Honors Senior Theses/Projects

Date of Award

Fall 2019

Exit Requirement

Undergraduate Honors Thesis/Project

Department

Honors Program

Faculty Advisor

Matthew Nabity

Honors Program Director

Dr. Gavin Keulks

Abstract

This project will explore why, when using an iterative algorithm, specifically Newton’s Method, to solve nonlinear equations, certain functions can be observed to behave predictably while others behave chaotically. In attempting to answer this query my project will elaborate on what Newton’s Method is and how its used as well as demonstrate that Newton’s Method itself behaves predictably via mathematical proof. In this context, I will examine real-valued functions with solutions then introduce complex-valued functions. Following a proof of Newton’s Method for complex functions, the project will compare the behavior of these complex-valued functions with the previously mentioned real-valued functions. The project will examine the convergence behavior of Newton’s Method when analyzing complex-valued functions and determine if the behavior is chaotic. Upon observing this chaotic 4 behavior, my project will seek to find complex functions that do not exhibit chaotic convergence behavior. Subsequently, I will analyze my discoveries and discuss their implications. As of yet, there has been no comprehensive study of iterative methods in the context of solving complex valued equations. Ultimately, my project will produce an analytic discussion of the behavior of several functions within Newton’s Method along with computational experiments. From this process we might find some distinguishing factor that determines whether behavior will be predictable or chaotic.

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