{
  "abstract": "Short-recurrence Krylov methods that are equivalent in exact arithmetic often diverge in floating-point arithmetic. To illustrate this, we provide an exactly representable two-cycle for steepest descent with a recursively updated residual. A complementary convergence theorem gives a sufficient condition under which the stored residual decreases geometrically. We then show that a second positive definite family yields different outcomes for a Hestenes--Stiefel Conjugate Gradient (CG) implementation and a direct Lanczos--Galerkin approach. CG finds the exact solution after four updates, whereas the two-step projected system becomes inconsistent. Under stated perturbation and transfer hypotheses, a common spectral enclosure gives comparable convergence bounds. For a dense, left-to-right evaluation order, we prove a sufficient precision bound for a prescribed backward error within $n$ updates for an $n\\times n$ matrix, together with a computable stopping test and specified exponent-range assumptions. We use polynomial estimates and numerical experiments to examine the trade-off between working precision and the number of CG iterations needed to achieve a prescribed backward error. We also compare results on inexact matrix-vector products and preconditioning with the frameworks of Paige and Greenbaum. For block Lanczos, we analyze Householder orthogonalization and singular-value truncation as the block size changes. Under componentwise and normwise error bounds, the computed coefficients satisfy a controlled local recurrence and an exact block Lanczos relation for a nearby symmetric problem in a larger space. A block form of Paige's identity bounds the overlap with a Ritz vector along its residual coordinate direction. An inter-block recurrence describes the evolution of overlap, and a Gram-matrix argument gives a count of additional nearby Ritz values.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08066",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohit Sinha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7bvvgqw8ar1de.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08066",
  "fulltext_url": null,
  "id": "7bvvgqw8ar1de",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08066",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7212050710",
  "pdf_url": "https://arxiv.org/pdf/2609.08066",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7bvvgqw8ar1de.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08066",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08066"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Finite-Precision Symmetric Krylov Methods: Exact Rounding Examples, Block Paige Identities,and a Variable-Block Lanczos Model",
  "title_text": "Finite-Precision Symmetric Krylov Methods: Exact Rounding Examples, Block Paige Identities,and a Variable-Block Lanczos Model",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/7bvvgqw8ar1de",
  "venue": "arXiv",
  "versions": []
}