Constraints on admissible behavior of GMRES applied to tridiagonal Toeplitz systems

Abstract

The result of Greenbaum, Pták, and Strakoš that for a given set of eigenvalues, any convergence curve is possible [SIMAX 1996] and the subsequent parameterization of such matrix-right-hand side pairs (A,b) of Arioli, Pták, and Strakoš [BIT 1998] demonstrated that the behavior of the GMRES could not be completely characterized by the eigenvalues of A alone. In this paper, we consider how to use this theory to understand the admissible and attainable \gmres behavior for matrices with constrained structure, focussing on non-Hermitian (non-symmetric) tridiagonal Toeplitz matrices. We show that Toeptliz structure necessarily constrains the how the theory from these papers can manifest but that a continuum of admissible behaviors is still attainable.

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BibTeX RIS

BibTeX

@misc{chen2026constraints,
  title = {{Constraints on admissible behavior of GMRES applied to tridiagonal Toeplitz systems}},
  author = {Fei Chen and Kirk M. Soodhalter},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.24751},
  eprint = {2609.24751},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24751},
}

RIS

TY  - UNPB
TI  - Constraints on admissible behavior of GMRES applied to tridiagonal Toeplitz systems
AU  - Fei Chen
AU  - Kirk M. Soodhalter
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.24751
UR  - https://arxiv.org/abs/2609.24751
AB  - The result of Greenbaum, Pták, and Strakoš that for a given set of eigenvalues, any convergence curve is possible [SIMAX 1996] and the subsequent parameterization of such matrix-right-hand side pairs (A,b) of Arioli, Pták, and Strakoš [BIT 1998] demonstrated that the behavior of the GMRES could not be completely characterized by the eigenvalues of A alone. In this paper, we consider how to use this theory to understand the admissible and attainable \gmres behavior for matrices with constrained structure, focussing on non-Hermitian (non-symmetric) tridiagonal Toeplitz matrices. We show that Toeptliz structure necessarily constrains the how the theory from these papers can manifest but that a continuum of admissible behaviors is still attainable.
ER  -

Abstract

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