.; Reichel, L., Krylov subspace methods for the Sylvester equation, Linear algebra appl., 174, 283-314, (1992) · Zbl 0777.65028
[20] Jaimoukha, I.M.; Kasenally, E.M., Krylov subspace methods for solving large Lyapunov equations, SIAM J. matrix anal. appl., 31, 227-251, (1994) · Zbl 0798.65060
[21] Jaimoukha, I.M.; Kasenally, E.M., Oblique projection methods for large scale model reduction, SIAM J. matrix anal. appl., 16, 602-627, (1995) · Zbl 0827.65073
[22] Bao, L.; Lin, Y.; Wei, Y., A new projection method for solving large Sylvester equations, Appl. numer. math., 57, 521-532, (2007) · Zbl 1118.65028
[23] Zhou, B.; Duan, G.R., A new solution to the generalized Sylvester matrix equation AV-EVF=BW, Syst. contr. lett., 55, 193-198, (2006) · Zbl 1129.15300
[24] Zhou, B.; Duan, G.R., On the generalized Sylvester mapping and matrix equations, Syst. contr. lett., 57, 200-208, (2008) · Zbl 1129.93018
[25] Robbé, M.; Sadkane, M., Use of near-breakdowns in the block Arnoldi method for solving large Sylvester equations, Appl. numer. math., 58, 486-498, (2008) · Zbl 1136.65046
[26] Zhou, B.; Li, Z.Y.; Duan, G.R.; Wang, Y., Solutions to a family of matrix equations by using the Kronecker matrix polynomials, Appl. math. comput., 212, 327-336, (2009) · Zbl 1181.15020
[27] Zhou, B.; Duan, G.R., An explicit solution to the matrix equation AX-XF=BY, Linear algebra appl., 402, 345-366, (2005) · Zbl 1076.15016
[28] Zhou, B.; Yan, Z.B., Solutions to right coprime factorizations and generalized Sylvester matrix equations, Trans. inst. meas. contr., 30, 397-426, (2008)
[29] Zhou, B.; Duan, G.R., Solutions to generalized Sylvester matrix equation by Schur decomposition, Int. J. syst. sci., 38, 369-375, (2007) · Zbl 1126.65034
[30] Zhou, B.; Li, Z.Y.; Duan, G.R.; Wang, Y., Weighted least squares solutions to general coupled Sylvester matrix equations, J. comput. appl. math., 224, 759-776, (2009) · Zbl 1161.65034
[31] Zhou, B.; Lam, J.; Duan, G.R., On Smith-type iterative algorithms for the Stein matrix equation, Appl. math. lett., 22, 1038-1044, (2009) · Zbl 1179.15016
[32] Ding, F.; Chen, T., Hierarchical gradient-based identification of multivariable discrete-time systems, Automatica, 41, 315-325, (2005) · Zbl 1073.93012
[33] Ding, F.; Chen, T., Hierarchical least squares identification methods for multivariable systems, IEEE trans. autom. contr., 50, 397-402, (2005) · Zbl 1365.93551
[34] Dehghan, M.; Hajarian, M., An iterative algorithm for solving a pair of matrix equations AYB=E, CYD=F over generalized centro-symmetric matrices, Comput. math. appl., 56, 3246-3260, (2008) · Zbl 1165.15301
[35] Dehghan, M.; Hajarian, M., An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation, Appl. math. comput., 202, 571-588, (2008) · Zbl 1154.65023
[36] Dehghan, M.; Hajarian, M., On the reflexive solutions of the matrix equation AXB+CYD=E, Bull. Korean math. soc., 46, 511-519, (2009) · Zbl 1170.15004
[37] Dehghan, M.; Hajarian, M., Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation \(A_1 X_1 B_1 + A_2 X_2 B_2 = C\), Math. comput. model., 49, 1937-1959, (2009) · Zbl 1171.15310
[38] M. Dehghan, M. Hajarian, The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations, Rocky Mountain J. Math., in press. · Zbl 1198.15011
[39] Dehghan, M.; Hajarian, M., Efficient iterative method for solving the second-order Sylvester matrix equation \(\mathit{EVF}^2 - \mathit{AVF} - \mathit{CV} = \mathit{BW}\), IET control theory appl., 3, 1401-1408, (2009)
[40] M. Dehghan, M. Hajarian, On the reflexive and anti-reflexive solutions of the generalized coupled Sylvester matrix equations, Int. J. Syst. Sci., in press. · Zbl 1196.65081
[41] Peng, Y.X.; Hu, X.Y.; Zhang, L., An iteration method for the symmetric solutions and the optimal appromation solution of the matrix equation AXB=C, Appl. math. comput., 160, 763-777, (2005) · Zbl 1068.65056
[42] Liang, M.L.; You, C.H.; Dai, L.F., An efficient algorithm for the generalized centro-symmetric solution of matrix equation \(\mathit{AXB} = C\), Numer. algor., 4, 173-184, (2007) · Zbl 1129.65030
[43] Zhou, B.; Duan, G.R., Closed-form solutions to the matrix equation AX - EXF = BY with F in companion form, Int. J. autom. comput., 6, 204-209, (2009)
[44] Ramadan, M.A., Necessary and suffcient conditions for the existence of positive definite solutions of the matrix equation X + ATX−2A = I, Int. J. comput. math., 82, 865-870, (2005) · Zbl 1083.15020
[45] Ramadan, M.A.; Danaf, T.S.E.; Shazly, N.M.E., Iterative positive definite solutions of the two nonlinear matrix equations X±ATX−2A = I, Appl. math. comput., 164, 189-200, (2005)
[46] Ramadan, M.A.; Shazly, N.M.E., On the matrix equation \(X + A^T \sqrt[2^m]{X^{- 1}} A = I\), Appl. math. comput., 173, 992-1013, (2006) · Zbl 1089.65037
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