Abstract
This paper deals with a general type of linear matrix equation problem. It presents new iterative algorithms to solve the matrix equations of the form A i X B i = F i . These algorithms are based on the incremental subgradient and the parallel subgradient methods. The convergence region of these algorithms are larger than other existing iterative algorithms. Finally, some experimental results are presented to show the efficiency of the proposed algorithms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Datta, B.N.: Numerical Methods for Linear Control Systems Design and Analysis. Elseiver, Amsterdam (2003)
Ding, F., Chen, T.W.: Gradient based iterative algorithms for solving a class of matrix equations. IEEE Trans. Automat. Control. 50, 1216–1221 (2005)
Ding, F., Chen, T.W.: On iterative solutions of general coupled matrix equations, SIAM. J. Control Optim. 44, 2269–2284 (2006)
Ding, F., Chen, T.W.: Iterative least squares solutions of coupled Sylvester matrix equations. Syst. Control Lett. 54, 95–107 (2005)
Ding, F., Chen, T.W.: Hierarchical gradient-based identification of multivariable discrete-time systems. Automation 41, 95–107 (2005)
Ding, F., Chen, T.W.: Hierarchical least squares identification methods for multivable systems. IEEE Trans. Automat. Control. 50, 397–402 (2005)
Ding, F., Chen, T.W.: Hierarchical identification of lifted state-space models for general dual-rate systems. IEEE Trans. Circ. Syst. 52, 1179–1187 (2005)
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)
Dehghan, M., Hajarian, M.: On the relexive solutions of the matrix equation A X B + C Y D = E. Bull. Korean Math. Soc. 46(3), 511–519 (2009)
Xie, L., Liu, Y.J., Yang, H.: Gradient based and least squares based iterative algorithms for matrix equations A X B + C X T D = F. Appl. Math. Comput. 217, 2191–2199 (2010)
Hajarian, M., Dehghan, M.: The generalized centro-symmetric and least squares generalized centro-symmetric solutions of the matrix equation AYB + C Y T D = E. Math. Meth. Appl. Sci. 34, 1562–1579 (2011)
Liang, K.F., Liu, J.: Iterative algorithms for the minimum-norm solution and the least-squares solution of the linear matrix equations. Appl. Math. Comput. 218, 3166–3175 (2011)
On Hermitian and skew-Hermitian splitting iteration methods for continous Sylvester equations. J. Comput. Math. 2, 185–198 (2011)
Wang, X., Dai, L., Liao, D.: A modified gradient based algorithm for solving Sylvester equations. Appl. Math. Comput. 218, 5620–5628 (2012)
Hansen, P.C., Nagy, J.M., O’Leary, D.P.: Deblurring Images: Matrices, Spectra, and Filtering. SIAM, Philadelphia (2006)
Ding, J., Liu, Y.J., Ding, F.: Iterative solutions to matrix equations of the form A i X B i = F i . Comput. Math. Appl. 59, 3500–3507 (2010)
Bertsekas, D.P.: A new class of incremental gradient methods for least squares problems. SIAM J. Optimiz. 7(4), 913–926 (1997)
Nedić, A., Bertsekas, D.P.: Incremental subgradient methods for nondifferentiable optimization. SIAM J. Optimiz. 12, 109–138 (2001)
Dos Santos, L.T.: A parallel subgradient projections method for the convex feasibility problem. J. Comput. Appl. Math. 18(3), 307–320 (1987)
Combettes, P.L., Puh, H.: Iterations of parallel convex projections in Hilbert spaces. Numer. Funct. Anal. Optimiz. 15, 225–243 (1994)
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. J. Comput. Appl. Math. 212, 231–244 (2008)
Liang, M.L., You, C.H., Dai, L.F.: An efficient algorithm for the generalized centro-symmetric solution of matrix equation A X B = C. Numer. Algor. 44, 173–184 (2007)
Peng, Z.Y.: New matrix iterative methods for constraint solutions of the matrix equation A X B = C. J. Comput. Appl. Math. 235(3), 726–735 (2010)
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)
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). Appl. Math. Comput. 189(2), 1350–1358 (2007)
Cai, J., Chen, G.L.: An iterative algorithm for the least squares bisymmetric solutions of the matrix equations \(A_{1} X B_{1} = C_{1}, A_{2} X B_{2} = C_{2}\). Math. Comput. Model. 50, 1237–1244 (2009)
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. Comput. Math. Appl. 56, 3246–3260 (2008)
Ruszxzynski, A.: Nonlinear Optimization. Princeton University Press, New Jersey (2006)
Polyak, B.T.: Introduction to Optimization. Optimization Softwarse, New York (1987)
Su, M., Xu, H.K.: Remarks on the gradient-projection algorithm. J. Nonlinear Anal. Opt. 1(1), 35–43 (2010)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert spaces. Springer, New York (2011)
Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Bauschke, H.H., Burachik, R., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer, New York (2009)
Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 20, 103–120 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tang, Y., Peng, J. & Yue, S. Cyclic and simultaneous iterative methods to matrix equations of the form A i X B i = F i . Numer Algor 66, 379–397 (2014). https://doi.org/10.1007/s11075-013-9740-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-013-9740-9