Rehabilitation of the Gauss-Jordan algorithm

Abstract

Summary.In this paper a Gauss-Jordan algorithm with column interchanges is presented and analysed. We show that, in contrast with Gaussian elimination, the Gauss-Jordan algorithm has essentially differing properties when using column interchanges instead of row interchanges for improving the numerical stability. For solutions obtained by Gauss-Jordan with column interchanges, a more satisfactory bound for the residual norm can be given. The analysis gives theoretical evidence that the algorithm yields numerical solutions as good as those obtained by Gaussian elimination and that, in most practical situations, the residuals are equally small. This is confirmed by numerical experiments. Moreover, timing experiments on a Cyber 205 vector computer show that the algorithm presented has good vectorisation properties.

Simultaneous backward stability of Gauss and Gauss–Jordan elimination

“…If A is an n × n matrix, this pivoting strategy requires O(n 2 ) operations. Let us recall that in Reference [5] it was shown that GJE with this pivoting strategy presents a satisfactory bound for the residual norm, in contrast to the usual partial pivoting. Let us deÿne another pivoting strategy.…”

On the backward stability of Gauss-Jordan elimination

“…The notion of equivalent perturbation (see [5]) is introduced for every piece of data (input, intermediate and output) in contrast to the Wilkinson's backward analysis, where equivalent perturbations of the inputs only are considered. Then a system of linear equations Be = ~7 (1) with respect to the vector of all the equivalent perturbations is derived. The solution of this system (if it exists) gives a first order approximation to the equivalent perturbations.…”

Solving dense linear systems by Gauss-Huard's method on a distributed memory system

“…Email: walter@fwi.uva.nl pivoting strategy [6]. After the stability of GaussJordan's method had been properly established in 1989 [1], a stabilizing pivoting strategy for GaussHuard's method could be introduced. In 1992 it was proven that Gauss-Huard's method in combination with this pivoting strategy is numerically stable [3].…”