Symmetric Exchange of Variables and Stepwise Regression Methods

A step in Gauss-Jordan elimination may be seen as exchanging a dependent variable for an independent one. If at the same time one of them changes sign a symmetric system matrix remains symmetric with the pivot reversing sign 2]. This variant can efficiently be exploited inverting symmetric positive definite matrices but also for obtaining a most complete set of relevant statistical quantities in stepwise regression via normal equations proposed by Efroymson 1960 and improved by Breaux 1968. We compare this method with numerically more stable ones applying orthogonal transformations, in particular that of Eldéen 1972 using Householder transformations and that of Gragg-LeVeque-Trangenstein 1979 using modified Gram-Schmidt. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)