{
  "abstract": "This paper presents several parallel versions of Jacobi's method for the symmetric eigenvalue problem and the SVD. A continuation of [Demmel, Luo, Schneider, & Wang 2025], we develop parallel Jacobi algorithms whose arithmetic cost is optimal and whose bandwidth or latency can match the corresponding lower bounds of parallel matrix multiplication. Our focus is a standard distributed-memory setting with variable processor layouts, including both 2D and 2.5D processor grids. In the 2D case, we demonstrate that a standard implementation of parallel Jacobi achieves a perfect speedup in arithmetic cost -- i.e., complexity $O(n^3/P)$ when done with $P$ processors -- while hitting the 2D matrix-multiplication lower bound for bandwidth and (nearly) the lower bound for latency. By employing a 2.5D processor grid and leveraging 2.5D matrix multiplication, equivalently by increasing the memory per processor, we demonstrate that parallel Jacobi can achieve even lower bandwidth/latency, though we also prove that these costs cannot simultaneously match the best-known bounds for parallel matrix multiplication in any Jacobi algorithm. Finally, we extend our results to one-sided Jacobi SVD.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28952",
  "author_total": 4,
  "authors": [
    {
      "name": "James Demmel"
    },
    {
      "name": "Hengrui Luo"
    },
    {
      "name": "Ryan Schneider"
    },
    {
      "name": "Yifu Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7njnjnrj703wa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28952",
  "fulltext_url": null,
  "id": "7njnjnrj703wa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28952",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7204902032",
  "pdf_url": "https://arxiv.org/pdf/2608.28952",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7njnjnrj703wa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.28952",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28952"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Minimizing the Arithmetic and Communication Complexity of Jacobi's Method for Eigenvalues and Singular Values: Part Two -- Parallel Algorithms",
  "title_text": "Minimizing the Arithmetic and Communication Complexity of Jacobi's Method for Eigenvalues and Singular Values: Part Two -- Parallel Algorithms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/7njnjnrj703wa",
  "venue": "arXiv",
  "versions": []
}