【作者】 段雪峰；

【导师】 廖安平；

【作者基本信息】 湖南大学 ， 应用数学， 2008， 博士

【摘要】 非线性矩阵方程是数值代数领域和非线性分析领域中研究和探讨的重要课题之一.它在控制理论,运输理论,动态规划,梯形网络,统计过滤和统计学等科学和工程计算领域中有着广泛的应用.本篇博士论文系统地研究了如下几类非线性矩阵方程的理论与数值方法.基于不动点定理和Banach空间的序列原理,系统地研究了矩阵方程的Hermitian正定解,其中A为n×n阶非奇异复矩阵,Q为n×n阶正定矩阵,q≥1.给出了该矩阵方程存在正定解的一些新的充分条件和必要条件,构造了求解的数值方法.还对该矩阵方程进行了扰动分析,得到了新的正定解的扰动界.基于Brouwer不动点定理和Banach不动点定理,系统地研究了矩阵方程的Hermitian正定解的存在性,其中A为n×n阶非奇异复矩阵,Q为n×n阶正定矩阵,且s,t是正整数.给出了该矩阵方程存在正定解的一些新的充分条件,必要条件及充要条件.并对该矩阵方程进行了扰动分析,得到了新的正定解的扰动界.数值例子说明了所得结论的正确性.基于单调算子的动力学性质,研究了矩阵方程的Hermitian正定解,其中A₁,A₂,…,A_m是n×n阶复矩阵,Q为n×n阶正定矩阵,m是正整数.给出了该矩阵方程的Hermitian正定解的存在性定理及数值求解方法,并对其进行扰动分析,得到了新的正定解的扰动界.基于正规锥上单调和混合单调算子的不动点定理,研究矩阵方程的Hermitian正定解,其中A₁,A₂,…,A_m是n×n阶复矩阵,Q为n×n阶正定矩阵,0<|δ_i|<1,i=1,2,…,m.首次证明了该矩阵方程总是存在唯一正定解.首次提出了求解该矩阵方程的多步定常迭代方法,利用正规锥上序列的性质得到了相应的收敛性定理,并用数值例子验证了此方法的可行性.基于摄动引理和Ostrowski定理,研究矩阵方程的非奇异解,即研究矩阵A的非奇异平方根.当矩阵A非奇异时,对其等价方程构造Newton迭代法,并结合Samanskii技术得到了一种修正Newton法.给出了新Newton法及其修正方法的局部收敛性定理.证明了这两种方法具有较好的数值稳定性.数值实验表明,新Newton法及其修正方法具有精度高和迭代步数少等优点.当矩阵A是一类上三角Toeplitz矩阵时,提出了一种待定系数法求其平方根.数值实验表明,该方法是可行的.更多还原

【Abstract】 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.更多还原