The largest 5th pivot may be the root of a 61st degree polynomial

TL;DR

This paper develops new techniques combining numerical and exact computations to precisely identify the maximum growth factor in Gaussian elimination with complete pivoting for small matrices, revealing it as a root of a 61st degree polynomial.

Contribution

It introduces a novel approach using JuMP, Groebner bases, and interval arithmetic to exactly determine the maximum growth factor, providing the first exact polynomial characterization for n=5.

Findings

Exact maximum growth factor for n=5 is the root of a 61st degree polynomial.

The maximum value matches previous numerical results of approximately 4.1325.

The paper proposes a conjecture that this value is the true maximum.

Abstract

This paper introduces a number of new techniques in the study of the famous question from numerical linear algebra: what is the largest possible growth factor when performing Gaussian elimination with complete pivoting? This question is highly complex, due to a complicated set of polynomial inequalities that need to be simultaneously satisfied. This paper introduces the JuMP + Groebner basis + discriminant polynomial approach as well as the use of interval arithmetic computations. Thus, we are introducing a marriage of numerical and exact mathematical computations. In 1988, Day and Peterson performed numerical optimization on $n=5$ with NPSOL and obtained a largest seen value of $4.1325...$. This same best value was reproduced by Gould with LANCELOT in 1991. We ran extensive comparable experiments with the modern software tool JuMP and also saw the same value $4.1325...$. While the combinatorial explosion of possibilities prevents us from knowing whether there may not be a larger maximum, we succeed in obtaining the exact mathematical value: the number $4.1325...$ is exactly the root of a 61st degree polynomial provided in this work, and is a maximum given the equality constraints seen by JuMP. In light of the numerics, we pose the conjecture that this lower bound is indeed the maximum. We also apply this technique to $n = 6$, $7$, and $8$. Furthermore, in 1969, an upper bound of $4\frac{17}{18}\approx 4.94$ was produced for the maximum possible growth for $n = 5$. We slightly lower this upper bound to $4.84$.