{
  "abstract": "We focus on solving the Lyapunov equation $AX + XA^T = F$, where $A$, $X$ and $F$ are square matrices, $A$ is symmetric positive definite (SPD) and sparse, and $F$ is symmetric and of a low-rank. The solution $X$ is then also symmetric and in general dense, but it can be approximated by a low-rank matrix. If $n$ is large, $X$ cannot be computed directly, but it is accessible by using the so-called low-rank arithmetics (LRA). The equation is solved by matrix reformulation of the method of conjugate gradients (MCG). We are interested in the behavior of ranks of the solution approximations $X_\\ell$, residuals $R_\\ell$, and direction vectors $P_\\ell$ during iterations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39308",
  "author_total": 2,
  "authors": [
    {
      "name": "Jana Lungová"
    },
    {
      "name": "Martin Plešinger"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/exd6sa54w235h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39308",
  "fulltext_url": null,
  "id": "exd6sa54w235h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39308",
  "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": "W7215109991",
  "pdf_url": "https://arxiv.org/pdf/2609.39308",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/exd6sa54w235h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39308",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39308"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "On bounds for rank growth of iterands in conjugate gradients for Lyapunov equation",
  "title_text": "On bounds for rank growth of iterands in conjugate gradients for Lyapunov equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/exd6sa54w235h",
  "venue": "arXiv",
  "versions": []
}