Math and Computer Science Honors Theses

Elusive Zeros Under Newton's Method

Date of Creation

5-31-2005

Document Type

Departmental Honors Thesis - Restricted Access

Department

Mathematics

First Advisor

Gareth E. Roberts

Abstract

Given that Newton's method is an iterative process, it is natural to study it as a discrete dynamical system. Of particular interest are the various open sets of initial seeds that fail to converge to a root under Newton's method. Thus we examine certain "bad" polynomials that contain extraneous, attracting periodic cycles. In particular, we chose to examine Newton's method applied to a particular family of fourth degree polynomials that rely on only one parameter value. These are polynomials of the form: P λ(z) = (z + 1)(z - 1)(z - λ)(z - ƛ) where λ ∈ C. We extensively analyze the parameter plane for this family of polynomials in cases where>. is both real and purely imaginary. More specifically, we have developed and implemented computer programs to locate λ values for which Newton's method fails on an open, often relatively large, set of initial conditions. In doing so, we have discovered some rather surprising dynamical figures in the λ-parameter plane, including Mandelbrot-like sets, tricorns, and swallowtails. Through symmetry and by restricting to the imaginary axis, we have uncovered certain analytic and numerical evidence that aids in explaining the existence of such figures.

Comments

Reader: David B. Damiano

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