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“The Weierstrass root finder is not generally convergent”
15.7.2021, 16:15 - 17:15
Speaker:Prof. Dr. Dierk Schleicher, Institut de Mathématiques de Marseille
Location:Mathematisches Institut, Bunsenstr 3-5Hybrid ( Geo Map
Organizer:Mathematisches Institut
One of the classical problems in all areas of mathematics is to find roots of complex polynomials. It is well known that this can be done only by methods of approximation. We discuss three classical methods: the Newton, Weierstrass, and Ehrlich-Aberth methods; these are complex analytic maps that, under iteration, are supposed to converge to one root, resp. all roots of the polynomial. Locally, these methods converge fast, but the global dynamical properties are hard to describe.

From the practical point of view of numerical analysis, the methods of Weierstrass (also known as Durand-Kerner) and Ehrlich-Aberth are known as “best practice” methods that have passed the test of time, and that are “empirically known” as generally convergent.

We prove that this expectation is false, at least for the Weierstrass method: there are open sets of polynomials for which open sets of initial points fail to converge. This result, obtained jointly with my PhD student Bernhard Reinke and with Michael Stoll (Bayreuth), thus disproves "empirically well known” facts, but establishes a conjecture by Steven Smale to the contrary that is 40+ years old. We also prove that both methods may have orbits that are defined forever but converge to infinity, rather than to roots.

The proof combines ideas from numerical analysis, dynamical systems, algebra, and computer algebra. We believe that an analogous result should also hold for the Ehrlich-Aberth method, but there are reasons to assume that this is beyond the possibilities of current computer algebra systems.
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Host:Prof. Dr. Laurent Bartholdi
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