

This is a Newton pattern given by the dynamical system
z_{n+1 }=
z_{n} g(z_{n})/g'(z_{n}),
with
g(z) = z^{5} + 2z^{3} + 3z^{3}  4z^{3}  5z  6,
plotted on a small rectangular neighborhood of the point (3.077956, 2.8834923).
Advanced Method for Newton Patterns:
If we can find the roots of a given polynomial prior to the fractal plotting, we can choose a color for each initial value z_{0} (which is a pixel in the canvas for a Julia pattern) depending upon which root the corresponding sequence z_{n} converges to. In plotting the picture above, the computer is programmed to find all five roots of g through the method called Müller's algorithm and then color an initial value red, for instance, if the sequence converges to the first root. This causes the canvas to be separated into the five regions that are painted red, orange, skyblue, blue, and green, called the basins of attraction of the roots of g.

