The explicit formula for Gauss-Jordan elimination and error analysis

Nam Van Tran, Júlia Justino, Imme van den Berg

Published 2020-05-21Version 1

The explicit formula for the elements of the successive intermediate matrices of the Gauss-Jordan elimination procedure is used for error analysis in the case that the procedure is applied to systems of linear equations. Stability conditions in terms of relative precision 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. We model imprecisions with the help of non-standard analysis. A direct proof by induction is given of the explicit formula for the intermediate matrices.