{
  "abstract": "We study uniform approximation of $\\exp(tz)$, $t>0$, on $(-\\infty,0]$ by rational functions $P_m(z)/(q_m-z)^m$ with a prescribed repeated pole $q_m>0$. A Möbius transformation reduces the problem to polynomial approximation of $F_λ(x)=\\exp\\!\\left(-λ\\frac{1-x}{1+x}\\right)$ on $[-1,1]$, where $λ=tq_m$. We derive a uniform two-saddle asymptotic formula for the Chebyshev coefficients, including explicit amplitude and phase and a relative remainder for each localized complex saddle contribution, covering fixed, sublinear, and linear pole scalings away from saddle coalescence. Because the two saddle contributions can cancel in a single coefficient, we pass to a growing block of neighboring coefficients and prove that the whole block cannot cancel. This transfers the coefficient asymptotics to approximation errors. For $q_m=(α/t)m$, $0<α<3\\sqrt3/2$, the best uniform error has two-sided order $m^{-1/2}H_e(α)^m$; at the optimal ratio $α=1/\\sqrt2$ this becomes $m^{-1/2}(\\sqrt2-1)^m$. The normalized Chebyshev-weighted $L^2$ projection error has an explicit bounded oscillatory profile. These prefactor-resolved estimates yield a two-term precision-to-work law and quantify mismatch between pole-design and stopping degrees. Finally, the scalar error gives an exact worst-case matrix-action benchmark for self-adjoint negative semidefinite matrices and dimension-independent shift-and-invert Krylov bounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21028",
  "author_total": 2,
  "authors": [
    {
      "name": "Fei Xue"
    },
    {
      "name": "Tianqi Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cheh0ex0javgy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21028",
  "fulltext_url": null,
  "id": "cheh0ex0javgy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21028",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.21028",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cheh0ex0javgy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21028",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21028"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Uniform Chebyshev asymptotics for repeated-pole rational approximation of the exponential",
  "title_text": "Uniform Chebyshev asymptotics for repeated-pole rational approximation of the exponential",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/cheh0ex0javgy",
  "venue": "arXiv",
  "versions": []
}