Numerical Solution of Eigenvalue and Singular Value Problems with Applications

这项研究是关于特征值和奇异值问题的数值算法的发展和分析，并将其应用于最小二乘和总体最小二乘问题。许多工作源于广义奇异值问题的摄动理论。该理论也适用于某些广义本征值问题。因此，该理论将应用于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有：1.稀疏广义奇异值分解计算方法的发展与分析。2.计算矩阵的双对角化简的一种更精确的方法。这是计算奇异值分解的最有效的稳定方法的第一步。3.提出了一种计算对称箭头矩阵特征向量的算法，该算法在分布式体系结构上需要较少的通信。4.利用显阶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.