Theories And Methods For Several Nonlinear Matrix Equations

Solving nonlinear matrix equations is one of important topics in the fields of numerical algebra and nonlinear analysis. Actually, it is widely used in areas of science and engineering computation, such as control theory, transport theory, dynamic programming, ladder networks, stochastic filtering and statistics. This dissertation studies systematacially the theories and numerical methods of the following nonlinear matrix equations.Basing on the fixed point theorem and sequence theory in Banach space, we study the Hermitian positive definite solution of the matrix equationwhere A is an n×n complex matrix, Q is an n×n Hermitian positive definite matrix and q≥1.We give some new sufficient conditions and necessary conditions for the existence of a positive definite solution, and propose two iterative methods to compute the positive definite solution. We also derive some new perturbation bounds of the positive definite solution.Basing on Brouwer's fixed point theorem and Banach's fixed point theorem, we study the existence of the Hermitian positive definite solution of the matrix equationwhere A is an n×n nonsingular matrix, Q is an n×n Hermitian positive definite matrix, s and t are positive integers. We give some new sufficient conditions and necessary conditions for the existence of a positive definite solution, and derive a new perturbation bound of the positive definite solution. The results are illustrated by numerical examples.Basing on the dynamics property of the monotone operator, we study the Hermitian positive definite solution of the general matrix equationwhere A₁,A₂,...,A_m are n×n complex matrix,Qisan n×n positive definite matrix and m is a positive integer. We give some sufficient conditions and necessary conditions for the existence of a positive definite solution, and construct an iterative method to solve it. We also derive a new perturbation bound of the positive definite solution.Basing on fixed point theorems for monotone and mixed monotone operators in a normal cone, we study the Hermitian positive definite solution of the matrix equation where A₁,A₂,...,A_m axe nxn complex matrix, Q is an n×n Hermitian positive definite solution, and 0<|δ_i|<1, i=1,2,...,m.We firstly prove that the matrix equation always has a unique positive definite solution. We firstly propose a muti-step stationary iterative method to compute the unique positive definite solution, and the convergence theorem is proved by the property of sequence in normal cone. The results are illustruted by numerical examples.Basing on perturbation lemma and Ostrpwski theorem, we study the nonsingular solution of the matrix equationThat is to say, we investigate the nonsingular square root of the matrix A. When A is an n×n nonsingular complex matrix, we apply Newton's method to its equivalent equation for computing the nonsingular square root of the matrix A. We also derive a modified Newton's method by using Samanskii technique. We give local convergence theorem for these new methods, and we also prove that these new methods have good numerical stability. Numerical examples show that these new methods are accurate and effective when they are used to compute the matrix nonsingular square root. When A is a kind of the upper triangular Toeplitz matrix, we propose a method of undeterminated coefficients to compute its square root. Numerical examples show that this numerical method is feasible.