{
  "abstract": "We develop Toeplitz infinite GMRES for solving large sparse analytic parameterized systems $A(s)x(s)=z$ at many parameter values. Our method exploits a block upper triangular Toeplitz structure in the companion Krylov sequence to construct the Arnoldi process and recover solution approximations without storing full Arnoldi vectors. We derive incremental recurrences requiring $\\mathcal{O}(np^2+p^3)$ arithmetic and $\\mathcal{O}(np+p^2)$ storage for $p$ Arnoldi steps, excluding factorization setup and assuming linear-cost coefficient actions and triangular solves. A dynamic generator-refreshing strategy addresses cancellation in this basic recurrence. Moreover, we show that the Arnoldi basis can be simply represented using one block row and a nilpotent matrix. For $A(s)$ admitting a split form, we further use this structure to develop an implicitly rebased method. A residual lower bound for square-summable Taylor coefficients explains the effect of scaling outside the unit disk. Theoretical analysis and numerical experiments illustrate the computational efficiency of the proposed methods.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01740",
  "author_total": 2,
  "authors": [
    {
      "name": "Weiguo Gao"
    },
    {
      "name": "Feiyang Jiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3hb28h3xm9qq1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01740",
  "fulltext_url": null,
  "id": "3hb28h3xm9qq1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01740",
  "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": "W7216174508",
  "pdf_url": "https://arxiv.org/pdf/2610.01740",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3hb28h3xm9qq1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.01740",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01740"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Toeplitz Infinite GMRES for Parameterized Linear Systems",
  "title_text": "Toeplitz Infinite GMRES for Parameterized Linear Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/3hb28h3xm9qq1",
  "venue": "arXiv",
  "versions": []
}