Newtonian Graph

Newton's method for finding roots of a complex polynomial entails iterating the function , which can be viewed as applying the Euler backward method with step size unity to the so-called Newtonian vector field . The rescaled and desingularized vector field then has sinks at roots of and has saddle points at roots of that are not also roots of . The union of the closures of the unstable manifolds of the saddles of defines a directed graph whose vertices are the roots of and of , and whose edges are the unstable curves oriented by the flow direction. This graph, along with the labelling of each vertex with the multiplicity of as a root of , is defined to be the Newtonian graph of (Smale 1985, Shub et al. 1988, Kozen and Stefánsson 1997).