Numerical Solution of Eigenvalue Problems and Related Least Squares Problems

本研究旨在提高求解某些特定特征值问题的方法的准确性。第一个是经典的对称特征值问题。最近的结果表明，存在一大类对称本征值问题，其中小的结构扰动导致本征值在相对意义上的小变化。这类矩阵被称为“行为良好”。我们正在研究一个矩阵“行为良好”的充要条件的问题。必须考虑一些相关的问题，包括：分析更新不定特征值问题的方法；研究了奇异值分解和广义奇异值分解的ULV逼近和ULLV逼近的更新方法；并对稀疏广义奇异值分解的计算方法进行了分析。后两个问题在求解病态最小二乘问题和全最小二乘问题中是非常重要的。目前正在研究所得结果对这些问题的影响。

This research seeks to improve the accuracy of methods to solve some particular eigenvalue 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 necessary and sufficient condition for a matrix to be ``well-behaved'' is being investigated. A number of related issues have to be considered, including: analysis of methods for updating indefinite eigenvalue problems; investigation of methods for updating the ULV and ULLV approximations to the singular value decomposition and generalized singular value decomposition; and development of analysis of methods for computing the sparse generalized singular value decomposition. The last two issues are extremely important in the solution of ill-posed least squares problems and total least squares problems. The implications of derived results for these problems are being investigated.