Numerical Solution of Least Squares Eigenvalue Problems

该项目解决了一些悬而未决的问题 最小二乘和特征值问题的精确解。 这两 问题领域是相互关联的，但每个问题领域都有单独的问题 其中之一 对于最小二乘问题的研究， 在算法的分析和实现中。 两 等式约束方程的直接迭代解法 将考虑最小二乘问题。 关于直接方法的工作 涉及稀疏问题的错误分析和实现问题 以及用于消息传递体系结构。 迭代的努力 方法涉及一类预处理器的开发， 在约束最小二乘问题上效果很好。 本征值问题的研究是建立在理论基础上的 问题. 最近扰动界的相对误差在 一类对称对角占优矩阵的特征值 已经开发了矩阵。 已经表明，一些算法 实现这些边界。 对于一类重要的算法， 分而治之的算法，它是不知道是否相对 可以达到误差界限。 由于这类算法具有 在几种分布式存储器架构上显示出良好的性能， 这一未决问题解决将对 如何解决特征值问题。

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.