On The Positive Definite Solutions Of Several Classes Of Nonlinear Matrix Equations

Nonlinear matrix equation is widely used in many theoretical and applied areas for a long time,since nonlinear phenomena exists almost in every aspects of our life.In recent years,study of many important problems in different fields such as statistics,stochastic filtering,dynamic programming,nano theory,etc.,leads to solving a special kind of symmetric non-linear matrix equations X±A*X-nA=Q,n=1,2.There have been many results on positive definite solutions of these equations.Moreover,these standard symmetrical nonlin-ear matrix equations are extended to several more general forms,including,but not limited to:Xs±A*X-tA=Q(s,t are positive integers),X±A*X-qA=Q(where q>0),X-A*eXA=I and so on.This thesis considers positive definite solutions of the following general nonlinear matrix equations:(1)X±A*X-1A=Q;(2)X-?i=1mAi*X-qAi=Q;(3)X+?i=1mAi*X-qAi-?j=1nBj*X-rBj=Q(0<q,r?1).For each type of these equations,we focus on three aspects:solvable theory,numerical algorithms and perturbation analysis of the positive definite solutions.1.Chapter 2 concerns with the positive definite solutions of the nonlinear matrix equation X-A*X-1A=Q,where A,Q are given complex matrices with Q positive definite.It is showed that this type of matrix equation always has a unique positive definite solution and if A is nonsingular,it also has a unique negative definite solution.Moreover,based on Sherman-Morrison-Woodbury formula,we derive elegant relationships between the unique positive definite solution of X-A*X-1A=I and the maximal positive definite solution of the well-studied standard nonlinear matrix equation Y+B*Y-1B=Q,where B,Q are uniquely determined by A.Then several effective numerical algorithms for the unique pos-itive definite solution of X-A*X-1A=Q with linear or quadratic convergence rate such as inverse-free fixed-point iteration,structure-preserving doubling algorithm,Newton algo-rithm are proposed.Similarly,we derive an elegant relationship between the unique positive definite solution of the minus sign equation X-A*X-1A=I and the maximal positive definite solution of a corresponding plus sign equation Y+B*Y-1B=Q.Then several new effective numerical algorithms for the unique positive definite solution of X-A*X-1A=I are easily obtained.Numerical examples are presented to illustrate the effectiveness of all the theoretical results and the behavior of the considered algorithms.2.Chapters 3 and 4 consider nonlinear matrix equation(?) where 0<q?1,A1,A2,…,Am,Q are n×n nonsingular complex matrices with Q Hermi-tian positive definite,and A*is the conjugate transpose of a matrix A.For the minus sign of this kind of equation,it has been proved that it always has a unique positive definite solution in case q=1.So we discuss only the perturbation of the unique positive definite solution in Chapter 3.Chapter 4 considers the equation X+?im=1Ai*X-qAi=Qwith 0<q?1,then we prove that if it has a positive definite solution,it must have the maximal positive definite solution XL.Moreover,when 0<q<1,we discuss the perturbation of the max-imal positive definite solution XL,and several numerical examples are offered to illustrate the effectiveness of the theoretical results.3.Chapter 5 considers positive definite solution of the nonlinear matrix equation X+?im=1Ai*X-qAi-?j=1nBj*X-rBj=Q(0<q,r?1,Q positive definite).Based on the Bhaskar-Lakshmi kantham coupled fixed point theorem,we derive some sufficient con-ditions for the existence and uniqueness of the positive definite solution to such equations respectively in two cases:q=r=1 and 0<q,r<1.Iterative methods,perturbation esti-mates and the explicit expressions of Rice condition number for the unique positive definite solution are obtained in these two cases,respectively.Numerical examples show that our theoretical results are effective and easy to compute.