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Numerical Methods for the Unsymmetric Tridiagonal EigenvalueProblem

非对称三对角特征值问题的数值方法

基本信息

批准号：
9109785
负责人：
Elizabeth Jessup
金额：
$ 4.66万
依托单位：
University of Colorado at Boulder
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
1991
资助国家：
美国
起止时间：
1991-07-01 至 1994-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9109785&HistoricalAwards=false
关键词：
Numerical Methods Unsymmetric Tridiagonal EigenvalueProblem

项目摘要

Large order eigenproblems arise in a variety of applications. The long computing times and large storage requirements of these problems motivate development of fast serial and efficient parallel eigensolvers. New numerical methods will be investigated for computing all eigenvalues and eigenvectors of real unsymmetric tridiagonal matrices that arise in applications and from reduction of general real matrices. The immediate goal is to provide new methods for the tridiagonal case. Because current methods for the general eigenproblem can suffer from parallel inefficiency or numerical instability, the larger goal is to develop accurate parallel methods that might ultimately be extended to the general eigenproblem without initial reduction to tridiagonal form. The research derives from methods that have been successful for the symmetric tridiagonal eigenproblem. The work will begin with a study of parallel methods for computing eigenvectors of an unsymmetric tridiagonal matrix given its computed eigenvalues. It will continue by developing divide and conquer methods to compute both eigenvalues and eigenvectors of an unsymmetric tridiagonal matrix. It will conclude with a mechanism for roughly locating the eigenvalues of any tridiagonal matrix. The latter technique is geared toward accelerating rootfinding methods for computing eigenvalues. Serial and parallel algorithms will be designed, implemented, and tested for all approaches.

大阶本征值问题在各种应用中都会出现。这些问题的长计算时间和大存储需求推动了快速、串行和高效的并行特征解算器的发展。新的数值方法将被用来计算实非对称三对角阵的所有特征值和特征向量，这些特征值和特征向量是由一般实矩阵的约化而产生的。目前的目标是为三对角线的情况提供新的方法。由于目前求解一般特征问题的方法可能存在并行效率低或数值不稳定的问题，因此更大的目标是发展精确的并行方法，这些方法最终可能扩展到一般特征问题而不需要初始化为三对角线形式。该研究源于已成功求解对称三对角线特征问题的方法。这项工作将首先研究计算非对称三对角矩阵的特征向量的并行方法，给出其计算的特征值。它将继续开发分而治之的方法来计算非对称三对角矩阵的特征值和特征向量。它将以一种粗略定位任意三对角矩阵的特征值的机制来结束。后一种技术旨在加速计算特征值的寻根方法。将针对所有方法设计、实施和测试串行和并行算法。