Newton's method for finding roots of a complex polynomial f entails iterating the function z - [f(z)/f'(z)], which can be viewed as applying the Euler backward method with step size unity to the so-called Newtonian vector field N_f(z) = - f(z)/f'(z). The rescaled and desingularized vector field V_f(z) = - f(z)(f'(z))^_ then has sinks at roots of f and has saddle points at roots of f' that are not also roots of f. The union of the closures of the unstable manifolds of the saddles of V_f defines a directed graph whose vertices are the roots of f and of f', and whose edges are the unstable curves oriented by the flow direction. This graph, along with the labelling of each vertex w with the multiplicity m(w)>=0 of w as a root of f, is defined to be the Newtonian graph of f (Smale 1985, Shub et al. 1988, Kozen and Stefánsson 1997).

Newtonian vector field | Newton's method | vector field