{
  "abstract": "Alefeld (1981) recast Halley's cubic convergence as Newton's method on $g=f/\\sqrt{f'}$, linked to the Schwarzian derivative. We generalize this to matrix gradient fields $F=\\nablaϕ$ ($X\\in\\R^{m\\times n}$) via a matrix Schwarzian derivative interpreted through information geometry ($α$-connections). Simplifying Palmore (1994), we construct this operator square-root-free derivative from the third Fréchet derivative of the Newton map. Our main theorem proves local cubic convergence with an explicit error constant, without self-adjointness, commutativity, or gradient-field assumptions. A matrix-free algorithm (Hessian-vector products only) validates the theory. We contrast this with a power-Newton family (whose naive matrix extension fails) and the matrix Laguerre family (which requires an operator square root). A case study on the Oja-type system $\\dot X=AXB-XBX^TAX$ reveals that convergence depends on target eigenvalue gaps, cubic gains grow with ill-conditioning, and 70-digit tests confirm exact theoretical orders alongside a working-precision accuracy budget. Finally, we note coupled formulations excel primarily when sequential deflation is inapplicable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06592",
  "author_total": 1,
  "authors": [
    {
      "name": "Shintaro Yoshizawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3mv0tcay9vsvr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06592",
  "fulltext_url": null,
  "id": "3mv0tcay9vsvr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06592",
  "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",
      "2612|Iterative Methods for Nonlinear Equations"
    ]
  ],
  "openalex": "W7212055596",
  "pdf_url": "https://arxiv.org/pdf/2609.06592",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3mv0tcay9vsvr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06592",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06592"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Halley's Method for Rectangular Matrix Variables and the Matrix Schwarzian Derivative",
  "title_text": "Halley's Method for Rectangular Matrix Variables and the Matrix Schwarzian Derivative",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/3mv0tcay9vsvr",
  "venue": "arXiv",
  "versions": []
}