How Newton's Method can be used to find all roots of all polynomials

Abstract

In this paper we will discuss Newton’s Method, its limitations and a theorem which deals with these limitations. We will be looking in particular at John Hubbard, Dierk Schleicher and Scott Sutherland’s theorem on finding all roots of complex polynomials by Newton’s method. It will look at constructing a finite set of points so that for every root of every polynomial of fixed degree, at least one of the points will converge to a root under Newton’s map.