Newton Basins **Newton Basins**

Newton's method is an iterative method for finding solutions of equations. It employs one of the basic strategies of mathematics--linearization. That is, given an estimate

*z*_{n} for a solution of an equation

*f(z) = 0*, it generates the estimate

*z*_{n+1}, which would be exact if

*f* were linear. This works out to give the formula

*z*_{n+1} = z_{n} - f(z_{n})/f'(z_{n}). Thus, beginning with an initial estimate *z*_{0}, we obtain a series of estimates *z*_{0}, z_{1}, z_{2}, ..., 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 *z*_{0}, not only whether the series *z*_{0}, z_{1}, z_{2}, ... 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 *z*_{0} 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

*f(z) = (z*^{4} - 1)(z^{2} - 4) with roots

*1, i, -1, -i, 2, -2*.

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*