非奇异H-矩阵的若干判定法

发布时间：2018-07-12 21:42

本文选题：严格对角占优矩阵 + 非奇异H-矩阵　；参考：《吉首大学》2017年硕士论文

【摘要】：非奇异H-矩阵是计算数学、数学物理、控制论和矩阵理论中较为活跃的研究领域,它在经济学、生物学、动力系统理论及智能科学等许多学科中都有着广泛的应用.但是判定一个矩阵是否为H-矩阵是较为困难的问题,因此,研究H-矩阵的简捷实用的判定条件,构造高效快速的判别算法,具有十分重要的理论价值和实际应用价值.本文主要研究了非奇异H-矩阵的直接判定法,递进判定法和交叉迭代判定算法,改进了近期的一些结果,主要内容如下:首先,介绍了非奇异H-矩阵判定问题的研究背景和现状,给出了有关的符号和定义,以及本文所做的主要工作.其次,研究了非奇异H-矩阵的直接判定法.根据非奇异H-矩阵的定义和性质,通过添加新的系数来构造正对角矩阵变换因子,利用不等式放缩技巧,得到了一种新的判定条件,改进了近期的一些结果,并用数值例子验证了这个方法的优越性.再次,研究了非奇异H-矩阵的递进判定.递进判定法主要通过对正对角矩阵因子进行递进选取,给出了几个新的递进判定法.数值实例验证了新的递进判定法的判定范围更加广泛,改进了近期的结果.最后,研究了非奇异H-矩阵的算法判定,给出了非奇异H-矩阵的一种新的无参数交叉迭代判定算法.对于一个给定的不可约矩阵A,总能通过有限步迭代来判断这个矩阵是否为非奇异H-矩阵.并用数值实例验证了新算法的有效性和优越性,新算法比旧算法迭代次数更少、应用范围更广,并且对于所给矩阵为非H-矩阵的情况,同样能够更快地进行判断,改进了近期的结果.
[Abstract]:Non-singular H-matrix is an active research field in computational mathematics, mathematical physics, cybernetics and matrix theory. It has been widely used in many disciplines such as economics, biology, dynamic system theory and intelligent science. However, it is difficult to determine whether a matrix is a H-matrix or not. Therefore, it is very important to study the simple and practical judgment conditions of H- matrix and to construct an efficient and fast discriminant algorithm, which has very important theoretical value and practical application value. In this paper, the direct decision method, the progressive decision method and the cross iterative decision algorithm of the nonsingular H-matrix are studied, and some recent results are improved. The main contents are as follows: first of all, In this paper, the research background and present situation of nonsingular H-matrix decision problem are introduced, and the related symbols and definitions are given, as well as the main work done in this paper. Secondly, the direct decision method of nonsingular H-matrix is studied. According to the definition and properties of nonsingular H-matrices, the transformation factors of diagonal matrices are constructed by adding new coefficients. By using the technique of inequality scaling, a new judgment condition is obtained, and some recent results are improved. A numerical example is given to verify the superiority of this method. Thirdly, the progressive decision of nonsingular H-matrix is studied. In this paper, several new progressive decision methods are given by selecting the positive diagonal matrix factors. Numerical examples show that the new progressive decision method is more extensive and improves the recent results. Finally, the algorithm of nonsingular H-matrix is studied, and a new algorithm of nonsingular H-matrix is presented. For a given irreducible matrix A, it is always possible to determine whether the matrix is a nonsingular H-matrix by finite step iteration. The effectiveness and superiority of the new algorithm are verified by numerical examples. The new algorithm has fewer iterations than the old one and has a wider range of applications. The new algorithm can also be used to judge the non-H-matrix more quickly. The recent results have been improved.
【学位授予单位】：吉首大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O151.21

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