基于特征对灵敏度分析的二次特征值问题的条件数

发布时间：2018-05-11 03:25

本文选题：二次特征值问题 + 条件数　；参考：《华东理工大学》2017年硕士论文

【摘要】：矩阵特征值的条件数反映了特征值对于矩阵元素变化的敏感性,它对于衡量特征值问题数值算法的稳定性有重要作用。本文以正则二次特征值问题半单特征值的解析扰动为基础,研究了正则二次特征值问题半单特征值的条件数。我们从半单特征值的方向导数出发,给出了正则二次特征值问题半单特征值条件数的多种定义。利用奇异值分解和酉不变范数的性质,导出了条件数的计算表达式。和已有结果相比较,本文定义的条件数不仅可以衡量重特征值扰动的最坏情形,而且能反映重特征值扰动后产生的不同特征值的相应灵敏度。另一方面,本文还研究了二次特征值问题半单特征值的病态扰动,给出了半单特征值重数发生改变时系数矩阵的扰动上界。
[Abstract]:The condition number of matrix eigenvalue reflects the sensitivity of eigenvalue to the change of matrix element, and it plays an important role in evaluating the stability of numerical algorithm for eigenvalue problem. Based on the analytic perturbation of semi-simple eigenvalues of regular quadratic eigenvalue problems, the condition number of semi-simple eigenvalues for regular quadratic eigenvalue problems is studied. Based on the directional derivatives of semi-simple eigenvalues, we give several definitions of semi-simple eigenvalue conditions for regular quadratic eigenvalue problems. By using the properties of singular value decomposition and unitary invariant norm, the expression of conditional number is derived. Compared with the existing results, the condition number defined in this paper can not only measure the worst-case of the repeated eigenvalue perturbation, but also reflect the sensitivity of different eigenvalues produced by the repeated eigenvalue perturbation. On the other hand, we also study the ill-conditioned perturbation of semi-simple eigenvalue for quadratic eigenvalue problems, and give the upper bound of the perturbation of coefficient matrix when the multiplicity of semi-simple eigenvalue changes.
【学位授予单位】：华东理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.6

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