An Introduction to Numerical Methods and Analysis

James F. Epperson

• Publisher: Wiley
• Year: 2013
• ISBN:
  9781118407462
• Edition: 2
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

Praise for the First Edition

"... outstandingly appealing with regard to its style, contents, considerations of requirements of practice, choice of examples, and exercises."— Zentralblatt MATH

"... carefully structured with many detailed worked examples."— The Mathematical Gazette

The Second Edition of the highly regarded An Introduction to Numerical Methods and Analysis provides a fully revised guide to numerical approximation. The book continues to be accessible and expertly guides readers through the many available techniques of numerical methods and analysis.

An Introduction to Numerical Methods and Analysis, Second Edition reflects the latest trends in the field, includes new material and revised exercises, and offers a unique emphasis on applications. The author clearly explains how to both construct and evaluate approximations for accuracy and performance, which are key skills in a variety of fields. A wide range of higher-level methods and solutions, including new topics such as the roots of polynomials, spectral collocation, finite element ideas, and Clenshaw-Curtis quadrature, are presented from an introductory perspective, and the Second Edition also features:

• Chapters and sections that begin with basic, elementary material followed by gradual coverage of more advanced material
• Exercises ranging from simple hand computations to challenging derivations and minor proofs to programming exercises
• Widespread exposure and utilization of MATLAB
• An appendix that contains proofs of various theorems and other material

The book is an ideal textbook for students in advanced undergraduate mathematics and engineering courses who are interested in gaining an understanding of numerical methods and numerical analysis.

CHAPTER 1

INTRODUCTORY CONCEPTS AND CALCULUS REVIEW

It is best to start this book with a question: What do we mean by “Numerical Methods and Analysis”? What kind of mathematics is this book about?

Generally and broadly speaking, this book covers the mathematics and methodologies that underlie the techniques of scientific computation. More prosaically, consider the button on your calculator that computes the sine of the number in the display. Exactly how does the calculator know that correct value? When we speak of using the computer to solve a complicated mathematics or engineering problem, exactly what is involved in making that happen? Are computers “born” with the knowledge of how to solve complicated mathematical and engineering problems? No, of course they are not. Mostly they are programmed to do it, and the programs implement algorithms that are based on the kinds of things we talk about in this book.

Textbooks and courses in this area generally follow one of two main themes: Those titled “Numerical Methods” tend to emphasize the implementation of the algorithms, perhaps at the expense of the underlying mathematical theory that explains why the methods work; those titled “Numerical Analysis” tend to emphasize this underlying mathematical theory, perhaps at the expense of some of the implementation issues. The best approach, of course, is to properly mix the study of the algorithms and their implementation (“methods”) with the study of the mathematical theory (“analysis”) that supports them. This is our goal in this book.

Whenever someone speaks of using a computer to design an airplane, predict the weather, or otherwise solve a complex science or engineering problem, that person is talking about using numerical methods and analysis. The problems and areas of endeavor that use these kinds of techniques are continually expanding. For example, computational mathematics—another name for the material that we consider here—is now commonly used in the study of financial markets and investment structures, an area of study that does not ordinarily come to mind when we think of “scientific” computation. Similarly, the increasingly frequent use of computer-generated animation in film production is based on a heavy dose of spline approximations, which we introduce in §4.8. And modern weather prediction is based on using numerical methods and analysis to solve the very complicated equations governing fluid flow and heat transfer between and within the atmosphere, oceans, and ground.

There are a number of different ways to break the subject down into component parts. We will discuss the derivation and implementation of the algorithms, and we will also analyze the algorithms, mathematically, in order to learn how best to use them and how best to implement them. In our study of each technique, we will usually be concerned with two issues that often are in competition with each other:

• Accuracy: Very few of our computations will yield the exact answer to a problem, so we will have to understand how much error is made, and how to control (or even diminish) that error.
• Efficiency: Does the algorithm take an inordinate amount of computer time? This might seem to be an odd question to concern ourselves with—after all, computers are fast, right?—but there are slow ways to do things and fast ways to do things. All else being equal (it rarely is), we prefer the fast ways.

We say that these two issues compete with each other because, generally speaking, the steps that can be taken to make an algorithm m...