Hermitian Positive Definite Solutions And Numerical Methods Of Two Classes Of Nonlinear Matrix Equations

Solving nonlinear matrix equations is one of important topics in the matrix theory in recent years. Actually, it is widely applied in many natural science fields such as application physics, bioscience, engineer technology, economy theory, management science. This dissertation considers mainly the following nonlinear matrix equations: and Here, we study the problem of the existence of Hermitian positive definite solutions, the methods for solving the numerical solution and the perturbation analysis of the solution.The main results are as follows:Based on Unitray decomposing the necessary and sufficiency condition of the ex-istence of positive definite solutions of the nonlinear matrix equation =I are given. Through studying on scalar equations, the existence intervals of the positive definite solutions are given. Based on Banach's fixed point theory, the suf-ficiency conditions of the unique positive definite solution are studied. By the based fixed point iterative method, the positive definite solutions of that nonlinear equation in specific interval is obtained. Then, the numerical examples are given to illustrate the correctness of the methods.Based on CS decomposing we give the necessary and sufficiency condition and the bound of the existence of positive definite solutions of the nonlinear matrix equationX＋Through studying on scalar equations, the existence intervals of the positive definite solutions are given. Based on iterative method, we solve the unique positive definite solution. Then we derive the perturbation analysis of this nonlinear equation, get the perturbation bound and investigate its sensitivity property. At last, the numerical examples are given to illustrate the correctness of theoretical results and the effectiveness of iterative methods.