Computational Methods in Linear Algebra Using the Theory of Moments

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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.