{
  "abstract": "We construct an explicit quartic polynomial for which Newton's and Halley's root-finding methods both fail to converge to a root on nonempty open sets of initial values. The example can be found through a reproducible computational search within a one-parameter family of quartics designed to exhibit a prescribed Newton two-cycle. We then prove analytically that Newton's method has a superattracting two-cycle and that Halley's method has a distinct attracting extraneous two-cycle, using a two-interval contraction argument. This fills a degree-four gap in the simultaneous Newton--Halley failure problem. The computational approach also identifies another quartic with numerical evidence of simultaneous failure of Newton's, Halley's, and Schröder's methods.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03246",
  "author_total": 2,
  "authors": [
    {
      "name": "Alessio Basti"
    },
    {
      "name": "Tommaso Cremaschi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/010x9r599br0m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03246",
  "fulltext_url": null,
  "id": "010x9r599br0m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03246",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2612|Iterative Methods for Nonlinear Equations"
    ]
  ],
  "openalex": "W7220456609",
  "pdf_url": "https://arxiv.org/pdf/2610.03246",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/010x9r599br0m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03246",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03246"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Simultaneous Global Convergence Failure of Newton's and Halley's Methods in Degree Four",
  "title_text": "Simultaneous Global Convergence Failure of Newton's and Halley's Methods in Degree Four",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/010x9r599br0m",
  "venue": "arXiv",
  "versions": []
}