[论文解读] The largest 5th pivot may be the root of a 61st degree polynomial

James Chen, Alan Edelman+1 more|arXiv (Cornell University)|Feb 23, 2026

Polynomial and algebraic computation被引用 0

一句话总结

本论文引入 JuMP + Groebner 基础 + 判别式多项式方法，以确定在完全选主元的高斯消去法中精确的最大增长因子，并给出 n=5 时的最大值等于一个 61 次多项式的根，以及对 n=6–8 的结果。

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 $4.84$.

研究动机与目标

• Motivate and determine the maximum growth factor in Gaussian elimination with complete pivoting.
• Develop an exact, computable framework combining numerical optimization with algebraic techniques.
• Provide a rigorous upper/lower bound analysis and conjecture for the true maximum at n=5.

提出的方法

• Use JuMP to numerically optimize the growth factor under complete pivoting with equality constraints.
• Fix tight constraints from numerical results and solve for remaining variables using algebraic elimination.
• Apply Groebner basis computations to eliminate non-linear polynomially dependent variables.
• Use the discriminant polynomial to locate candidate maxima via a univariate polynomial in the growth factor.
• Derive an explicit 61st-degree polynomial whose real root in (4,5) equals the maximum growth factor for n=5.
• Extend the approach to n=6,7,8 to obtain lower bounds for those sizes.

Figure 1. The 61st degree polynomial with only 12 Mathematica input cells

实验结果

研究问题

• RQ1 What is the largest possible growth factor for Gaussian elimination with complete pivoting for n=5?
• RQ2 Can the maximum be characterized exactly as a root of a high-degree polynomial with integer coefficients?
• RQ3 How can numerical optimization be combined with exact algebraic methods to certify growth-factor maxima?
• RQ4 What are the implications and extensions of this approach to n>5 (e.g., n=6,7,8)?

主要发现

• The largest growth factor for 5x5 matrices under complete pivoting is the unique real root in [4,5] of a 61st-degree polynomial P5(g).
• The 61st-degree polynomial is explicitly obtained via a JuMP + Groebner basis + discriminant pipeline and factorization (the discriminant yields a polynomial in g with a single real root in (4,5)).
• The numerically observed value 4.1325… is exactly this root, with a high-precision corroboration (4.1325170786... and 4.132517078632472854223346853277).
• Lancelot and JuMP-based numerical experiments reproduce the same value, supporting it as a lower bound which the authors conjecture to be the maximum (Conjecture 2.1).
• The method provides exact lower bounds for n=6 (value 5), n=7 (root of a sixth-degree polynomial), and n=8 (value 8).
• An improved upper bound for n=5 of 4.84 is obtained via a combination of interval arithmetic and analysis, improving Tornheim’s 1968 bound.

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