Computational Methods in Linear Algebra Using the Theory of Moments

为解决应用中出现的大规模矩阵问题，发展了数值方法。这些应用相当多样化，包括统计建模和量子色动力学（QCD）等领域。这些计算的基本工具来自经典的矩理论和数值正交。这些问题等价于估计二次型的上界和下界。一个特别有趣的领域是计算广义交叉验证参数。该技术用于正则化方法求解不适定最小二乘问题。开发了一种数值程序，使人们能够分析非常大的数据集。在此基础上，提出了一种求解病态稀疏最小二乘问题中重要参数的计算方法。本研究的基本目标是开发新的鲁棒算法来解决非常大规模的线性问题。

Numerical methods are developed for solving large scale matrix problems arising in applications. These applications are quite diverse and include such areas as statistical modeling and quantum chromodynamics (QCD). The basic tool for these computations comes from the classical theory of moments and numerical quadrature. These problems are equivalent to estimating upper and lower bounds for a quadratic form. One area of particular interest is computing the generalized cross validation parameter. This technique is used in solving ill-posed least squares problems by regularization methods. A numerical procedure is developed which enables one to analyze very large data sets. Also, a method for computing a parameter which is of importance in solving ill-conditioned sparse least squares problems is developed. The basic objective of this research is to develop new and robust algorithms for solving very large scale linear problems.