Gaussian elimination corrects pivoting mistakes

TL;DR

Gaussian elimination is highly robust to pivoting mistakes, with later steps correcting earlier errors, enabling stable low-rank approximations even with non-optimal pivoting.

Contribution

The paper demonstrates that Gaussian elimination corrects pivoting mistakes during the process, ensuring stability despite non-optimal pivot choices.

Findings

GE corrects earlier pivoting mistakes in subsequent steps

GE remains numerically stable despite pivoting errors

This robustness supports iterative low-rank approximation methods

Abstract

Gaussian elimination (GE) is the archetypal direct algorithm for solving linear systems of equations and this has been its primary application for thousands of years. In the last decade, GE has found another major use as an iterative algorithm for low rank approximation. In this setting, GE is often employed with complete pivoting and designed to allow for non-optimal pivoting, i.e., pivoting mistakes, that could render GE numerically unstable when implemented in floating point arithmetic. While it may appear that pivoting mistakes could accumulate and lead to a large growth factor, we show that later GE steps correct earlier pivoting mistakes, even while more are being made. In short, GE is very robust to non-optimal pivots, allowing for its iterative variant to flourish.