Numerical Algorithms For Several Kinds Of Nonlinear Matrix Equation

The problem of solving nonlinear matrix equations is one of important issues in the field of numerical algebra and non-linear sciences in recent years. Actually, it is widely used in many fields such as structural design, system identification, dynamic programming,automatics control theory, vibration theory, statistic.This dissertation considers mainly the following nonlinear matrix equations:1. Symmetric nonlinear matrix equation,2. Square root of a matrix,X~2-A =0.3. Quadratic matrix equation,AX~2 + BX + C = 0.The main reaserch results on this dissertaton are as follows.1. We state the research significance and development of nonlinear matrix equations in Chapter 1. Chapter 2, Chapter 3 and Chapter 4 mainly concern about the sysmmetricnonliner matrix equtions. In Chapter 2, we study existence and uniqueness of its Herimitian positive definite solutions, Moreover, we reveal essential properties of these Herimitian positive definite solutions. Simultaneity, two numerical algorithms are given for obtaining the maximal positive definite solution. For the maximal Hermitian positive definite solution, we investigate its sensitivity property and backward error analysis in Chapter 3. In Chapter 4, we study the general symmetric nonlinear matrix equation.2. The problem of finding a square root of a matrix can be often met in scientific and engineering problems. Newton's algorithm is the popular algorithm for finding the square root of a matrix. However, it is expensive and difficult to obtain the exact solution from Lyapunov equation at each Newton iterative step. Simplified Newton's algorithm is cheaper than Newton algorithm and is poor numerical stability. According to these defects, A new algorithm is obtained by incorporating exact line searches into Newton's algorithm, it has not only the merit of Newton's algorithm, but also more efficiency than Newton algorithm. A new algorithm is gotten by incorporationg exact line searches into simplified Newton's algorithm. It has not only the merit of simplified Newton's algorithm, but also the better stability than simplified Newton's algorithm in chapter 5.3. Chapter 6, Chapter 7 and Chapter 8 mainly concern about the general quadratic matrix equtions. In Chapter 6, we propose a algorithm based on Newton algorithm, exact line searches and (?)amanskii technique. The algorithm has local cubic convergence order. it is more efficient algorithm than Newton algorithm. In Chapter 7, we recommend an inexact Newton algorithm. The algorithm, at each iterative step, is not to solve Sylvester equation exactly and superliear convergence. We propose the steepest decesent algorithm in Chapter 8. The algorithm avoid the impossible situation of the Newton algorithm for some iterative value.This dissertation is supported by the Natural Science Foundation of China 10571047 and Doctorate Foundation of the Ministry of Education of China 20060532014.This dissertation is typeset by software L~AT_EX2_ε.