On The Positive Definite Solution Of Nonlinear Matrix Equation

In recent years, with the development of science and technology, nonlinear matrix equation has showed a very important role in different fields, and the solution of the nonlinear matrix equation has also become a hot topic in the field of study.On the base of the existing results, we study the following two types of nonlinear matrix equation: X-1 + A* Xq A= Q, X + A* X-q A= Q.Here, we mainly study the necessary and sufficient condition for the existence of the positive definite solution, existence and uniqueness of solution and the convergence of iterative algorithm and error analysis, and some new conclusions are obtained.The main results are as follows:1.Mainly study the nonlinear matrix equation X-1 + A* Xq A= Q in O<q ≤1 and q≥ 1 two different cases; We obtain some necessary conditions and the sufficient and necessary conditions for the existence of the positive definite solutions of the matrix equation,and construct its solving iterative methods.By the way, the different convergence of these iterative methods and error analysis are proved which discussed,and conclude that in 0< q≤ 1 the corresponding iterative algorithm converges to the a minimal solution of the equation, in q≥ 1 the corresponding iterative algorithm converges to the a maximum solution of the equation.2.Mainly study the existence condition of the positive definite solutions of nonlinear matrix equation X + A* X-qA = Q. First of all, on the base of the existing results, we give a necessary condition for the existence of the positive definite solution in 0<g≤1, solution for the new estimates, the range of solution will be optimized. Give in 0< q ≤1 and q≥ 1 two kinds of cases, by constructing two different mapping function, the two sufficient conditions for the existence of solution and the solution of the uniqueness theorem are obtained by the fixed point theorem.