非线性矩阵方程X +A*X＾qA = Q(q > 0) 的正定解 Alternative Title positive definite solution of the nonlinear matrix equation X +A*X＾qA = Q(q > 0) 谢维维 Thesis Advisor 张国凤 2011-05-28 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 非线性矩阵方程 正定解 迭代方法 不动点定理 Abstract 非线性矩阵方程的求解问题是近年来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。非线性矩阵方程在控制理论、动态规划、统计、随机渗入、梯形网络等多个领域都有重要的应用。本文主要讨论了非线性矩阵方程X +A*X＾qA = Q(q > 0) 的正定解的存在性，给出了一种迭代方法，其中A是n×n阶复矩阵，Q是n×n阶正定矩阵。 第一部分介绍了非线性矩阵方程的发展和研究背景及其研究的主要成果，并介绍了本文所用的记号。 第二部分讨论了方程X +A*X＾qA = Q(q > 0)解的性质并得出了Hermitian正定解的存在性的充分条件。 第三部分给出了非线性矩阵方程X +A*X＾qA = Q(q > 0)的正定解的有效的迭代方法并证明了迭代方法的收敛性。接着对解的扰动分析进行了讨论。 第四部分给出了数值例子。利用数值例子验证了文中所得结论的正确性以及求解方法的有效性。 Other Abstract 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 investigate existence and effective iterative method of the Hermitian positive definite solution of the nonlinear matrix equation X +A*X＾qA = Q(q > 0),where A is a n×n complex matrix, Q is a n×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 solution 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. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/225452 Collection 数学与统计学院 Recommended CitationGB/T 7714 谢维维. 非线性矩阵方程X +A*X＾qA = Q(q > 0) 的正定解[D]. 兰州. 兰州大学,2011.
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