Diverging orbits for the Ehrlich–Aberth and the Weierstrass root finders
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- by Bernhard Reinke PDF
- Proc. Amer. Math. Soc. 150 (2022), 1287-1300 Request permission
Abstract:
We show that several well known higher dimensional methods for finding roots of univariate polynomials have infinite orbits that diverge to infinity: in particular, the “Weierstrass” (Durand–Kerner) and the “Ehrlich–Aberth” methods. This is possible for the Jacobi update scheme (all coordinates are updated in parallel) as well as Gauss–Seidel (any coordinate update is used for all subsequent coordinates).
These root finding methods are in active use in practice, but very little is known about their global dynamical properties, and diverging orbits were discovered only recently for one of them. Our results are established by a combination of methods from dynamical systems and computer algebra.
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Additional Information
- Bernhard Reinke
- Affiliation: Aix-Marseille Université, Institut de Mathématiques (I2M UMR CNRS7373), Campus de Luminy, 163 avenue de Luminy — Case 907, 13288 Marseille 9, France
- Address at time of publication: Sorbonne Université and Université de Paris, CNRS, IMJ-PRG, F-75005 Paris, France
- ORCID: 0000-0001-9024-2449
- Email: bernhard.reinke@imj-prg.fr
- Received by editor(s): December 31, 2020
- Received by editor(s) in revised form: May 21, 2021, and May 31, 2021
- Published electronically: December 7, 2021
- Additional Notes: This work was supported by the Advanced Grant 695 621 HOLOGRAM of the European Research Council.
- Communicated by: Filippo Bracci
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 1287-1300
- MSC (2020): Primary 65H04; Secondary 37F80, 37N30, 68W30
- DOI: https://doi.org/10.1090/proc/15715
- MathSciNet review: 4375722