Numerical Methods For Positive Definite Solutions Of Several Classes Of Nonlinear Matrix Equations

With the development of science and technology,many different types of nonlinear matrix equations have been derived in the fields of control theory,dynamic programming,Kalman filtering,optimal interpolation,engineering calculation and other fields.Thereby,the practical problems in these fields are transformed into the problem of solving nonlinear matrix equations,seeking feasible and effective numerical solutions to these equations has attracted the attention of scholars at home and abroad,and is a research hotspot in the field of matrix algebra.By using matrix theory and other related knowledge,this thesis discusses the Hermite positive definite solutions of three kinds of nonlinear matrix equations.The specific content consists of five chapters,which are summarized as follows:In chapter 1,the research background and development status of nonlinear matrix equa-tions at home and abroad are introduced,the main research content of this thesis is proposed,and some basic concepts and related lemmas are given.In chapter 2,the Hermite positive definite solution of the nonlinear matrix equation X^m-BX-C=0 is discussed.Firstly,the nonlinear matrix equation with symmetric structure and the same solution as the original equation is given.The corresponding iterative schemes are constructed for three cases of positive definite,negative definite and indefinite coefficient matrix B,respectively.The selection method of an iterative initial matrix is given according to the characteristics of each iteration.In chapter 3,the Hermite positive definite solution of the nonlinear matrix equation X±A^*(R+B^*X B)^-tA=Q is discussed.Firstly,the sufficient condition for the existence of a positive definite solution of the equation X+A^*(R+B^*X B)^-tA=Q is derived.The fixed point iteration and inversion-free iteration for solving this equation are constructed.Secondly,using the properties of the Thompson metric and the fixed point theorem of monotone operators on closed convex sets,some sufficient and necessary conditions for the positive definite solution of the equation X-A^*(R+B^*X B)^-tA=Q are given,the corresponding iterative scheme is established and its convergence is proved.In chapter 4,the Hermite positive definite solution of the nonlinear matrix equation X^m-A^*X^-sA+B^*X^-tB=Q is discussed.The original problem is transformed into a problem of solving an equivalent matrix equation through matrix transformation,and then the value range of a positive definite solution and three iterative schemes are constructed by using the coefficient matrix and its partial order.It is proved that the given iteration converges to the positive definite solution of the original equation under certain conditions.In chapter 5,we summarize the current research work and put forward some follow-up research ideas.