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PreprintarXiv licencearXiv

Generalized Pellet's theorem for analytic matrix-valued functions

  • Hiromochi Itoh
  • Kei Matsushima
  • Takayuki Yamada
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Abstract

Generalized Pellet's theorem is extended to analytic matrix-valued functions using Descartes' rule of signs for infinite series. Cauchy's estimate is incorporated to allow numerical computation of the bounds. To address the issue that Pellet's theorem and its generalizations may not always guarantee the existence of bounds, for the case of matrix size one, we employ the Dandelin- Lobachevsky-Graeffe method, which ensures the existence of the bounds for the roots and further improves them as the order of the transform increases. Without the need for any a priori information on the function, these theorems allow us to compute mathematically verified bounds for the roots in a certain region using an interval arithmetic implementation of the function. Numerical examples and an implementation are provided to illustrate the effectiveness of the proposed method.

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Cite

BibTeXRIS
BibTeX
@misc{itoh2026generalized,
  title = {{Generalized Pellet's theorem for analytic matrix-valued functions}},
  author = {Hiromochi Itoh and Kei Matsushima and Takayuki Yamada},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18941},
  eprint = {2610.18941},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18941},
}
RIS
TY  - UNPB
TI  - Generalized Pellet's theorem for analytic matrix-valued functions
AU  - Hiromochi Itoh
AU  - Kei Matsushima
AU  - Takayuki Yamada
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18941
UR  - https://arxiv.org/abs/2610.18941
AB  - Generalized Pellet's theorem is extended to analytic matrix-valued functions using Descartes' rule of signs for infinite series. Cauchy's estimate is incorporated to allow numerical computation of the bounds. To address the issue that Pellet's theorem and its generalizations may not always guarantee the existence of bounds, for the case of matrix size one, we employ the Dandelin- Lobachevsky-Graeffe method, which ensures the existence of the bounds for the roots and further improves them as the order of the transform increases. Without the need for any a priori information on the function, these theorems allow us to compute mathematically verified bounds for the roots in a certain region using an interval arithmetic implementation of the function. Numerical examples and an implementation are provided to illustrate the effectiveness of the proposed method.
ER  -