Good pivots for small sparse matrices

TL;DR

This paper investigates optimal pivot choices for small sparse matrices during Gaussian elimination, demonstrating slight improvements over traditional strategies and introducing a machine learning-based approach for better pivot selection.

Contribution

The paper introduces a machine learning-based pivot selection strategy that improves Gaussian elimination for small sparse matrices over classical methods.

Findings

ML-based pivot strategy outperforms classical methods

Optimal pivots are slightly better than standard choices

Small matrices benefit from tailored pivot selection

Abstract

For sparse matrices up to size $8 \times 8$, we determine optimal choices for pivot selection in Gaussian elimination. It turns out that they are slightly better than the pivots chosen by a popular pivot selection strategy, so there is some room for improvement. We then create a pivot selection strategy using machine learning and find that it indeed leads to a small improvement compared to the classical strategy.