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The Ehrlich--Aberth root finder is not generally convergent

  • Torben Grabbel
  • Nedim Srkalovic
  • Dierk Schleicher
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Abstract

We show that the well-known Ehrlich--Aberth root finding method is not generally convergent. More precisely, we show that there is an open subset of polynomials of any degree d≥ 7, and an open subset of initial conditions, such that the iteration of the Ehrlich--Aberth method converges to an attracting periodic cycle of period $4$, rather than to a vector of roots. The proof is constructive in the sense that we present explicit values for the polynomial and the $4$-cycle. The search is highly non-trivial: we found no attracting cycles for lower degrees/periods, and the size of the attracting domain relative to a ``unit cube'' in the space of polynomials and starting vectors is on the order of 10−127.

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Cite

BibTeXRIS
BibTeX
@misc{grabbel2026ehrlichaberth,
  title = {{The Ehrlich--Aberth root finder is not generally convergent}},
  author = {Torben Grabbel and Nedim Srkalovic and Dierk Schleicher},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02143},
  eprint = {2611.02143},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02143},
}
RIS
TY  - UNPB
TI  - The Ehrlich--Aberth root finder is not generally convergent
AU  - Torben Grabbel
AU  - Nedim Srkalovic
AU  - Dierk Schleicher
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.02143
UR  - https://arxiv.org/abs/2611.02143
AB  - We show that the well-known Ehrlich--Aberth root finding method is not generally convergent. More precisely, we show that there is an open subset of polynomials of any degree d≥ 7, and an open subset of initial conditions, such that the iteration of the Ehrlich--Aberth method converges to an attracting periodic cycle of period $4$, rather than to a vector of roots. The proof is constructive in the sense that we present explicit values for the polynomial and the $4$-cycle. The search is highly non-trivial: we found no attracting cycles for lower degrees/periods, and the size of the attracting domain relative to a ``unit cube'' in the space of polynomials and starting vectors is on the order of 10^−127.
ER  -