Section 4: Newton's Method and the Julia Sets
A map is an iteration scheme. One of the most important iteration schemes is Newton's Method for finding the roots of algebraic equations. In general, Newton's method works impressively fast (with quadratic convergence). But it requires a good initial guess, which normally needs to be close to one of the roots. Otherwise, strange things may happen.

First, let us consider the simple example:
figure 1
Suppose xc is a critical point of ƒ (i.e. ƒ' (xc) = 0), and we apply Newton's method to the equation ƒ(x) = 0. If our initial guess of x0 is equal to xc then the iteration will diverge and will not produce any roots. If x0 is slightly smaller or larger than xc , the outcome of the iteration will be drastically different. This is called sensitive dependence on the initial conditions.

Next, we discuss Newton's method applied to the simple cubic equation
z3 = 1 (4.1)
The iteration is
figure 2 (4.2)
or
figure 3 (4.3)
This map has three superattracting fixed points which are the three roots of equation (4.1): 1, e 2πi/3, e4πi/3.  If the initial point z0 of the map (4.2) is near one of these three fixed points,  ƒn (z0) will converge to it.

We define the attraction basins of these fixed points as
A1 = {z:ƒn(z) → 1},

A2 = {z:ƒn(z) → e2πi/3},

A3 = {z:ƒn(z) → e4πi/3}.

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Section 4: Newton's Method and Julia Sets