Numerical Control Structures for the Computation of Large Eigenvalue and Singular Value Problems

在数值程序的组成部分中， 对精度影响可以称为数字控制 结构：收敛检验，停止准则，收缩准则， 和识别需要特殊 治疗 为了保证计算的高精度， 这些控制结构必须是代数上可靠的 并且在有限精度运算中表现良好。 这个项目将利用一个问题的结构和几何形状， 设计数控结构的目的。 结构 奇异值分解将用于确定 方向的错误和发展的扰动理论;和 子空间的几何形状和它们之间的角度将用于设计 数控结构 由此产生的扰动理论和控制结构的预期 以导致用于计算的高度精确的实现， 对称矩阵的特征向量与奇异向量的逆 迭代，用于计算奇异值的（扩展） Golub-Reinsch算法，并通过 QL算法 通过设计，（扩展）Golub的数控结构， Reinsch算法在列缩放下是不变的。 因为最近 线性方程组解的误差界，奇异值 分解，和特征分解表现出相同的不变性， 将采用数控结构来解释 这些误差界的意义和特定形式。 目标是 对问题条件反射进行更现实的评估， 算法稳定性，这对于高精度 解决大问题。

Among the components of a numerical program that have the greatest influence on accuracy are what can be called numerical control structures: convergence tests, stopping criteria, deflation criteria, and identification of problem instances that require special treatment. In order to guarantee high accuracy in computations with large matrices these control structures must be algebraically sound and perform well in finite precision arithmetic. This project will exploit the structure and geometry of a problem for the purpose of designing numerical control structures. The structure of the singular value decomposition will be used to determine the directions of errors and develop a perturbation theory; and the geometry of subspaces and angles between them will be used to design numerical control structures. The resulting perturbation theory and control structures are expected to lead to highly accurate implementations for the computation of eigenvectors of symmetric matrices and singular vectors by inverse iteration, for the computation of singular values by the (extended) Golub-Reinsch algorithm, and for the computation of eigenvalues by the QL algorithm. By design, the numerical control structures of the (extended) Golub- Reinsch algorithm are invariant under column scaling. Since recent error bounds for linear system solution, singular value decompositions, and eigendecompositions exhibit the same invariance, the numerical control structures will be employed to explain the significance and particular form of these error bounds. The goal is to produce more realistic assessments of problem conditioning and algorithm stability, which is especially important for highly accurate solution of large problems.