Numerical Solution of Eigenvalue and Singular Value Problems with Applications

本研究涉及特征值和奇异值问题的数值求解算法的发展和分析，并应用于最小二乘和总最小二乘问题。许多工作是由摄动理论发展而来的，用于解决广义奇异值问题。该理论也适用于某些广义特征值问题。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更精确的程序。这是计算奇异值分解最有效的稳定方法的第一步。3. 对称箭头矩阵特征向量的一种算法，该算法在分布式架构下需要较少的通信。4. 用秩揭示分解解递归总最小二乘问题的一种算法。

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.