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Elementary Numerical Analysis

Kendall Atkinson and Weimin Han
John Wiley
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[Reviewed by
Charles Ashbacher
, on

Some exposure to numerical analysis beyond the pittance commonly covered in calculus courses should be a mandatory part of the training of all mathematicians. As I am fond of telling my computer students, the algebra that they spent so much time learning does not apply to computer computations. I show them examples in which the associative, commutative and distributive properties fail. The reasons are simple, in algebra, we assume that all numbers are the same type and stored to infinite precision. In algebra, there is no difference between an integer and a floating point, which allows us to combine them in any way.

Next fall, I will be teaching numerical analysis for the first time, and I am looking forward to it with relish. Even more so now that I managed to obtain and examine a copy of this book. Atkinson taught me numerical analysis from an earlier edition many years ago and I see that he has lost nothing over the years. I had no trouble understanding his earlier text when I was a student and expect my students to have a similar experience when I use this one for my class. With so much to cover, it is succinct, yet there is enough detail so that the main points are clear.

Calculus up through the fundamentals of differential equations is necessary to understand the entire book, although differential equations do not appear until the end. There are chapters devoted to the solution of linear algebra problems, but I don’t think that a course in linear algebra is really necessary. Most students are exposed to the fundamentals of matrices and determinants before calculus, and nothing beyond that is needed. The second half of linear algebra courses is generally reserved for an introduction to mathematical proofs and proofs are rarely done here.

The authors use MATLAB to code the solutions and one can argue both ways about the wisdom of their choice. I find it a moot point, since most of the time the actions of the code are so obvious that the language of choice is largely irrelevant. If you don’t understand the math, then the language of the code changes nothing and if you understand the math, then a computer program is just a succinct restatement of an algorithm you already know. The material is segmented so that it is possible to teach a one-semester course using the initial chapters and a full year course using the entire book.

In my ideal world, every student in mathematics or computers will be required to read and understand chapter two on the fundamental errors in computer arithmetic. Since my desires will no doubt go unfulfilled, I will content myself with teaching a class and continuing to mention the basics every chance that I get.





Chapter 1. Taylor Polynomials.

Chapter 2. Error and Computer Arithmetic.

Chapter 3. Rootfinding.

Chapter 4. Interpolation and Approximation.

Chapter 5. Numerical Integration and Differentiation.

Chapter 6. Solution of Systems of Linear Equations.

Chapter 7. Numerical Linear Algebra: Advanced Topics.

Chapter 8. Ordinary Differential Equations.

Chapter 9. Finite Difference Method for PDEs.

Appendix A: Mean Value Theorems.

Appendix B: Mathematical Formulas.

Appendix C: Numerical Analysis Software Packages.

Appendix D: Matlab: An Introduction.

Appendix E: The Binary Number System.

Answers to Selected Problems.