## Title data

Reinke, Bernhard ; Schleicher, Dierk ; Stoll, Michael:

**The Weierstrass–Durand–Kerner root finder is not generally convergent.**

*In:* Mathematics of Computation.
Vol. 92
(2022)
Issue 340
.
- pp. 839-866.

ISSN 0025-5718

DOI: https://doi.org/10.1090/mcom/3783

Review: |

## Abstract in another language

Finding roots of univariate polynomials is one of the fundamental tasks of numerics, and there is still a wide gap between root finders that are well understood in theory and those that perform well in practice. We investigate the root finding method of Weierstrass, a root finder that tries to approximate all roots of a given polynomial in parallel (in the Jacobi version, i.e., with parallel updates). This method has a good reputation for finding all roots in practice except in obvious cases of symmetry, but very little is known about its global dynamics and convergence properties. We show that the Weierstrass method, like the well known Newton method, is not generally convergent: there are open sets of polynomials p of every degree d≥3 such that the dynamics of the Weierstrass method applied to p exhibits attracting periodic orbits. Specifically, all polynomials sufficiently close to Z^3+Z+180 have attracting cycles of period 4. Here, period 4 is minimal: we show that for cubic polynomials, there are no periodic orbits of length 2 or 3 that attract open sets of starting points. We also establish another convergence problem for the Weierstrass method: for almost every polynomial of degree d≥3 there are orbits that are defined for all iterates but converge to ∞; this is a problem that does not occur for Newton's method. Our results are obtained by first interpreting the original problem coming from numerical mathematics in terms of higher-dimensional complex dynamics, then phrasing the question in algebraic terms in such a way that we could finally answer it by applying methods from computer algebra.

## Further data

Item Type: | Article in a journal |
---|---|

Refereed: | Yes |

Keywords: | Weierstrass; root-finding methods; general convergence; attracting cycle; escaping points |

Subject classification: | 2010 Mathematics Subject Classification 65H04 (Primary), 37F80, 37N30, 68W30 (Secondary) |

Institutions of the University: | Faculties Faculties > Faculty of Mathematics, Physics und Computer Science Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra) Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra) > Chair Mathematics II (Computer Algebra) - Univ.-Prof. Dr. Michael Stoll |

Result of work at the UBT: | Yes |

DDC Subjects: | 500 Science > 510 Mathematics |

Date Deposited: | 22 Dec 2023 11:06 |

Last Modified: | 22 Dec 2023 11:06 |

URI: | https://eref.uni-bayreuth.de/id/eprint/87802 |