Finite Iterative Algorithm for Solving a Class of Complex Matrix Equation with Two Unknowns of General Form

Mohamed A. Ramadan; Mokhtar A. Abdel Naby; Talaat S. El-Danaf; Ahmed M. E. Bayoumi

doi:doi:10.11648/j.acm.20140305.23

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Finite Iterative Algorithm for Solving a Class of Complex Matrix Equation with Two Unknowns of General Form

Mohamed A. Ramadan, Mokhtar A. Abdel Naby, Talaat S. El-Danaf, Ahmed M. E. Bayoumi

Published in Applied and Computational Mathematics (Volume 3, Issue 5)

Received: 24 October 2014 Accepted: 6 November 2014 Published: 20 November 2014

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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.

Published in	Applied and Computational Mathematics (Volume 3, Issue 5)
DOI	10.11648/j.acm.20140305.23
Page(s)	273-284
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

General Sylvester-Conjugate matrix Equations, Finite Iterative Algorithm, Orthogonality, Inner Product Space, Frobenius norm

References

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[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.
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[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.
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[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.

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Mohamed A. Ramadan, Mokhtar A. Abdel Naby, Talaat S. El-Danaf, Ahmed M. E. Bayoumi. (2014). Finite Iterative Algorithm for Solving a Class of Complex Matrix Equation with Two Unknowns of General Form. Applied and Computational Mathematics, 3(5), 273-284. https://doi.org/10.11648/j.acm.20140305.23

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ACS Style

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. Appl. Comput. Math. 2014, 3(5), 273-284. doi: 10.11648/j.acm.20140305.23

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AMA Style

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. Appl Comput Math. 2014;3(5):273-284. doi: 10.11648/j.acm.20140305.23

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@article{10.11648/j.acm.20140305.23,
  author = {Mohamed A. Ramadan and Mokhtar A. Abdel Naby and Talaat S. El-Danaf and Ahmed M. E. Bayoumi},
  title = {Finite Iterative Algorithm for Solving a Class of Complex Matrix Equation with Two Unknowns of General Form},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {5},
  pages = {273-284},
  doi = {10.11648/j.acm.20140305.23},
  url = {https://doi.org/10.11648/j.acm.20140305.23},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140305.23},
  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.},
 year = {2014}
}

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TY  - JOUR
T1  - Finite Iterative Algorithm for Solving a Class of Complex Matrix Equation with Two Unknowns of General Form
AU  - Mohamed A. Ramadan
AU  - Mokhtar A. Abdel Naby
AU  - Talaat S. El-Danaf
AU  - Ahmed M. E. Bayoumi
Y1  - 2014/11/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140305.23
DO  - 10.11648/j.acm.20140305.23
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 273
EP  - 284
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140305.23
AB  - 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.
VL  - 3
IS  - 5
ER  -

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Author Information

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

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APA Style

Mohamed A. Ramadan, Mokhtar A. Abdel Naby, Talaat S. El-Danaf, Ahmed M. E. Bayoumi. (2014). Finite Iterative Algorithm for Solving a Class of Complex Matrix Equation with Two Unknowns of General Form. Applied and Computational Mathematics, 3(5), 273-284. https://doi.org/10.11648/j.acm.20140305.23

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ACS Style

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. Appl. Comput. Math. 2014, 3(5), 273-284. doi: 10.11648/j.acm.20140305.23

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AMA Style

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. Appl Comput Math. 2014;3(5):273-284. doi: 10.11648/j.acm.20140305.23

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@article{10.11648/j.acm.20140305.23,
  author = {Mohamed A. Ramadan and Mokhtar A. Abdel Naby and Talaat S. El-Danaf and Ahmed M. E. Bayoumi},
  title = {Finite Iterative Algorithm for Solving a Class of Complex Matrix Equation with Two Unknowns of General Form},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {5},
  pages = {273-284},
  doi = {10.11648/j.acm.20140305.23},
  url = {https://doi.org/10.11648/j.acm.20140305.23},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140305.23},
  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.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - Finite Iterative Algorithm for Solving a Class of Complex Matrix Equation with Two Unknowns of General Form
AU  - Mohamed A. Ramadan
AU  - Mokhtar A. Abdel Naby
AU  - Talaat S. El-Danaf
AU  - Ahmed M. E. Bayoumi
Y1  - 2014/11/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140305.23
DO  - 10.11648/j.acm.20140305.23
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 273
EP  - 284
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140305.23
AB  - 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.
VL  - 3
IS  - 5
ER  -

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