An accelerated gradient based iterative algorithm for solving systems of coupled generalized Sylvester-transpose matrix equations

Journal of Linear and Topological Algebra

مقاله 4، دوره 08، شماره 02، شهریور 2019، صفحه 117-126   

نوع مقاله: Research Paper

نویسندگان

A‎. ‎M‎. ‎E‎. ‎ Bayoumi¹؛ M. A. Ramadan²؛ M. Nili Ahmadabadi^{* 3}

¹Department of Mathematics‎, ‎Faculty of Education‎, ‎Ain Shams University‎, ‎Cairo‎, ‎Egypt

²Department of Mathematics‎, ‎Faculty of Science‎, ‎Menoufia University‎, ‎Shebeen El‎- ‎Koom‎, ‎Egypt

³Department of Mathematics‎, ‎Najafabad Branch‎, ‎Islamic Azad University‎, ‎Najafabad‎, ‎Iran

چکیده

‎In this paper‎, ‎an accelerated gradient based iterative algorithm for solving systems of coupled generalized Sylvester-transpose matrix equations is proposed‎. ‎The convergence analysis of the algorithm is investigated‎. ‎We show that the proposed algorithm converges to the exact solution for any initial value under certain assumptions‎. ‎Finally‎, ‎some numerical examples are given to demonstrate the behavior of the proposed method and to support the theoretical results of this paper‎.

کلیدواژه‌ها

Coupled matrix equations؛ Frobenius norm؛ relaxation parameters؛ gradient algorithm

مراجع

[1] A. M. E. Bayoumi, M. A. Ramadan, An accelerated gradient based iterative algorithm for solving extended Sylvester-conjugate matrix equations, Trans. of Ins. Measur. Contr. (2016), DOI: 10.1177/0142331216658092.

[2] F. Ding, T. Chen, Gradient based iterative algorithms for solving a class of matrix equations, IEEE Trans. Autom. Control. 50 (2005), 1216-1221.

[3] F. Ding, T. Chen, Hierarchical gradient-based identification of multivariable discrete-time systems, Automatic. 41 (2005), 315-325.

[4] F. Ding,T. Chen, Hierarchical least squares identification methods for multivariable systems, IEEE Trans. Autom. Control. 50 (2005), 397-402.

[5] F. Ding, T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control. Optim. 44 (2006), 2269-2284.

[6] F. Ding, P. X. Liu, J. Ding, Iterative solutions of the generalized Sylvester matrix equation by using the hierarchical identification principle, Appl. Math. Comput. 197 (2008), 41-50.

[7] G. X. Huang, F. Yin, K. Guo, An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB = C, Appl. Math. Comput. 212 (2008), 231-244.

[8] Q. Niu, X. Wang. L. Z. Lu, A relaxed gradient based algorithm for solving Sylvester equations, Asian J. of Control. 13 (3) (2011), 461-464.

[9] M. A. Ramadan, T. S. El-Danaf, A. M. E. Bayoumi, A relaxed gradient based algorithm for solving extended Sylvester-conjugate matrix equations, Asian J. of Control. 16 (5) (2014), 1-8.

[10] X. Wang, L. Dai, D. Liao, A modified gradient based algorithm for solving Sylvester equations, Appl Math. Comput. 218 (2012), 5620-5628.

[11] A. G. Wu, G. Feng, G. R. Duan, Finite iterative solutions to a class of complex matrix equations with conjugate and transpose of the unknowns, Math. Comput. Model, 52 (9-10) (2010), 1463-1478.

[12] A. G. Wu, X. Zeng, G. R. Duan, Iterative solutions to the extended Sylvester-conjugate matrix equation, Appl. Math. Comput. 217 (1) (2010), 130-142.

[13] A. G. Wu, G. Feng, G. R. Duan, W. Liu, Iterative solutions to the Kalman-Yakubovich-conjugate matrix equation, Appl. Math. Comput. 217 (2011), 4427-4438.

[14] Y. J. Xie, C. F. Ma, The accelerated gradient based iterative algorithm for solving a class of generalized Sylvester-transpose matrix equation, Appl. Math. Comput. 237 (2017), 1257-1269.