Applied and Computational Mathematics

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

Received: 24 October 2014    Accepted: 06 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.

DOI 10.11648/j.acm.20140305.23
Published in Applied and Computational Mathematics (Volume 3, Issue 5, October 2014)
Page(s) 273-284
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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), 2024. 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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[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.
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[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.
Author Information
  • Department of Mathematics, Faculty of Science, Menoufia University, Shebeen El- Koom, Egypt

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

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

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

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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://download.sciencepg.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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