{
  "abstract": "Gaussian elimination with partial pivoting is the standard algorithm for solving a dense $n\\times n$ system of linear equations $Ax=b$. The ``growth factor'' $ρ\\,$ for this process may be as large as $2^{n-1}$, and when $ρ$ is large, the algorithm is unstable. Nevertheless, long experience has shown that Gaussian elimination is resoundingly stable in practice. Experiments indicate that among random matrices with independent normally distributed entries, growth factors $ρ\\gg n^{1/2}$ are in a precise sense exponentially rare. Here we take a step toward confirming this observation by proving it not for the full growth factor, which depends on all the entries of the upper-triangular factor $U$, but for the corner entry $u_{nn}^{}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04761",
  "author_total": 1,
  "authors": [
    {
      "name": "Lloyd N. Trefethen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4x8042wrw3wxk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04761",
  "fulltext_url": null,
  "id": "4x8042wrw3wxk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04761",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7220659047",
  "pdf_url": "https://arxiv.org/pdf/2610.04761",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4x8042wrw3wxk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04761",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04761"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Instability of Gaussian elimination is exponentially rare (proof of partial result)",
  "title_text": "Instability of Gaussian elimination is exponentially rare (proof of partial result)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/4x8042wrw3wxk",
  "venue": "arXiv",
  "versions": []
}