Troy Siemers, Ph.D.

Introduction

Many concepts in the realm of mathematics are taken for granted by both students and teachers alike.  One of these concepts is the approximation of various functions.  Undergraduate calculus courses include approximation techniques such as Taylor series and Fourier series.  Unfortunately, insufficient time is spent with either, since approximating and building functions are the basic building blocks of math.

Aside from Taylor and Fourier approximation techniques, there exist two fairly “new” techniques that are lesser known.  One of these two techniques, entitled Wavelets, has in recent years become a widely researched topic in the world of mathematics.  Numerous books devoted solely to these approximating functions have been written and published just in the past decade, hoping to broaden the horizons of students and teachers alike.  The second of these lesser-known techniques is called the CORDIC Algorithm, the method used in hand-held calculators to compute values and create graphs for the stored functions (not Taylor series (!) which many people, including myself, had previously thought).

At the beginning of this research program, my mentor and I set out with a goal to investigate the strengths and weaknesses of these four approximation techniques and to determine which one was the “best” at approximating functions.  Specifically, we wanted to focus on the precision of the approximations using the software package Maple (version 8.0) and an error computation technique using L2 norms.  Through the course of this research, however, we found that in order to conduct a thorough investigation, a ten week period is not nearly enough time. 

During the past ten weeks, six of those were spent learning and understanding each of the four approximation techniques.  In the next sections of this paper, I explain these four techniques, explain the L2 norm, and give the results and conclusions of the 3 weeks of experimentation.  I end with sections on future goals, selected references, and an appendix containing the Maple programs.