{
  "abstract": "This article develops matrix-free Q-less QR algorithms for orthogonalizing tall linear operators. Our main algorithm, CQRRTO, uses randomized preconditioning to make a single Cholesky QR pass numerically safe for any numerically full-rank operator, overcoming the stability issues of Cholesky QR and related methods that operate on the Gram matrix. Compared with deterministic stabilized variants such as Q-less Cholesky QR2 (reorthogonalized Cholesky QR), it remains numerically stable under substantially weaker conditioning assumptions while requiring fewer operator applications. Matrix-free Q-less QR can be useful for improving the numerical robustness of least-squares solves via the normal equations, spectral approximations in a low-dimensional basis, and inversion of operators with low-rank updates via the Woodbury identity. We present a componentwise stability analysis of preconditioned Q-less Cholesky QR that applies to matrix-free operators and is modular with respect to the preconditioner, yielding guarantees for Q-less Cholesky QR, Cholesky QR2, shifted Cholesky QR3, and CQRRTO as special cases. All methods are implemented in the open-source RandLAPACK library.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16101",
  "author_total": 3,
  "authors": [
    {
      "name": "Oleg Balabanov"
    },
    {
      "name": "Maksim Melnichenko"
    },
    {
      "name": "Michael W. Mahoney"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0wpq90wpm2np8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16101",
  "fulltext_url": null,
  "id": "0wpq90wpm2np8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16101",
  "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",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7221817528",
  "pdf_url": "https://arxiv.org/pdf/2610.16101",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0wpq90wpm2np8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16101",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16101"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Stable Matrix-Free Q-less QR via Preconditioning and Randomization",
  "title_text": "Stable Matrix-Free Q-less QR via Preconditioning and Randomization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/0wpq90wpm2np8",
  "venue": "arXiv",
  "versions": []
}