Wilkinson's complete-pivoting growth-factor bound conjecture

Wilkinson’s conjecture asserted that complete pivoting never produces growth exceeding the matrix dimension, with equality only for Hadamard matrices. Gould found a dimension-$13$ counterexample in 1991, and Edelman confirmed it in exact arithmetic in 1992.

Known results

• Exact values are known for $n=1,2,3,4$; the fifth-dimensional maximum is strictly below $5$.
• Edelman and Urschel proved $g_n(\mathbb{R})\ge 1.0045n$ for every $n\ge 11$.
• They also proved $\limsup_{n\to\infty}g_n(\mathbb{R})/n\ge 3.317$.

2023 upper-bound improvement

Edelman, Urschel, and Bisain improved the general upper bound to approximately $n^{0.20781\log n+0.91}$, but the gap between upper and lower bounds remains substantial; this does not revive the refuted conjecture.

Current status (as of September 2026): The conjecture is refuted by an exact-arithmetic counterexample in dimension $13$ and by lower bounds exceeding $n$ for every $n\ge 11$; the asymptotic growth factor remains open.