Newton’s Basins of Attraction

This is a Newton fractal given by the dynamical system

zn+1 = zn- g(zn)/g'(zn),
g(z) = z5 + 2z3 + 3z3 - 4z3 - 5z - 6,

plotted on a small rectangular neighborhood of the point (3.077956, 2.8834923).

Advanced Method for Newton Fractals: If we know or can find the roots of a given polynomial prior to the fractal plotting, we can choose a color for each initial value z0 (which is a pixel in the canvas for a Julia fractal) depending upon which root the corresponding sequence zn converges to. In plotting the picture on the left, 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, indigo, and green, called the basins of attraction of the roots of g.

Julia Set, Chaos, and Fractal Plotting: The basins of attraction are intricately interlaced and separated by a nonempty set of points called the basin boundary. It is difficult to visualize the boundary as it happens to be a totally disconnected dusty Julia set often called Cantor dust in topology. We can generally find better (more intricate) Newton fractals near the Julia set but ironically it causes the Newton's root-finding algorithm to behave unpredictably because of chaos associated with the Julia set. It is one of the few shortcomings of the otherwise extremely powerful and widely applicable algorithm, and it explains why I use stable and "automatic" Müller's algorithm instead in finding the roots prior to the fractal plotting. Unlike Newton's algorithm however, Müller's algorithm only applies to polynomials.