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

The determination of the pseudo-rank of a system of linear equationsis complicated by ill-conditioning and the presence of noisein the matrix elements and/or data. A method is proposed forselecting a pseudo-rank suitable for calculating approximateleast-squares solutions by considering auto-correlation propertiesof the residual vector. The method is inspired by one proposedby Powell for the fitting of curves to data, and presupposesa given ordering of the equations. The method is applied toselected ill-conditioned systems with added random noise, andthe accuracy of the calculated solutions is investigated. Pseudo-ranksdetermined 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 determinationis a useful alternative to the method of generalized cross-validation,and that a combination of the two methods may yield more reliableselected pseudo-ranks.     

Quadratic Transformations of Triangular Finite Elements in Two Dimensions

The general quadratic isoparametric transformation of a triangleresults in an element with three parabolic curved sides. Thevertices are assumed fixed and the transformation is completelydetermined by the image positions of the non-vertex nodes. Inorder to study the bijectivity of this transformation, simplifiedsufficient conditions for the bijectivity of quadratic transformationsof any convex masterelement are derived. Properties of imageelements that result from the quadratic transformation of atriangle when the transformation Jacobians are linear are consideredin detail. Present address: Computer Research Laboratory, MS 50-662, Tektronix,Inc., PO Box 500, AFQ1, Beaverton, OR 97077, U.S.A.