Positive Definite Solutions Of The Nonlinear Matrix Equation X+A~*X~qA=Q(q>O)

The problem of solving the nonlinear matrix equations is one of important issues in the fields of numerical algebra and nonlinear analysis in recent years.Actually,the non-linear matrix equations are widely used in many fields such as control theory,dynamic programming,statistics,stochastic filtering and ladder networks. In this study, we investi-gate existence and effective iterative method of the Hermitian positive definite solution of the nonlinear matrix equation X+A*XqA=Q(q>0),where A is a n×n complex matrix, Qisan×n positive definite matrix.In the first section,the major achievements,development,research background of the matrix equation are stated and the marks used in this thesis are introduced.In the second section,some properties of the positive definite solution for the matrix equation are discussed and the sufficient and necessary conditions for the existence of the Hermitian positive solutions of the nonlinear matrix equation are derived.In the third section, the effective iterative method to obtain the positive definite so-lution of the equation is established.And the iterative method converging to the definite solution of the nonlinear matrix equation is certified.Then,the perturbation analysis of the matrix equation is discussed.The numerical examples are given in the last section.The numerical examples are given to illustrate the correctness of theoretical results and the effectiveness of the iterative methods.