产生于Euler方程数值解中的广义H-矩阵和矩阵谱估计研究

【作者】 申淑谦；

【导师】 黄廷祝；

【作者基本信息】 电子科技大学 ， 计算数学， 2004， 硕士

【摘要】 本文的研究涉及三类特殊矩阵: 广义正定矩阵, 广义H-矩阵和非负不可约矩阵. 全文共分五章三部分,创新成果着重体现在第二和第三部分.第一部分(第二章)给出了广义正定矩阵的一些新性质, 得到了两个有关广义正定矩阵的行列式不等式.第二部分(第三章)主要讨论了广义H-矩阵. 利用广义M-矩阵, 广义H-矩阵与M-矩阵和正定矩阵的关系, 获得了若干广义H-矩阵新的等价条件,并指出文[R. Nabben, On a class of matrices which arise in the numericalsolution of Euler equations, Numer. Math. 1992, 63: 411-431]中的两个结论是错误的,并进行了修正.第三部分(第四章和第五章)构造非负不可约矩阵Perron 根的新上界序列, 通过数值例子说明该序列在许多情况下优于一些著名结果. 最后, 利用非负矩阵的性质研究矩阵的最大奇异值和谱半径, 给出了若干不等式.更多还原

【Abstract】 This thesis presents a systematic research on some types of special matricessuch as generalized positive definition matrices, generalized H-matrices andnonnegative irreducible matrices. The thesis consists three parts with fivechapters.In part one (chapter two), we give some new properties of generalizedpositive definition matrices, and two determinant inequalities about generalizedpositive definition matrices are presented.The second part (chapter three) contributes to generalized H-matrices.Based on the relationships between generalized M-matrices and generalizedH-matrices with M-matrices and positive definition matrices, we obtain severalequivalent conditions for generalized H-matrices, some remarks on two resultsin [R. Nabben, on a class of matrices which arise in the numerical solution ofEuler equations, Numer. Math. 1992, 63: 411-431] are given.In the last part (chapter four and five), for the Perron root of nonnegativeirreducible matrices, a new sequence of upper bounds is presented, whichconvergence is discussed. The comparison of the new sequence with the knownones is supplemented with two numerical examples. Finally, by the properties ofnonnegative matrices, we give some inequalities about the largest singularvalues and spectral radius of matrices.更多还原