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Finding roots of univariate polynomials is one of the fundamental tasks ofnumerics, and there is still a wide gap between root finders that are wellunderstood in theory and those that perform well in practice. We investigatethe root finding method of Weierstrass, a root finder that tries to approximateall roots of a given polynomial in parallel (in the Jacobi version, i.e., withparallel updates). This method has a good reputation for finding all roots inpractice except in obvious cases of symmetry, but very little is known aboutits global dynamics and convergence properties. We show that the Weierstrassmethod, like the well known Newton method, is not generally convergent: thereare open sets of polynomials $p$ of every degree $d \ge 3$ such that thedynamics of the Weierstrass method applied to $p$ exhibits attracting periodicorbits. Specifically, all polynomials sufficiently close to $Z^3 + Z + 180$have attracting cycles of period $4$. Here, period $4$ is minimal: we show thatfor cubic polynomials, there are no periodic orbits of length $2$ or $3$ thatattract open sets of starting points. We also establish another convergenceproblem for the Weierstrass method: for almost every polynomial of degree $d\ge3$ 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 areobtained by first interpreting the original problem coming from numericalmathematics in terms of higher-dimensional complex dynamics, then phrasing thequestion in algebraic terms in such a way that we could finally answer it byapplying methods from computer algebra.

The Weierstrass root finder is not generally convergent