On the efficient global dynamics of Newton’s method for complex polynomials

Abstract

We investigate Newton’s method as a root finder for complex polynomials of arbitrary degrees. While polynomial root finding continues to be one of the fundamental tasks of computing, with essential use in all areas of theoretical mathematics, numerics, computer graphics and physics, known methods may have excellent theoretical complexity but cannot be used in practice, or are practically efficient but lack a successful theory behind them. We provide precise and strong upper bounds for the theoretical complexity of Newton’s method and show that it is near-optimal with respect to the known set of starting points that find all roots. This theoretical result is complemented by a recent implementation of Newton’s method that finds all roots of various polynomials of degree more than a billion, significantly faster than our upper bounds on the complexity indicate, and often much faster than established fast root finders. Newton’s method thus stands out as a method that has strong merits both from the theoretical and from the practical point of view. Our study is based on the known explicit set of universal starting points, for each degree d, that are guaranteed to find all roots of polynomials of degree d (appropriately normalized). We show that this set contains d points that converge very quickly to the d roots: the expected total number of Newton iterations required to find all d roots with precision ɛ is , which can be further improved to . The key argument shows that many root finding orbits are ‘R-central’ in the sense that they stay forever in a disk of radius R, and each iteration ‘uses up’ an explicit amount of area within this disk.