  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 $d \ge 3$ 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 $d\ge
3$ there are orbits that are defined for all iterates but converge to $\infty$;
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.
