Skip to content
DraconiticResearch
Search Sources API
190,869 studies31 sourcesUpdated 25 min ago
PreprintCC BY 4.0arXiv

On the cyclic limiting behavior of the restarted Arnoldi algorithm for real symmetric matrices with restart length s=2

  • Vance Faber
  • Jarek Liesen
  • Jörg Liesen
  • Petr Tichý
Open at sourcePDFDOICite

Abstract

We prove that the restarted Arnoldi algorithm for real symmetric matrices with restart length s=2 has a cyclic limiting behavior. In the terminology established in (arXiv:2209.14579), this is a proof of the ACI(s) conjecture for A=AT∈ ℝn× n and s=2. As explained in (arXiv:2209.14579), this also proves the Forsythe conjecture (Numerische Mathematik, 11 (1968), pp. 57-76) about the limiting behavior of the conjugate gradient method and positive definite A=AT∈ℝn× n for s=2. The proof we give is an extensively rewritten and simplified version of a proof found using Codex with GPT-5.6 Sol. At the end of this paper, we discuss its origin and evolution, as well as related recent developments around the Forsythe conjecture. Our motivation is to present a proof that is instructive and transparent, so that it can enhance the knowledge of interested human readers.

Related

  1. Limiting shape of alternating sign matrices
    Alexey Bufetov, Evgeny Obukhov
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsMathematical physicsarXiv
  2. Time-and-band limiting for symmetric classical Dunkl orthogonal polynomials
    M. M. Castro, F. A. Grünbaum
    arXivPreprintarXiv licenceFull textAnalysis & PDEsNumerical analysis & scientific computingarXiv
  3. Limiting moments of sample covariance matrices from large fold tensor vectors
    Wangjun Yuan
    arXivPreprintCC BY 4.0Full textProbabilityarXiv
  4. Ratio and limiting zero distribution asymptotics for symmetric multiple orthogonal polynomials
    Ana Loureiro, Walter Van Assche
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  5. On wellposedness and limiting behavior of generalized SQG equations
    Anuj Kumar
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{faber2026cyclic,
  title = {{On the cyclic limiting behavior of the restarted Arnoldi algorithm for real symmetric matrices with restart length $s=2$}},
  author = {Vance Faber and Jarek Liesen and Jörg Liesen and Petr Tichý},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16492},
  eprint = {2610.16492},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16492},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the cyclic limiting behavior of the restarted Arnoldi algorithm for real symmetric matrices with restart length s=2
AU  - Vance Faber
AU  - Jarek Liesen
AU  - Jörg Liesen
AU  - Petr Tichý
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16492
UR  - https://arxiv.org/abs/2610.16492
AB  - We prove that the restarted Arnoldi algorithm for real symmetric matrices with restart length s=2 has a cyclic limiting behavior. In the terminology established in (arXiv:2209.14579), this is a proof of the ACI(s) conjecture for A=A^T∈ ℝ^n× n and s=2. As explained in (arXiv:2209.14579), this also proves the Forsythe conjecture (Numerische Mathematik, 11 (1968), pp. 57-76) about the limiting behavior of the conjugate gradient method and positive definite A=A^T∈ℝ^n× n for s=2. The proof we give is an extensively rewritten and simplified version of a proof found using Codex with GPT-5.6 Sol. At the end of this paper, we discuss its origin and evolution, as well as related recent developments around the Forsythe conjecture. Our motivation is to present a proof that is instructive and transparent, so that it can enhance the knowledge of interested human readers.
ER  -