The explicit formula for Gauss-Jordan elimination and error analysis

TL;DR

This paper derives an explicit formula for Gauss-Jordan elimination, enabling detailed error analysis and stability conditions, ensuring solutions respect initial uncertainties and align with Cramer's Rule.

Contribution

It introduces an explicit formula for intermediate matrices in Gauss-Jordan elimination and develops an extended error calculus for stability analysis.

Findings

Stability conditions relate uncertainties to determinants.

Solutions match those of Cramer's Rule.

Error tracking is extended through scalar neutrices.

Abstract

The explicit formula for the elements of the successive intermediate matrices of the Gauss-Jordan elimination procedure for the solution of systems of linear equations is applied to error analysis. Stability conditions in terms of relative uncertainty and size of determinants are given such that the Gauss-Jordan procedure leads to a solution respecting the original imprecisions in the right-hand member. The solution is the same as given by Cramer's Rule. Imprecisions are modelled by scalar neutrices, which are convex groups of (nonstandard) real numbers. The resulting calculation rules extend informal error calculus, and permit to keep track of the errors at every stage.