Gallery of fractals created using the
Newton Raphson method

Written by Paul Bourke
August 1989

z4 + z3 + z - 1 = 0

Lovers            z 4 + 3 z3 + 2 z2 + 0.2 z + 1 = 0

z4 + z3 - 1 = 0

z3 + 2 z2 + z + 3 = 0

-2 z4 + z3 + z2 - 2 z - 1 = 0


This technique is based on the Newton Raphson method of finding the solution (roots) to a polynomial equation of the form

The method generates a series where the n+1'th approximation to the solution is given by

where f'(zn) is the slope (first derivative) of f(z) evaluated at zn. To create a 2D image using this technique each point in a partition of the plane is used as initial guess, zo, to the solution. The point is coloured depending on which solution is found and/or how long it took to arrive at the solution. A simple example is an application of the above to find the three roots of the polynomial z3 - 1 = 0. The following shows the appearance of a small portion of the positive real and imaginary quadrant of the complex plane. A trademark of chaotic systems is that very similar initial conditions can give rise to very different behaviour. In the image shown there are points very close together one of which converges to the solution very fast and the other converges very slowly.