“Instability of Gaussian elimination is exponentially rare” (pair of papers on arXiv this past week, one by Prof. Lloyd N. Trefethen, another by Prof. John Urschel)

"[Gaussian elimination] is the simplest way to solve linear systems of equations by hand, and also the standard method for solving them on computers. [It] transforms a full linear system into an upper-triangular one by applying simple linear transformations on the left.") A pivoting strategy (strategy for swapping rows &/or columns) in necessary for generic linear systems, otherwise applying Gaussian elimination to certain invertible matrices will result in dividing by zero. Enter Gaussian elimination with partial pivoting. At step k , when considering the k -th column, choose the the i -th row (i ≥ k) with the largest number in absolute value. (Partial pivoting is less computationally expensive than other pivoting strategies, and is commonly used in practice, see Wikipedia: [1] & [2] .) The question arises: Is Gaussian elimination with partial pivoting stable? That is, for a given matrix n -by- n matrix A , we compute its Gaussian elimination with partial pivoting: P A = L U , where P is a permutation matrix, L a unit lower-triangular matrix, and U an upper triangular matrix. Define the growth factor ρ as the ratio: ρ = (max | U ᵢ‚ⱼ|) / (max | A ᵢ‚ⱼ|), where the maximum is taken over all indices 1 ≤ i ≤ n and 1 ≤ j ≤ n . By "Is Gaussian elimination with partial pivoting stable?" we mean "Is ρ bounded?", for some sense of the word "bounded". Professor Lloyd N. Trefethen offered a $1,000 reward in 2012 for "for a proof that Gaussian elimination with partial pivoting is stable in [a certain] probabilistic sense", for Gaussian random matrices,† as he outlined in SIAM News: siam.org/publications/siam-news/articles/the-smart-money-s-on-numerical-analysts . This past week he posted a partial result on arXiv: Trefethen (2026), Instability of Gaussian elimination is exponentially rare (proof of partial result) , arxiv.org/abs/2610.04761 Yesterday, Professor John Urschel posted a full resolution on arXiv, "[proving] that the growth factor [ ρ ] of a Gaussian matrix is at most n 1/2 + o(1) with overwhelming probability, that is, 1 - n - α for any α ": Urschel (2026), On the Growth Factor of Random Matrices , arxiv.org/abs/2610.06785 ) Trefethen, L. N., & Bau III, D. (1997). Numerical Linear Algebra. Society for Industrial and Applied Mathematics (SIAM). † Random matrices with independent, normally distributed entries.