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On the existence of extremal positive definite solutions of the nonlinear matrix equation. (English) Zbl 1203.15011
The authors consider the matrix equation ${X}^{r}+{\sum }_{j=1}^{m}{A}_{j}^{*}{X}^{{\delta }_{j}}{A}_{j}=I$, $-1<{\delta }_{j}<0$, where ${A}_{j}$ are nonsingular $n×n$ matrices, $I$ is the identity matrix, $r$ and $m$ are positive integers, ${A}_{j}^{*}$ is the transpose conjugate of ${A}_{j}$. The authors derive a necessary condition for the existence of a positive definite solution and, based on the Banach fixed point theorem, a sufficient condition for the existence of a unique such solution. Iterative methods for obtaining the extremal (maximal-minimal) positive definite solutions of this equation are proposed. The rate of convergence of some proposed algorithms is proved and numerical examples are given to illustrate their performance and effectiveness.
##### MSC:
 15A24 Matrix equations and identities 65F30 Other matrix algorithms
##### References:
 [1] Duan, X.; Liao, A.; Tang, B.: On the nonlinear matrix equation X-$\sum$i=1mAi$*x\delta$iAi=Q, Linear algebra appl. 429, 110-121 (2008) · Zbl 1148.15012 · doi:10.1016/j.laa.2008.02.014 [2] El-Sayed, S. M.; Ramadan, M. A.: On the existence of a positive definite solution of the matrix equation X-A$*$X-12mA=I, Int. J. Comput. math. 76, 331-338 (2001) · Zbl 0972.65030 · doi:10.1080/00207160108805029 [3] Peng, Z. Y.; El-Sayed, S. M.; Zhang, X. L.: Iterative methods for the extremal positive definite solution of the matrix equation X+A$*X-\alpha$A=Q, Appl. math. Comput. 200, 520-527 (2007) · Zbl 1118.65029 · doi:10.1016/j.cam.2006.01.033 [4] Ramadan, Mohamed A.: On the existence of extremal positive definite solutions of a kind of matrix, Int. J. Nonlinear sci. Numer. simul. 6, No. 2, 115-126 (2005) [5] Ramadan, Mohamed A.: Necessary and sufficient conditions to the existence of positive definite solution of the matrix equation X+ATX-2A=I, Int. J. Comput. math. 82, No. 7, 865-870 (2005) · Zbl 1083.15020 · doi:10.1080/00207160412331336107 [6] Ramadan, Mohamed A.; El-Shazly, Naglaa M.: On the matrix equation X+A$*$X-12mA=I, Appl. math. Comput. 173, 992-1013 (2006) [7] Zhan, X.: Computing the extremal positive definite solutions of a matrix equation, SIAM J. Sci. comput. 17, No. 5, 1167-1174 (1996) · Zbl 0856.65044 · doi:10.1137/S1064827594277041 [8] Liu, X. G.; Gao, H.: On the positive definite solutions of a matrix equations $Xs±$ATX-ta=In, Linear algebra appl. 368, 83-97 (2003) · Zbl 1025.15018 · doi:10.1016/S0024-3795(02)00661-4 [9] Duan, X.; Liao, A.: On the existence of Hermitian positive definite solutions of the matrix equation xs+A$*$X-ta=Q, Linear algebra appl. 429, 673-687 (2008) · Zbl 1143.15011 · doi:10.1016/j.laa.2008.03.019 [10] Yueting, Y.: The iterative method for solving nonlinear matrix equation xs+A$*$X-ta=Q, Appl. math. Comput. 188, 46-53 (2007) · Zbl 1131.65039 · doi:10.1016/j.amc.2006.09.085 [11] N.M.A. El-Shazly, A theoretical and numerical study of solving a class of nonlinear matrix equations, Ph.D. Sc. Thesis, Menoufia University, Faculty of Science, Math. Dep., 2006 [12] Bhatia, R.: Matrix analysis, Matrix analysis 169 (1997)