Newton’s Method and Chaotic Behavior

Title

Newton’s Method and Chaotic Behavior

Author

Aubrey Manculich

Faculty Sponsor

Matthew Nabity

Gavin Keulks

Date

10/1/2019

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.

Type

Text

Honors Thesis

Department

Honors Program

Language

eng

Rights

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Identifier

honors_theses/207