【作者】 王广彬；

【导师】 蒋尔雄；

【作者基本信息】 上海大学 ， 计算数学， 2004， 博士

【摘要】 本文研究了在理论和实际应用中有重要用途的H矩阵和Hermite正半定矩阵的相关问题，包括非奇H矩阵的判据、非奇H矩阵在线性系统的稳定性中的应用、逆H矩阵的基本性质以及系数矩阵分别为H矩阵和奇异Hermite正半定矩阵的线性代数方程组迭代法的收敛性。全文共分为八章。 第一章为前言，介绍了选题背景。第二章给出了非奇H矩阵的判据，并给出了数值例子以表明所给判据的优越性。第三章讨论了非奇H矩阵在线性定常系统的稳定性中的应用，给出了线性定常系统稳定度的判据。第四章在文[16]给出的逆H矩阵定义的基础上，进一步得到了逆H矩阵的新的性质。第五章给出了系数矩阵为非奇H矩阵的线性代数方程组广义交替迭代法的收敛性定理，并给出了当系数矩阵为非奇M矩阵(非奇H矩阵的子类)时迭代矩阵的谱半径的比较定理。第六章给出了系数矩阵为非奇H矩阵的线性代数方程组并行交替迭代法的收敛性定理，并给出了当系数矩阵为非奇M矩阵(非奇H矩阵的子类)时迭代矩阵的谱半径的比较定理。第七章给出了系数矩阵为非奇H矩阵的线性代数方程组两级多分裂迭代法的收敛性定理。第八章给出了系数矩阵分别为奇异H矩阵和奇异Hermite正半定矩阵的线性代数方程组外插迭代法的收敛性定理。更多还原

【Abstract】 In this dissertation, some criterea of nonsingular H-matrix, some new basic properties of inverse H-matrix and the convergence theorems of some iterative methods for the system of linear equations are given. There are eight chapters altogether.In chapter 1, the background of this dissertation is introduced. In chapter 2, some critera for nonsingular H-matrix are given and some examples are presented to show the superiority of the critera. In chapter 3, the critera for the degree of stability of linear ordinary system are given. In chapter 4, some new basic propeties of inverse H-matrix are studied on the basis of paper [16]. In chapter 5, the convergence of generalized alterating iterative method for the system of linear equations is studied when the coefficient matrix is a nonsingular H-matrix and some comparison theorems of spectral radius of the iterative matrix are given when the coefficient matrix is a nonsingular M-matrix (a subclass of nonsingular H-matrix). In chapter 6, the convergence of parallel alterating iterative method for the system of linear equations is studied when the coefficient matrix is a nonsingular H-matrix and some comparison theorems of spectral radius of the iterative matrix are given when the coefficient matrix is a nonsingular M-matrix (a subclass of nonsingular H-matrix). In chapter 7, the convergence of two-stage multisplitting iterative method for the system of linear equations is studied when the coefficient matrix is a nonsingular H-matrix. In chapter 8, the convergence of the extrapolated iterative methods for the system of linear equations is studied when the coefficient matrix is a singular H-matrix and a singular Hermitian positive semidefinite matrix, respectively.更多还原