The Hermitian Positive Definite Solutions Of Nonlinear Eqution X+A^*X^-nA=Q(n>0)

The problem of solving the nonlinear matrix equation, is mainly to determine the solution by the information of the parameters of equation. In practice, the equation X+A*X-nA=Q arises in various areas of applications, including control theory, ladder networks, dynamic programming, statistic, the finite difference application point of view, the Hermitian positive solution is more important, about which are concerned. In the sequel, a solution always means a Hermitian positive definite one. The study of this kind of problem has three basic problems: (1) the theoretic issue on solvability, ie., the necessary and sufficient conditions for the existence of the solution; (2) the numerical solution, ie., the effective numerical ways;(3)the analysis of the perturbation.First, if the matrix equation has a solution, we can get some propertise of the solution according to the following theorems.Theorem 2.1 Suppose that Eq. (1) has a positive definite solution X, A is invertible, thenTheorem 2.3 If equation (1) has a positive definite solution X, 0＜n≤1, then whereμand v are minimal and maximal eigenvalues of AQ-1A*.Theorem 2.4 If equation(1)has a positive definite solution X, n＞1,then whereωandθare minimal and maximal eigenvalues of Q.Second,we get some properties of the quasi-maximal solution of equation in this paper, we offer the conditions for the existence of the solution of the equation (2).Theorem 4.1.2 There is at most one X is a hermitian positive definite solution of the matrix equation (2) such that x＞(?)I, Therefore, if X is a hermitian positive definite solution of the matrix equation (2) satisfies that X＞(?)I, then X is a quasi-maximal solution of the matrix equation(2).Theorem 4.1.3 If X is a hermitian positive definite solution of the matrix equation (2) satisfies ||X-1||-(n+1)||A||2＜1/n,then x＞(?)I,and so X is a quasi-maximal solution of the matrix equation (2).Theorem 4.1.4 If then the solution XL is unique and there is no solution greater than XL, that is the quasi-maximal solution XL such that XL＞(?)I. Moreover,we have Here 0＜η＜1/n satisfies ||A||2(1+η)n+1=η.Moreover, about the numerical solution of X+A*X-2A=I, (3) this paper offers four different iterative methods to approximate the quasi-maximal solution of the equation and the covergence. MethodⅠMethodⅡMethodⅢTheorem 4.2.2 If Moreover,(i)XL is the solution of the eqution(3).That is the quasi-maximal solution XL such that XL＞(?)I.(ii)we have ||XL-1||≤1+η.Here 0＜η＜1/n satisfies ||A||2(1+η)n+1=η,||XL-1||≤1+η.(iii)||Ys-YL||≤ρ||Yk-Yk-1||,Hereρ=3||A||2.(1+η)2＜1.Theorem 4.2.4 Let A in equation (3) satisfy then there existsδ＞0,such that ||X0-XL||＜δ,the iterates {Xk} generated by MethodⅣsatisfies that for all k=1,2,…,that is,the iterates {Xk} converge quadratically to the quasi-maximal solution XL.Last conclude and look to the future for the matrix equation.