Thus, beginning with an initial estimate z0, we obtain a series of estimates z0, z1, z2, ..., which, one hopes, converges to a solution of f(z) = 0.
The question of convergence turns out to be a most interesting one. Given a function f with f(z) = 0 having more than one solution, we may ask, for an initial value z0, not only whether the series z0, z1, z2, ... converges to a solution, but, if so, which solution it converges to. In this way we arrive at the notion of a Newton basin: The Newton basin of a solution is the set of starting points z0 for which Newton's method converges to that solution. Here we consider functions f(z) defined for complex numbers z.
We may illustrate the Newton basins for f by coloring each basin a different color, and coloring those points for which Newton's method does not converge black. Here is the complex plane, colored to show the Newton basins of the polynomial
This graphic was produced by using David Joyce's Newton basin generator. It allows you to experiment--you can pick the roots of a polynomial and it will find, and color, the Newton basins.
-- Steven Weintraub