Numerical Analysis: An Introduction

Book Preface

This book is intended for math majors interested in learning tools for solving differential equations, eigenvalue problems, linear systems of equations, nonlinear systems of equations, and to perform curve fitting. The major topics for a first course in scientific computing are covered, and we emphasize fundamental ideas, simple codes, and mathematical proofs. We do not get bogged down with low level details of the fifty different methods there are to solve a particular problem. Instead,we try to give an undergraduate math major an idea of what scientific computing and numerical analysis is all about, along with exposure to the fundamental ideas of the proofs. Some key questions we aim to address are:
– How do we best approximate mathematical processes/operations that cannot be exactly represented on a computer?
– How accurate are our approximations?
– How efficient are our approximations?

It should be no surprise that we want to quantify accuracy as much as possible. Moreover, when a method fails, we want to know why it fails. In this book, we will see how to mathematically analyze the accuracy of many numerical methods. Concerning efficiency, we can never have the answer fast enough, but often there is a trade-off between speed and accuracy. Hence, we also analyze efficiency so that we can “choose wisely” when selecting an algorithm. Thus, to put it succinctly, the purpose of this course is to:
– Introduce students to some basic numerical methods and how to use them on a computer.
– Analyze these methods for accuracy and efficiency.
– Implement these methods and use them to solve problems.

We assume a knowledge of calculus, differential equations, and linear algebra, and also that students have some programming experience (e. g., an introductory computer programming course) and have used MATLAB in at least some capacity.

This book is related to a sophomore engineering course in scientific computing written by the authors and published in 2015 by DeGruyter.1 Some parts of this book are very similar, including roundoff error. But most of the chapters have been expanded to include more in-depth discussion and mathematical theory wherever possible, and there is a significant amount of new material.

Lastly, despite countless hours of effort, there are bound to be remaining typos and mistakes. We would greatly appreciate any users of this book to point them out to us. We would also appreciate any other constructive comments and criticisms regarding the presentation of the material.

Software

Algorithms given in the text are written in the language of MATLAB and Octave. Currently at Clemson, all students have free access to MATLAB. Octave is a free version of MATLAB, which has almost all of the same functionality. Newer versions of MATLAB have more bells and whistles, but for the purposes of this book, either MATLAB or Octave can be used.