The Weierstrass root finder is not generally convergent – Bernhard Reinke

Bernhard Reinke
I2M, Aix-Marseille Université

Date(s) : 23/10/2020   iCal
11 h 00 min - 12 h 00 min

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 will give an overview of root-finding methods and their interpretation as complex dynamical systems. The main focus will be the Weierstrass/Durand-Kerner method and its similarities and differences to the Newton and the Ehrlich-Aberth methods.

In particular, we show how to use methods from computer algebra to investigate (and/or establish) the existence of attracting periodic cycles, as well as diverging orbits, and present explicit examples of both phenomena for the Weierstrass method.
We thereby settle the conjecture by Smale that the Weierstrass root finder is not generally convergent.

This talk is based on joint work with Dierk Schleicher and Michael Stoll.

FRUMAM, St Charles (3ème étage)


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