α-Nekrasov矩阵一些性质的研究

发布时间：2018-12-18 04:48

【摘要】：Nekrasov矩阵是一类具有重要理论意义和实际应用背景的特殊矩阵,其结构的特殊性使得Nekrasov矩阵具有许多良好的性质,使其在数值代数、控制理论、电力系统理论、经济数学、统计学等众多重要领域有着广泛且深刻的应用,因此受到众多数学学者的关注.本文基于Nekrasov矩阵的独特结构提出了α-Nekrasov矩阵、S-α-Nekrasov矩阵、链α-Nekrasov矩阵的概念,探讨了这些矩阵的性质,并在此基础上提出了若干非奇异H-矩阵的充分条件.第一章主要介绍了Nekrasov矩阵的应用背景和研究现状,概述了本文将要做的主要工作和相关的符号说明及定义等.第二章考察了Nekrasov矩阵的独特结构,构造出类似狄利克雷函数形式的矩阵结构,提出了α-Nekrasov矩阵、S-α-Nekrasov矩阵的概念,证明两类矩阵都是非奇异H-矩阵的子类,并通过划分指标集、二次划分指标集的方法,结合不等式的放缩技巧给出了新的判定矩阵为非奇异H-矩阵的方法.本章所给的判定条件改进和推广了前期的研究成果,最后用数值例子说明所给判定条件的有效性和优越性.第三章构造出另一类类似狄利克雷函数形式的矩阵类链α-Nekrasov矩阵,并证明其为非奇异H-矩阵的子类,之后通过划分指标集的方法,结合不等式的放缩技巧给出了新的判定矩阵为非奇异H-矩阵的方法.通过比较分析,此判别方法放宽了对矩阵各行元素的条件限制,并用例子说明了此方法的有效性.
[Abstract]:Nekrasov matrix is a kind of special matrix which has important theoretical significance and practical application background. The particularity of its structure makes Nekrasov matrix have many good properties, such as numerical algebra, control theory, power system theory, economic mathematics, etc. Statistics and other important fields have a wide range of deep applications, so many mathematical scholars pay attention to them. In this paper, the concepts of 伪-Nekrasov matrix, S- 伪-Nekrasov matrix and chain 伪-Nekrasov matrix are proposed based on the unique structure of Nekrasov matrix. The properties of these matrices are discussed, and some sufficient conditions for nonsingular H-matrices are proposed. The first chapter mainly introduces the application background and research status of Nekrasov matrix, and summarizes the main work to be done in this paper, as well as the related symbolic description and definition. In the second chapter, we investigate the unique structure of Nekrasov matrix, construct a matrix structure similar to the form of Delikley function, put forward the concepts of 伪-Nekrasov matrix and S- 伪-Nekrasov matrix, and prove that both kinds of matrices are subclasses of nonsingular H- matrix. By dividing the index set, the quadratic partition index set, and combining with the scaling technique of inequality, a new method of determining the matrix to be a nonsingular H-matrix is given. The decision conditions given in this chapter improve and generalize the previous research results. Finally, numerical examples are used to illustrate the effectiveness and superiority of the given conditions. In chapter 3, we construct another class of matrix chain 伪-Nekrasov matrix, which is similar to Delikley function, and prove that it is a subclass of nonsingular H-matrix. Based on the scaling technique of inequality, a new method of determining the matrix to be a nonsingular H-matrix is given. Through comparison and analysis, this method relaxed the conditions for each row element of the matrix, and an example was given to illustrate the effectiveness of the method.
【学位授予单位】：湘潭大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O151.21

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