Gallery of fractals created using the Newton Raphson method
Written by Paul Bourke
August 1989
z^{4} + z^{3} + z  1 = 0
Lovers
z ^{4} + 3 z^{3} + 2 z^{2} + 0.2 z + 1 = 0
z^{4} + z^{3}  1 = 0
z^{3} + 2 z^{2} + z + 3 = 0
2 z^{4} + z^{3} + z^{2}  2 z  1 = 0
Algorithm
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'(z_{n})
is the slope (first derivative) of f(z) evaluated at z_{n}. To create
a 2D image using this technique each point in a partition of the plane is used
as initial guess,
z_{o}, 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 z^{3}  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.
