【作者】 郭志军；

【导师】 刘建州；

【作者基本信息】 湘潭大学 ， 应用数学， 2005， 硕士

【摘要】 广义严格对角占优矩阵具有很广的实际背景,这类特殊矩阵在数值代数、控制论、电力系统理论、经济数学及弹性力学等众多领域中有着重要的实用价值。但实际中对此类矩阵进行有效判别,尤其是对大型矩阵的判别,还存在许多困难。经国内外许多学者不懈努力,已获得一些重要结果。本文根据α-对角占优矩阵的性质,从三个方面获得了一些新的判定条件,改进了某些已有判定条件的范围,并用数值例子进行了比较。在第一章中,首先引述了α-对角占优矩阵的性质及已有的一些判定条件,给出了判定广义严格对角占优矩阵的几个新的结论,最后说明了这些结论的有效性。在第二章中,利用矩阵某些元素,构造出了几个乘积因子,然后利用α-对角占优矩阵的一些性质,结合放缩不等式的技巧,给出了广义严格对角占优矩阵的几个新的判定条件,改善了已有的某些结果。在第三章中,首先由α-对角占优矩阵的定义,引进了两类局部双α对角占优矩阵,并利用它们及α-对角占优矩阵的性质,结合放缩不等式的技巧,讨论了局部双α对角占优矩阵与广义严格对角占优矩阵的关系,并由此得到判定广义严格对角占优矩阵的几个实用准则。更多还原

【Abstract】 Generalized strictly diagonally dominant matrix play an important role in numerical algebra 、control theory 、electric system、economic mathematics and elastic dynamics and so on. But it isn’t easy to determine whether the matrix is or not generalized strictly diagonally dominant matrix in reality, especially for the large matrices. Under scholar abroad and home hard work there are many important fruits. In this paper we obtain some new determination conditions from three aspects according to some exists theory and the property of αdiagonally dominant matrix, improve the determination conditions and give some numerical example to show our theory is effective. In chapter one, firstly, we introduce the properties of αdiagonally dominant matrix and some exists determination conditions of generalized strictly diagonally dominant matrix, then we give some new results for the criteria of generalized strictly diagonally dominant matrix, finally, we show the validity of these conclusions. In chapter two, by using the elements of the matrix we first construct some multiplier factors, then, use the properties of αdiagonally dominant matrix and the techniques of inequalities, we give some new determination conditions for generalized strictly diagonally dominant matrix, these theory have improved some existing results. In chapter three, at first we introduces two kinds locally double αdiagonally dominant matrix from the concept of αdiagonally dominant matrix, by using this conception and the properties of αdiagonally dominant matrix and the techniques of inequalities, we discuss the relation of locally double αdiagonally dominant matrix and generalized strictly diagonally dominant matrix, according to these relations we obtain some effective criteria for generalized strictly diagonally dominant matrix.更多还原