Numerical Solution of Eigenvalue Problems

特征值问题的数值解

• 批准号: 9201612
• 负责人: Jesse Barlow
• 金额: $ 19.68万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-09-01 至 1996-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9201612&HistoricalAwards=false
• 关键词: Numerical; Solution; Eigenvalue; Problems

This research concerns the improvement of the accuracy of methods to solve two particular eignenvalue problems. The first is the classical symmetric eigenvalue problem. Recent results have shown that there is a large class of symmetric eigenvalue problems for which small structured perturbations result in small changes in the eigenvalues in the relative sense. This class of matrices is called "well- behaved". The problem of finding a neccessary and sufficent condition for a matrix to be "well-behaved" will be investigated. Another concern is that only a few algorithms have been shown to achieve relative accuracy on a subset of the set of "well- behaved" matrices. The investigation will try to discover if that set of algorithms can be expanded. The divide and conquer approaches of Cuppen, Dongarra, Jessup, Sorensen, and Tang are considered. Good absolute error bounds on these algorithms have already been established and it remains to show tighter relative error bounds for diagonal dominant problems and for the singular value decomposition. The other large problem area concerns the eigenvector problem for non-symmetric matrices. The problem arises in the numerical solution of Markov modeling problems. Accurate general bounds on Gaussian elimination have been proven. These allow for the straighforward application of standard sparse matrix techniques to this problem. However, better structured perturbation bounds need to be proved for the nearly uncoupled case.

本研究涉及提高的准确性， 方法来解决两个特殊的特征值问题。 第一个是经典的对称特征值问题。 最近的研究结果表明，有一大类 对称特征值问题 扰动导致特征值的微小变化， 相对意义。 这类矩阵被称为“好- 行为”。 找到一个必要的和足够的 矩阵“行为良好”的条件是 研究了 另一个令人担忧的问题是，只有少数算法已经显示 为了在“井”的集合的子集上实现相对准确性， 行为”矩阵。 调查将试图发现， 这组算法可以扩展。 的分而 征服Cuppen，Dongarra，杰瑟普，索伦森的方法，和 唐认为。 这些数据具有良好的绝对误差界 算法已经建立，它仍然是 示出了对角占优的更严格的相对误差界 问题和奇异值分解。 另一个大的问题领域涉及特征向量问题 对于非对称矩阵。 问题出现在 马尔可夫建模问题的数值解。 准确 高斯消去法的一般界限已被证明。 这些允许直接应用标准 稀疏矩阵技术来解决这个问题。 然而，更好地 结构扰动界需要证明的 几乎没有耦合的情况下。