Computing the Hermitian Positive Definite Solutions of a Nonlinear Matrix Equation
DOI:
https://doi.org/10.15377/2409-5761.2016.03.01.4Keywords:
Nonlinear matrix equation, hermitian positive definite solution, iterative methodAbstract
In this paper, we consider a nonlinear matrix equation. We propose necessary and sufficient conditions for the existence of Hermitian positive definite solutions. Some necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions of this equation is also derived. Based on the Banach fixed point theorem, the existence and the uniqueness of the Hermitian positive definite solution are studied. An iterative method for obtaining the Hermitian positive definite solution of this equation is proposed. Finally, some numerical examples are presented to illustrate the performance and efficiency of the proposed algorithm.
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References
Parodi M. La localisation des valeurs caracerisiques desamatrices etses applications, Gauthuervillar, Pairs 1959.
Bhatca R. Matrix analysis, Berlin: Springer 1977.
Zhan XZ. Computing the extremal positive solutions of a matrix equation. SIAMJ Sci Comput 1996; 17(4): 1167-1174.http://dx.doi.org/10.1137/S1064827594277041 DOI: https://doi.org/10.1137/S1064827594277041
Wang JF, Zhang YH and Zhu BR. The Hermitian positive definite solutions of matrix equation. Math Numer Sinica 2004; 26: 31-73.
Duan X and Liao A. On the existence of Hermitian positive definite solutions of matrix equation Xs+ A*X-tA=Q(q<1),Linear Algebra Appl 2008; 429: 637-687.http://dx.doi.org/10.1016/j.laa.2008.03.019 DOI: https://doi.org/10.1016/j.laa.2008.03.019
Furuta T. Operator inequalities associated with Holder-Mc-Carthy and Kantorovich inequalities. J Inequal Appl 1998; 6: 137-148. DOI: https://doi.org/10.1155/S1025583498000083
Long Jianhui, Hu Xiyan and Zhang lei. On the existence of Hermitian positive definite solutions of matrix equation X + A* X-1 A + B* X-1 B = I. Bull Braz. Math Soc New Series 2008; 39(3): 371-386.http://dx.doi.org/10.1007/s00574-008-0011-7 DOI: https://doi.org/10.1007/s00574-008-0011-7
Cui Xiaomei and Liu Libo. On the existence of Hermitian positive definite solutions of matrix equation X + A*X-1 A + B* X-t B = I. Jounal of Changchun university of Technology 2014; 35(6): 622-624.
Cui Xiaomei, Liu Libo and Gao Han. On the existence of Hermitian positive definite solutions of matrix equation X + A* X- A + B* X- B = I. J of Math 2014; 34(6): 1149-1154.
Sarhan AM, Nahlaa M El-Shazly and Enas M Shehata. On the existence of extremal positive definite solutions of thenonlinear matrix equation Xr + mi=1 Ai* Xi Ai = I. Math Computer Modelling 2010; 51(2): 1107-1117.http://dx.doi.org/10.1016/j.mcm.2009.12.021 DOI: https://doi.org/10.1016/j.mcm.2009.12.021
Du Fuping. The matrix equation X - A* X- A - B* X- B = I. Journal of Jilin University 2010; 1(48): 26-32.
Liu Aijing and Chen Guoliang. On the existence of Hermitian positive definite solutions of nonlinear matrix equation Xs + mi=1 Ai* Xti Ai = Q. Applied Mathematics and Computation 2014; 243: 950-959.http://dx.doi.org/10.1016/j.amc.2014.05.090 DOI: https://doi.org/10.1016/j.amc.2014.05.090
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Copyright (c) 2016 Minghui Wang, Luping Xu
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