The Fundamental Theorem for Algebra says that every polynomial (in one variable) factors over the complex numbers into linear factors, and the Ruffini-Abel-theorem says that the roots have to be found in general by iterative methods. This situation has been known for centuries, and it is obviously of significant relevance in all areas of mathematics and engineering, but to this day no root finding method is known that is supported by theory and has been shown to work well in practice, especially for polynomials of large degrees. In this talk, we describe three well known iterative root finders: those named after Newton, Weierstrass (a.k.a. Durand-Kerner), and Ehrlich-Aberth. All of these provide interesting dynamical systems. We describe some of their properties, some classical and some recent. In particular, we show that the Weierstrass method is not generally convergent, which confirms a conjecture of Smale and disproves a long-standing assumption among numerical analysists. Moreover, we show how to develop good theoretical complexity results for Newton’s method, and present numerical evidence that the method works surprisingly efficiently.
