The Positive Definite Solution Of Nonlinear Matrix Equations

In recent years, with the development of science and technology, nonlinear matrix equation theory is widely used in science and engineering calculation field. The research about nonlinear matrix equations is also increasingly subject to people’s attention, more and more scholars begin to focus on numerical method for solving the matrix equations, which has become a hot topic in the field of algebra. Nonlinear matrix equation has a profound theoretical significance and widespread application background in the field of numerical solution of nonlinear analysis. There have been many results on positive definite solutions of nonlinear matrix equation in the literature. On the base of the existing results, we study the following two types of nonlinear matrix equation systematically. （1）X^a+A*X^-βA= I （where A is a invertible matrix）; （2） X^s+A*X-’A= Q （s,t are positive numbers）. Here, we mainly study the necessary and sufficient conditions of a positive definite solution existence of the above matrix equations and methods for solving these matrix equations.In this thesis, the work and the specific research contents are as follows:The first chapter is for introduction. Firstly, the background significance, development process and trends, research route and method of the nonlinear matrix equation of this subject are simply introduced; secondly, the main content studied in this paper are given; finally, the basic definitions and theorems on the matrix, characteristics such as the positive definite matrix, Hermite matrix, the matrix value, triangular matrix decomposition theorem and CS of Hermite matrix, norm of a vector, matrix norm, and functional analysis in the field of Brouwer fixed point set, contraction mapping theorem and so on, which provide the relevant theoretical basis for research content.The second chapter combined with the recently got positive definite solution of on the nonlinear matrix equation X^a+A*X-PA= I both at home and abroad, we first obtain a new necessary and sufficient conditions of this equation. Then, by using the Brouwer not fixed point theorem, iterative method, matrix decomposition, the results are discussed with the necessary and sufficient conditions for the solution of matrix equation obtained; finally, provided the sufficient condition of the positive definite solution and the numerical solution for matrix equation with two cases:α≥1,0<β<1 and 0<a≤1, β≥1.In the third chapter, we consider the range of the existence of solutions positive definite solution of the matrix equation X5+A’X-’A= Q with two cases:s> 1,0< t< 1 and 0<s≤1, t≥1. And the existence conditions of positive definite solutions in these two cases are proved. In particular, when the proper value of the matrix A, Q satisfy a new estimate of positive definite solutions is obtained. In the end, we propose the iterative method for computing the positive definite solutions and the convergence of the method. Finally, the numerical algorithm and the convergence rate of the matrix equation are given, and the disturbance of the matrix is analyzed.