On the speed of convergence of Newton's method for complex polynomials

We investigate Newton's method for complex polynomials of arbitrary degree d, normalized so that all their roots are in the unit disk. For each degree d, we give an explicit set S-d of 3.33d log(2) d(1 + o(1)) points with the following universal property: for every normalized polynomial of degree d there are d starting points in S-d whose Newton iterations find all the roots with a low number of iterations: if the roots are uniformly and independently distributed, we show that with probability at least 1-2/d the number of iterations for these d starting points to reach all roots with precision epsilon is O(d(2) log(4) d + d log vertical bar log epsilon vertical bar). This is an improvement of an earlier result by Schleicher, where the number of iterations is shown to be O(d(4) log(2) d + d(3) log(2) d vertical bar log epsilon vertical bar) in the worst case (allowing multiple roots) and O(d(3) log(2) d(log d + log delta) + d log vertical bar log epsilon vertical bar) for well-separated (so-called delta-separated) roots. Our result is almost optimal for this kind of starting points in the sense that the number of iterations can never be smaller than O(d(2)) for fixed e. It provides theoretical support for an empirical study, by Schleicher and Stoll, in which all roots of polynomials of degree 10(6) and more were found efficiently.