广义H-矩阵一些新的判定

发布时间：2018-07-15 09:26

【摘要】：H-矩阵在经济学,统计学,工程技术等领域有着广泛的应用。作为H-矩阵在正定条件下的一种推广形式,广义H-矩阵在流体力学计算中2-D或3-D欧拉方程数值解,动力系统的不变环面的计算中和数值代数以及矩阵分析中有着重要的理论和实际意义。在本文中,我们应用广义特征值与广义瑞利商之间的关系,以及广义M-矩阵与M-矩阵之间的联系,结合相应的矩阵分块和子矩阵谱半径估计等技巧,得出了广义H-矩阵的一些判定定理,改进了近期的相关结果。第一章主要介绍了广义H-矩阵的理论背景,应用背景和研究现状,并给出本文相关的符号说明,定义,引理等。第二章应用广义M-矩阵与M-矩阵之间的联系以及子矩阵谱半径,以及广义特征值与广义瑞利商之间的关系,结合矩阵分块和相关不等式放缩等技巧,得出广义H-矩阵的一些较为简捷的判定定理,并给出相应的数值例子说明判定的有效性。第三章在第二章的基础上,我们采取递进的方式选取正对角矩阵D使得AD为按偏序严格块对角占优矩阵,给出了广义H-矩阵的一些递进判定方法,这些方法为第四章提出的一些广义H-矩阵迭代判别算法提供了理论基础。第四章在第三章的理论基础上,构造相应的迭代矩阵,给出了广义H-矩阵的一些迭代判别算法,并对各个算法的收敛性进行了理论证明。最后,我们运用合适的数值例子说明算法的有效性。
[Abstract]:H-matrix is widely used in economics, statistics, engineering technology and so on. As a generalized form of H-matrix under positive definite condition, the numerical solution of 2-D or 3-D Euler equation in hydrodynamic calculation of generalized H-matrix is obtained. The neutralization of the invariant torus of the dynamical system and the matrix analysis have important theoretical and practical significance. In this paper, we apply the relation between generalized eigenvalue and generalized Rayleigh quotient, and the relation between generalized M- matrix and M- matrix, combining with the corresponding matrix block and the estimation of spectral radius of submatrix, etc. Some judgment theorems of generalized H-matrix are obtained, and some recent results are improved. The first chapter mainly introduces the theoretical background, application background and research status of generalized H-matrix, and gives the symbol description, definition, Lemma and so on. In chapter 2, the relation between generalized M- matrix and M- matrix, the spectral radius of submatrix, the relationship between generalized eigenvalue and generalized Rayleigh quotient, and the techniques of matrix partitioning and inequality scaling are used. Some simple judgment theorems of generalized H-matrix are obtained, and corresponding numerical examples are given to illustrate the validity of the decision. In chapter 3, on the basis of the second chapter, we select the positive diagonal matrix D in a progressive way so that AD is a diagonally dominant matrix strictly in partial order, and give some progressive judgment methods of generalized H-matrix. These methods provide a theoretical basis for some generalized H-matrix iterative discriminant algorithms proposed in Chapter 4. In chapter 4, on the basis of the theory in chapter 3, the corresponding iterative matrix is constructed, and some iterative discriminant algorithms of generalized H-matrix are given, and the convergence of each algorithm is proved theoretically. Finally, we use appropriate numerical examples to illustrate the effectiveness of the algorithm.
【学位授予单位】：湘潭大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O151.21

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