{
  "abstract": "We prove that the restarted Arnoldi algorithm for real symmetric matrices with restart length $s=2$ has a cyclic limiting behavior. In the terminology established in (arXiv:2209.14579), this is a proof of the ACI($s$) conjecture for $A=A^T\\in \\mathbb{R}^{n\\times n}$ and $s=2$. As explained in (arXiv:2209.14579), this also proves the Forsythe conjecture (Numerische Mathematik, 11 (1968), pp. 57-76) about the limiting behavior of the conjugate gradient method and positive definite $A=A^T\\in\\mathbb{R}^{n\\times n}$ for $s=2$. The proof we give is an extensively rewritten and simplified version of a proof found using Codex with GPT-5.6 Sol. At the end of this paper, we discuss its origin and evolution, as well as related recent developments around the Forsythe conjecture. Our motivation is to present a proof that is instructive and transparent, so that it can enhance the knowledge of interested human readers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16492",
  "author_total": 4,
  "authors": [
    {
      "name": "Vance Faber"
    },
    {
      "name": "Jarek Liesen"
    },
    {
      "name": "Jörg Liesen"
    },
    {
      "name": "Petr Tichý"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bd6hv4rwfe34g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16492",
  "fulltext_url": null,
  "id": "bd6hv4rwfe34g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16492",
  "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": "W7221793319",
  "pdf_url": "https://arxiv.org/pdf/2610.16492",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bd6hv4rwfe34g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16492",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16492"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "On the cyclic limiting behavior of the restarted Arnoldi algorithm for real symmetric matrices with restart length $s=2$",
  "title_text": "On the cyclic limiting behavior of the restarted Arnoldi algorithm for real symmetric matrices with restart length s=2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/bd6hv4rwfe34g",
  "venue": "arXiv",
  "versions": []
}