{
  "abstract": "In this note, we provide a short and direct proof that approximately solving $Ax=b$ to relative error $\\varepsilon$, where $A$ has condition number $κ$ and unrestricted dimension, requires $Ω(κ\\log(1/\\varepsilon))$ matrix-vector multiplications in the worst case, even for randomized algorithms. This essentially recovers the lower bound of Dereziński, Epperly and Meyer [2026] for this setting, whose elegant and more general approach inspired us to seek a short direct proof. A straightforward reduction implies the classical $Ω(\\sqrtκ\\log(1/\\varepsilon))$ lower bound for optimizing strongly convex quadratic functions, applicable to randomized algorithms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10874",
  "author_total": 1,
  "authors": [
    {
      "name": "Ohad Shamir"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cmpn505cheqcb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10874",
  "fulltext_url": null,
  "id": "cmpn505cheqcb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10874",
  "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"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7212279511",
  "pdf_url": "https://arxiv.org/pdf/2609.10874",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cmpn505cheqcb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.10874",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10874"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A Simple Complexity Lower Bound for Solving $Ax=b$",
  "title_text": "A Simple Complexity Lower Bound for Solving Ax=b",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/cmpn505cheqcb",
  "venue": "arXiv",
  "versions": []
}