{
  "abstract": "We show that the well-known Ehrlich--Aberth root finding method is not generally convergent. More precisely, we show that there is an open subset of polynomials of any degree $d\\ge 7$, and an open subset of initial conditions, such that the iteration of the Ehrlich--Aberth method converges to an attracting periodic cycle of period $4$, rather than to a vector of roots. The proof is constructive in the sense that we present explicit values for the polynomial and the $4$-cycle. The search is highly non-trivial: we found no attracting cycles for lower degrees/periods, and the size of the attracting domain relative to a ``unit cube'' in the space of polynomials and starting vectors is on the order of $10^{-127}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02143",
  "author_total": 3,
  "authors": [
    {
      "name": "Torben Grabbel"
    },
    {
      "name": "Nedim Srkalovic"
    },
    {
      "name": "Dierk Schleicher"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bgjks4pck06xa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02143",
  "fulltext_url": null,
  "id": "bgjks4pck06xa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02143",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02143",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bgjks4pck06xa.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02143",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02143"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "The Ehrlich--Aberth root finder is not generally convergent",
  "title_text": "The Ehrlich--Aberth root finder is not generally convergent",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/bgjks4pck06xa",
  "venue": "arXiv",
  "versions": []
}