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 zn for a solution of an equation f(z) = 0, it generates the estimate zn+1, which would be exact if f were linear. This works out to give the formula
zn+1 = zn - f(zn)/f'(zn).
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
f(z) = (z4 - 1)(z2 - 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
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