论文标题:一类矩阵的SAOR迭代收敛性及第二型quasi非负分裂的半收敛性分析
The Analysis on Convergence of the SAOR for One Type Matrix and Semiconvergence of the Second Quasi-nonnegative Splitting
论文作者
论文导师 畅大为,论文学位 硕士,论文专业 计算数学
论文单位 陕西师范大学,点击次数 270,论文页数 37页File Size1293K
2007-04-01论文网 http://www.lw23.com/lunwen_1241292/
SAOR iterative method; the second quasi-nonnegative splitting; semiconvergence; spectral radius; the optimum parameter
数学、物理、力学等学科和工程技术中许多问题的解决最终都归结为解一个或一些大型稀疏矩阵的线性方程组，而对这种方程组一般采用迭代法求解．研究迭代法的关键是迭代格式的收敛性和收敛速度．迭代不收敛的格式自然不能用，虽然收敛但收敛很慢的格式使用起来不仅人工和机器的时间比较浪费，而且还不一定能得出结果，因此必须寻求收敛速度比较快的格式和确定格式中的某些参数(如SOR迭代法的松弛因子)．一般来说，迭代法的收敛性与方程组系数矩阵的性质有着密切的关系，例如非负阵、循环阵、M阵、H阵等等．矩阵不同，迭代法的研究方法也会有所差异．因此，讨论某种迭代法时，往往是在指定矩阵类型的前提下进行的．本文题目中的一类矩阵指的是(1，1)相容次序矩阵． 本文共分为三章，各章的主要内容如下： 第一章预备知识．这部分主要是为第二章和第三章作准备．介绍了矩阵的一些基本概念：相容次序矩阵、矩阵范数、非负矩阵及著名的Perron-Frobenius定理等．这些都是研究矩阵谱性质的重要依据．同时还对Drazin逆的基本知识作了简单的介绍． 第二章相容次序矩阵的SAOR迭代法的收敛性分析．这部分是本文的主要结论部分．利用SAOR迭代矩阵S_(γ，ω)的特征值λ和Jacobi迭代矩阵的特征值μ之间的关系式，对SAOR迭代法的收敛性进行了讨论．当系数矩阵A为(1，1)相容次序矩阵且Jacobi特征值全为实数时，计算出实参数γ和ω相应的取值范围，即SAOR方法的收敛区间．最后讨论了在γ=2，ω为复数的条件下，当Jacobi特征值全为实数或纯虚数时，SAOR迭代法的收敛性和最优参数分析． 第三章第二型quasi非负分裂的半收敛分析．先介绍了奇异矩阵线性方程组的一些背景知识，给出了半收敛的概念及其等价条件．在此基础上，引入了第二型quasi非负分裂这个新的概念，它是由quasi非负分裂和第二弱分裂结合而来的．最后讨论了第二型quasi非负分裂半收敛的等价定理和比较定理．
The solutions of many problems in mathematics, physics, mechanics, engineering and so on are sumed up to the solutions of one or some large sparse linear systems which usually are solved by iterative method. The critical matter of the study of iterative methods is the convergence and the convergence rate of the iterative methods. Of course, the divergent methods will not be adopted. If the method has a low rate of convergence, the time of the human and machines will be wasted and the answer are not surely attainable. So, we must look for the methods with the high rate of convergence or try to settle some parameters of the iteration methods (for instance the overrelaxation parameter of SOR iterative method). In general, the convergence of the iterative method is closely related to the property of the coefficient matrix of linear systems, for instance, nonnegative matrices, cyclic matrices, M matrices, H matrices and so on. If the matrices are different, then the research methods of iterative methods will also be different. Therefore, it"s often to study the convergence for the given matrices, of the iterative method. One type matrix in the title this paper just is (1, 1) consistently ordered matrix. This paper contains three chapters. The main results of chapters as following: Chapter 1 Preliminaries. This part mainly makes preparations for Chapter 2 and Chapter 3. Chapter 1 mainly introduce some elementary concepts about matrix: consistently ordered matrix, matrix norm, nonnegative matrix and famous Perron-Frobenius theorem. They are important theoretical foundation of the study about nonnegative matrices" spectral radius. Meanwhile, basic knownledge about Drazin inverse is simplely presented. Chapter 2 The convergence analysis on the SAOR interative method for consistently ordered matrices. The chapter is the main results part of this paper. The convergence of the SAOR interative method is discussed according to the formula about the relation between eigenvalues of the SAOR interative matrix and eigenvalues of its Jacobi matrix. When the coefficient matrix of a linear system is (1, 1) consistently ordered and all eigenvalues of its Jacobi matrix are real numbers, we compute the range of real parametersγandω, that is convergence area of the SAOR iterative method. At last, in the case thatγis 2 andωis complx, when all eigenvalues of its Jacobi matrix are real numbers or all pure imaginaries, we discuss the convergence and the optimum parameters of its SAOR method. Chapter 3 The semiconvergence analysis on the second quasi-nonnegative splitting. Firstly, we introduce some background knowledge about singular linear systems, give the definition and equivalence conditions of the semiconvergence. Based on this, we introduce a new concept of the second quasi-nonnegative splitting that is from quasi-nonnegative splitting and the second weak splitting. Finally, we study the equivalence theorem and the comparison theorem of semiconvergence for the second quasi-nonnegative splitting.
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