The Use of Auto-correlation for Pseudo-rank Determination in Noisy III-conditioned Linear Least-squares Problems

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Abstract
The determination of the pseudo-rank of a system of linear equations is complicated by ill-conditioning and the presence of noise in the matrix elements and/or data. A method is proposed for selecting a pseudo-rank suitable for calculating approximate least-squares solutions by considering auto-correlation properties of the residual vector. The method is inspired by one proposed by Powell for the fitting of curves to data, and presupposes a given ordering of the equations. The method is applied to selected ill-conditioned systems with added random noise, and the accuracy of the calculated solutions is investigated. Pseudo-ranks determined by generalized cross-validation are used for comparison, and the relative accuracy of the computed solutions is considered. Results indicate that auto-correlated pseudo-rank determination is a useful alternative to the method of generalized cross-validation, and that a combination of the two methods may yield more reliable selected pseudo-ranks.

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