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Numerical Solution of Least Squares Eigenvalue Problems

最小二乘特征值问题的数值求解

基本信息

批准号：
9000526
负责人：
Jesse Barlow
金额：
$ 5.5万
依托单位：
Pennsylvania State Univ University Park
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
1990
资助国家：
美国
起止时间：
1990-08-15 至 1993-01-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9000526&HistoricalAwards=false
关键词：
Numerical Solution Least Squares Eigenvalue

项目摘要

This project addresses a number of open questions concerning the accurate solution of least squares and eigenvalue problems. These two problem areas are interrelated, but there are separate issues for each one of them. For the work on least squares problems, there remain important issues in both the analysis and implementation of the algorithms. Both direct and iterative methods for the solution of equality constrained least squares problems will be considered. The work on direct methods concerns error analysis and implementation issues for sparse problems and for message passing architectures. The effort on iterative methods concerns the development of a class of preconditioners that would work well on constrained least squares problems. The investigation of the eigenvalue problems is based upon theoretical questions. Recently perturbation bounds on the relative errors in the eigenvalues of a certain class of symmetric diagonally dominant matrices have been developed. It has been shown that some algorithms achieve these bounds. For one important class of algorithms, divide-and-conquer algorithms, it is not known whether the relative error bounds can be achieved. Since this class of algorithms has shown good performance on several distributed memory architectures, the resolution of this open question will have a significant impact on how eigenvalue problems are solved.

该项目解决了有关最小二乘和特征值问题的精确解决方案的许多悬而未决的问题。这两个问题领域是相互关联的，但每个领域都有不同的问题。对于最小二乘问题的研究，算法的分析和实现仍然存在重要问题。将考虑求解等式约束最小二乘问题的直接方法和迭代方法。直接方法的工作涉及稀疏问题和消息传递架构的错误分析和实现问题。迭代方法的工作涉及开发一类可以很好地解决约束最小二乘问题的预处理器。特征值问题的研究基于理论问题。最近，已经开发了一类对称对角占优矩阵的特征值的相对误差的扰动界。事实证明，一些算法可以达到这些界限。对于一类重要的算法，即分而治之算法，尚不清楚是否可以达到相对误差界限。由于此类算法在多种分布式内存架构上表现出了良好的性能，因此这个悬而未决的问题的解决将对如何解决特征值问题产生重大影响。