Finite Iterative Algorithm for Solving a Class of Complex Matrix Equation with Two Unknowns of General Form
Applied and Computational Mathematics
Volume 3, Issue 5, October 2014, Pages: 273-284
Received: Oct. 24, 2014; Accepted: Nov. 6, 2014; Published: Nov. 20, 2014
Views 2574      Downloads 123
Authors
Mohamed A. Ramadan, Department of Mathematics, Faculty of Science, Menoufia University, Shebeen El- Koom, Egypt
Mokhtar A. Abdel Naby, Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt
Talaat S. El-Danaf, Department of Mathematics, Faculty of Science, Menoufia University, Shebeen El- Koom, Egypt
Ahmed M. E. Bayoumi, Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt
Article Tools
Follow on us
Abstract
This paper is concerned with an efficient iterative algorithm to solve general the Sylvester-conjugate matrix equation of the form ∑_(i= 1)^s▒〖A_i V B_i 〗+ ∑_(j=1)^t▒〖C_j W D_j 〗=∑_(l=1)^m▒〖E_1 V ̅ 〗 F_1+C The proposed algorithm is an extension to our proposed general Sylvester-conjugate equation of the form ∑_(i= 1)^s▒〖A_i V 〗+ ∑_(j=1)^t▒〖B_j W 〗=∑_(l=1)^m▒〖E_1 V ̅ 〗 F_1+C When a solution exists for this matrix equation, for any initial matrices, the solutions can be obtained within finite iterative steps in the absence of round off errors. Some lemmas and theorems are stated and proved where the iterative solutions are obtained. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm.
Keywords
General Sylvester-Conjugate matrix Equations, Finite Iterative Algorithm, Orthogonality, Inner Product Space, Frobenius norm
To cite this article
Mohamed A. Ramadan, Mokhtar A. Abdel Naby, Talaat S. El-Danaf, Ahmed M. E. Bayoumi, Finite Iterative Algorithm for Solving a Class of Complex Matrix Equation with Two Unknowns of General Form, Applied and Computational Mathematics. Vol. 3, No. 5, 2014, pp. 273-284. doi: 10.11648/j.acm.20140305.23
References
[1]
Dehghan, M., Hajarian, M. (2008), An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation, Appl. Math. Comput., 202 , pp 571–588.
[2]
Dehghan, M., Hajarian, M. (2008), An iterative algorithm for solving a pair of matrix equations over generalized Centro-symmetric matrices, Comput. Math. Appl. ,56, pp 3246–3260.
[3]
Wang, Q.W. (2005), Bisymmetric and Centro symmetric solutions to systems of real quaternion matrix equations, Comput. Math. Appl., 49, pp 641–650.
[4]
Wang, Q.W., Zhang, F. (2008), The reflexive re-nonnegative definite solution to a quaternion matrix equation, Electron. J. Linear Algebra ,17, pp88–101.
[5]
Wang, Q.W., Zhang, H.S., Yu, S.W. (2008), On solutions to the quaternion matrix equation , Electron. J. Linear Algebra, 17, 343–358.
[6]
Ding,F. , Liu, P.X. , Ding, J. (2008) ,Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Appl. Math. Comput. ,197 ,pp 41–50.
[7]
Peng, X.Y. , Hu, X.Y, Zhang, L. (2007) , The reflexive and anti-reflexive solutions of the matrix equation , J. Comput. Appl. Math. ,186, pp638–645.
[8]
Ramadan, M. A., Abdel Naby, M A. , Bayoumi, A. M. E. (2009), On the explicit solution of the Sylvester and the yakubovich matrix equations, Math. Comput. Model. ,50,pp1400-1408.
[9]
Wu, A. G., Duan, G.R., Yu, H.H. (2006), On solutions of the matrix equations and , Appl. Math. Comput. ,183 ,pp932–941.
[10]
Zhou, B., Li, Z.Y., Duan, G.R., Wang, Y. (2009), Weighted least squares solutions to general coupled Sylvester matrix equations, J. Comput. Appl. Math. ,224,pp 759–776.
[11]
Zhou, B., Duan, G.R. (2008), On the generalized Sylvester mapping and matrix equations, Systems Control Lett., 57 ,pp200–208.
[12]
Ding,F., Chen, T. (2005) , Gradient based iterative algorithms for solving a class of matrix equations, IEEE Trans. Automat. Control ,50 ,pp1216–1221.
[13]
Ding,F., Chen, T. (2005) , Hierarchical gradient-based identification of multivariable discrete-time systems, Automatica, 41, pp 315–325.
[14]
Ding,F., Chen, T. (2005) , Iterative least squares solutions of coupled Sylvester matrix equations, Systems Control Lett. ,54 ,pp 95–107.
[15]
Ding,F., Chen, T. (2006) , on iterative solutions of general coupled matrix equations, SIAM J. Control Optim. ,44, pp 2269–2284.
[16]
Ding,F., Chen, T. (2005) , Hierarchical least squares identification methods for multivariable systems, IEEE Trans. Automat. Control ,50, pp 397–402.
[17]
Wang, X., Wu, W. H. (2011), A finite iterative algorithm for solving the generalized (P, Q)-reflexive solution of the linear systems of matrix equations, Mathematical and Computer Modelling, 54,pp 2117-2131.
[18]
Wu, A. G., Li, B. , Zhang, Y. ,Duan, G.R. (2011), Finite iterative solutions to coupled Sylvester-conjugate matrix equations. Applied Mathematical Modelling, , 35(3),pp 1065-1080.
[19]
Wu, A. G., Lv, L., Hou, M.-Z. (2011), Finite iterative algorithms for extended Sylvester-conjugate matrix equation, Math. Comput. Model. ,54,pp2363-2384.
[20]
Ramadan, M. A., Abdel Naby, M A. , Bayoumi, A. M. E. (2014), Iterative algorithm for solving a class of general Sylvester-conjugate Matrix equation ∑_(i= 1)^s▒〖A_i V 〗+ ∑_(j=1)^t▒〖B_j W 〗=∑_(l=1)^m▒〖E_1 V ̅ 〗 F_1+C , J. Appl. Math. and Comput, 44(1-2),pp 99-118.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186