Title:
The Weierstrass root finder is not generally convergent.
Abstract:
Finding roots of univariate polynomials is one of the fundamental tasks
in numerical analysis. We investigate the root finding method of
Weierstrass, a root finder that tries to approximate all roots of a
given polynomial in parallel. This method has a good reputation for
finding all roots in practice, but very little is known about its global
dynamics and convergence properties.
We show that the Weierstrass 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.
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.