Newton's method is a useful tool for finding roots of functions when analytical methods fail. The goal of our research was to understand the dynamics of Newton's method on cubic polynomials with real coefficients. Usually iterations will converge quickly to the root. However, there are more interesting things that can happen. When we allow initial values to be chosen from the complex plane, we find that the points that converge are bounded by fractals. For some polynomials we found interesting phenomena including chaos and attracting periodic cycles. We classified which polynomials could have attracting periodic cycles.

Author Bio

I graduated in May, 2000 from Davidson College in Davidson, North Carolina with a BS with Honors in Mathematics, Magna cum laude, and a minor in Music. I began this project in the summer of 1999 at the Summer Undergraduate Institute at Carnegie Mellon University. My advisor at CMU was Dr. Tim Flaherty, and I worked with two other students, Zalenda Cyrille and Kari Whitcomb. I continued the project for my senior honors thesis at Davidson with Dr. Richard Neidinger as my advisor. I am currently a graduate student in Physics at West Virginia University. My hobbies include playing the cello and swimming.