{
  "abstract": "We give a deterministic algorithm that solves a nonsingular linear system $Ax=b$, where $A\\in\\mathbb{R}^{n\\times n}$ has $m$ nonzero entries and condition number $κ$, to any relative residual tolerance $0<ε\\le1/2$ using $\\widetilde{O(}mn\\log(κ/ε))$ bit operations for inputs with logarithmically many bits per entry. For sparse, polynomially conditioned systems with $m=\\widetilde{O}(n)$, this gives an $\\widetilde{O}(n^2)$ algorithm for any inverse-polynomial accuracy, improving on the algorithm of Peng and Vempala, as sharpened by Nie, whose running time in this regime is approximately $O(n^{2.2707})$ with the best current matrix multiplication exponent, and largely closing a gap between the idealized performance of the conjugate gradient method in exact arithmetic and the running time achievable with finite-precision computation that has persisted for over 70 years. The algorithm is surprisingly simple. For integer inputs, we apply Dixon's lifting algorithm to the perturbed system $(A+I/R)x=b$ for a suitable integer $R$. After scaling, the matrix of this system is $RA+I\\equiv I\\pmod R$, so its modular inverse is trivial, and each lifting step needs only a sparse matrix-vector product with $A$ on $O(\\log R)$-bit numbers. Fast rational reconstruction then recovers the exact solution of the perturbed system, which is an $ε$-accurate solution of the original one. Normalization and rounding extend the result to fixed-point and floating-point inputs, with floating-point outputs represented using short integer significands and a common encoded exponent. A computable certificate removes the need for prior knowledge of $κ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38101",
  "author_total": 1,
  "authors": [
    {
      "name": "Jonathan A. Kelner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/19bad72n9598n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38101",
  "fulltext_url": null,
  "id": "19bad72n9598n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38101",
  "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.CC"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7214926906",
  "pdf_url": "https://arxiv.org/pdf/2609.38101",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/19bad72n9598n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:36Z",
      "native_id": "2609.38101",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38101"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Solving Linear Systems in $\\widetilde{O}(mn \\log \\fracκε)$ Bit Operations",
  "title_text": "Solving Linear Systems in Õ(mn log κ/ε) Bit Operations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:36Z",
  "url": "https://research.draconitic.com/paper/19bad72n9598n",
  "venue": "arXiv",
  "versions": []
}