Numerical Solution of Eigenvalue and Singular Value Problems with Applications

本研究关注的发展和分析的数值算法的解决方案的特征值和奇异值的问题，与应用程序的最小二乘和总最小二乘问题。 大部分的工作源于广义奇异值问题的摄动理论。这一理论也适用于某些广义特征值问题。因此，该理论将被应用到各种奇异值和最小二乘问题的算法开发。 需要解决的具体问题是 1.稀疏广义奇异值分解计算方法的发展与分析。 2.计算矩阵双对角约简的更精确的过程。这是计算奇异值分解的最有效的稳定方法的第一步。 3.一种在分布式体系结构上计算对称箭阵特征向量的算法，该算法需要较少的通信。 4.使用秩显示分解求解递归总体最小二乘问题的算法。

This research concerns the development and analysis of numerical algorithms for the solution of eigenvalue and singular value problems, with applications to least squares and total least squares problems. Much of the work grows out of perturbation theory for generalized singular value problems. This theory is also applicable to certain generalized eigenvalue problems. The theory will thus be applied to algorithm development for various singular value and least squares problems. Specific problems to be addressed are 1. Development and analysis of methods for computing the sparse generalized singular value decomposition. 2. A more accurate procedure for computing a bidiagonal reduction of a matrix. This is the first step in the most efficient stable methods for computing the singular value decomposition. 3. An algorithm for the eigenvectors of symmetric arrow matrices that requires less communication on distributed architectures. 4. An algorithm for the solution of recursive total least squares problems using rank--revealing decompositions.