Numerical Solutions Of Some Nonlinear Matrixe Quations

The numerical solution of nonlinear matrix equation has a wide and profound application in physics,engineering,control theory and other fields.With the rapid development of computer in recent years,this field has gradually become a very hot research topic.The study of these numerical methods not only contributes to the development of equation theory itself,but also has a very important significance in practical application.This paper mainly studies the numerical solution of a class of coupled Riccati equations,coupled nonlinear equations and Chandrasekhar equations.In the second chapter,the coupled algebraic Riccati equations in control theory are studied.Newton's method and the fixed point iteration method are used to solve these equations.Under certain conditions,it is proved that these two kinds of iterative methods monotonically converge to the minimum nonnegative solution with practical significance.Numerical experiments indicate the effectiveness of Newton's method and the fixed point method.The third chapter focuses on the numerical solution of the coupled nonlinear matrix equations from nano devices.Firstly,a sufficient condition for the existence of solutions of coupled nonlinear equations is proposed,and then the coupling doubling algorithm is developed into the coupling tripling algorithm.The decoupled low-rank tripling algorithmis presented subsequently for large-scale problems.Numerical experiments show that the new presented algorithm has the same preprocessing complexity as the low rank doubling algorithm,but it can maintain a lower residual level within fewer iterations through the extra ignored iteration time.In Chapter 4,we focus on the structured Shamanskii method for solving the Riccati equation corresponding to Chandrasekhar equation in particle radiation.Firstly,the efficiency index of Newton iteration is analyzed in detail,and a structured Shamanskii method is designed to solve the minimum positive solution.The monotone convergence of the algorithm is established.Numerical experiments show that the new two-step structured Shamanskii method is superior to Newton method in CPU time and iteration times,without losing the accuracy.