The exact fifth complete-pivoting growth factor
For a nonsingular $A \in \mathbb{R}^{5 \times 5}$ let $\rho_{\mathrm{CP}}(A)$ be the growth factor of Gaussian elimination with complete pivoting in exact arithmetic: the largest entry modulus of the successive active Schur complements $S_1 = A, S_2, \dots, S_5$, divided by that of $A$. Let $$g_5 = \sup \{\rho_{\mathrm{CP}}(A) : A \in \mathbb{R}^{5 \times 5} \text{ nonsingular}\},$$ the supremum being taken over all complete-pivoting paths, that is, over every choice among tied pivots. Let $\alpha \approx 4.1325170786$ be the unique real root in $(4, 5)$ of the integer polynomial $P_5$ of degree $61$ given by Chen, Edelman and Urschel. For orders $1, 2, 3, 4$ the corresponding suprema are $1, 2, 9/4$ and $4$. The largest growth factor found for order $5$ by numerical optimisation, by Day and Peterson in 1988 and since, is $4.1325\ldots$. Chen, Edelman and Urschel showed that this value is exactly $\alpha$, that it is attained, so $g_5 \ge \alpha$, and that it is the largest growth factor among completely pivoted matrices satisfying the same pattern of equality constraints as the numerical optimum; they also proved $g_5 \le 4.84$.
Problem. Is $g_5 = \alpha$?
Source: Chen, Edelman and Urschel (2026), Conjecture 2.1, with P₅ their equation (2.15).
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- NA — Numerical Analysis
- Keywords
- Gaussian elimination, growth factors, OpenProblemsInNLA
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@article{chen2026,
author = {Chen, James and Edelman, Alan and Urschel, John},
title = {The largest 5th pivot may be the root of a 61st degree polynomial},
journal = {IMA J. Numer. Anal.},
year = {2026},
pages = {drag039},
doi = {10.1093/imanum/drag039},
eprint = {2602.20390},
archivePrefix = {arXiv},
note = {Advance article},
}
@article{day1988,
author = {Day, Jane and Peterson, Brian},
title = {Growth in {G}aussian elimination},
journal = {Amer. Math. Monthly},
volume = {95},
number = {6},
year = {1988},
pages = {489--513},
doi = {10.1080/00029890.1988.11972038},
}
@misc{openproblemsinnla-ie11,
author = {Townsend, Alex},
title = {{IE-11}: The exact fifth complete-pivoting growth factor},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/linear-systems-and-elimination/IE-11/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00189, "The exact fifth complete-pivoting growth factor", clopen.solutions.