Newton’s Method Fractals

Newton’s Method is a popular and simple way to approximate solutions for systems of equations. You start by taking a guess as to what the solution might be, and then you pop that guess into a formula, and pop the result of that back into the formula, and again …, until you have the desired level of accuracy. When there are multiple solutions to the system the solution obtained by the above procedure will depend on the initial guess. We would expect that the solution obtained would be the one closest to the starting guess. However we observe that this isn’t true.

Take a look at this picture.

a1.png
Solutions for z = f(z) = z^3 + 1 = 0

This picture shows the complex plane. The areas that are colored cyan contain points that have a specific property, that if any of those points is used as an initial guess the solution obtained would be the same (in this case these would produce the solution z = -1). And the same idea is true of the points colored yellow and magenta. What we observe is that in between any two points that converge to any two solutions you can find a point that will converge to a different solution. Fascinating. Also the image obtained is pretty interesting. The inquisitive mind of course would want to know what images would be obtained by applying the same idea to other functions.

a2
Solutions for z = f(z) = z^3 + 1.5*z^2 – 1.5*z + 1 = 0

Using this approach with the more basic functions is usually not nearly as interesting as the visualizations of the functions themselves.

sin
z = f(z) = sin(z) = 0
tan
z = f(z) = tan(z) = 0
logplus1
z = f(z) = log(z) – 1 = 0. This is a little more interesting.
2.png
z = f(z) = cos(z)*sin(z) = 0. This is also pretty interesting.

The real fun started when I made a mistake. I attempted to draw the picture for z = f(z) = log(z^2 – z). But I made a mistake (this procedure involves the derivative of the function, and I erred in computing the derivative).

log
This is what the picture should have looked like.
logmistake
This is what I got instead. Much more interesting, no?

Here are a few variations of the first image shown on this page. These are obtained by fiddling around with the expression used in place of the derivative.

12345

 

Note: All these pictures were generated using a program I wrote in C++, using the SFML library.

 

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