Abstract:
Newton's Method is an algorithm for finding the roots and/or zeros of a function. In other words, it is a way to find where a function crosses the x-axis with out knowing where it happens or even what the function looks like. Given an initial guess that is sufficiently close to a zero or root the algorithm will converge on it until the desired level of accuracy is reached. My project will examine only polynomial function with positive whole number exponents. For polynomial functions every zero is a root in the real plane. However, many functions have imaginary roots in the complex plane and this is where it gets interesting. Newton's Method exhibits seemingly chaotic behavior when the initial guess is not close enough to a root to guarantee convergence. In these indeterminate regions it is impossible to know what root the algorithm will find without iterating it many times. This is where a computer becomes quite useful. By applying Newton's Method to many points in the complex plane one finds that the behavior of Newton's Method is fractal rather than purely chaotic. For example the picture above is the function x^4 -1. The different colors each represent a different root and the different levels shading correspond to how many iterations it took to find the root. The actual locations of the roots are the lightest circles in the center of each of the large solid color regions.
Project Description:
My project is to create a program which will produce pictures like the one above in the complex plane given any polynomial function. It is intended to be flexible and give the user control over much of the program. The user will be able to define any polynomial function with positive integer exponents. Also the user will be able to control what colors are used and the gradient levels for shading. One can also control the the window size, resolution, and the accuracy which all affect the speed of the program. It will also include zoom features to allow the user to navigate and explore the fractals interactively.
Executable files and source code
Bibliography and links to related sources