Publication Date

Spring 2018

Advisor(s) - Committee Chair

Dr. Mark Robinson (Director), Dr. Dominic Lanphier, and Dr. Ngoc Nguyen

Degree Program

Department of Mathematics

Degree Type

Master of Science

Abstract

Iterative methods have been a very important area of study in numerical analysis since the inception of computational science. Their use ranges from solving algebraic equations to systems of differential equations and many more. In this thesis, we discuss several iterative methods, however our main focus is Newton's method. We present a detailed study of Newton's method, its order of convergence and the asymptotic error constant when solving problems of various types as well as analyze several pitfalls, which can affect convergence. We also pose some necessary and sufficient conditions on the function f for higher order of convergence. Different acceleration techniques are discussed with analysis of the asymptotic behavior of the iterates. Analogies between single variable and multivariable problems are detailed. We also explore some interesting phenomena while analyzing Newton's method for complex variables.

Disciplines

Dynamical Systems | Non-linear Dynamics | Numerical Analysis and Computation

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