My research project is a numerical optimization problem. My task is to find a convincing visualization of the behavior of various numerical solvers for evaluation/rating of forces and weaknesses.
The first step is to first find a set of sample points to iterate over. This involves a few substeps: First, I reduce a multi-dimensional variable to two dimensions. Then I look at all points within a two-dimensional neighborhood of the reduced point. Finally I map these points back to higher dimensional variables. The second step is to use the Newton-Rhapson method to observe how each sample point approaches the root. I then present a graphical analysis of the iterative process, showing the path followed by these points as they approach the root, as well as the number of iterations it takes a particular point to reach the root. In many cases, the resulting figure is a fractal.
I am writing the program in MATLAB, and the code can easily be modified for iterative methods other than the Newton-Rhapson method.