An Introduction to Numerical Methods and Analysis, 2nd Edition

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”) ...