From the very beginning Gautschi’s book distinguishes itself from scores of other numerical analysis texts. His book, he says, is designed for a graduate program that would include a basic introductory course followed by more specialized courses. In particular, the first four chapters of his book would, in his view, serve as a text for the introductory course, and the remaining three chapters could provide material for an advanced course in numerical solution of ordinary differential equations. Gautschi notes that he breaks the tradition of incorporating all the major topics of numerical analysis in one text; his justification is that major subdisciplines such as numerical linear algebra and numerical solution of partial differential equations have become so substantial that they can no longer be practically or adequately treated in a single book.
Having thus set the metes and bounds of his book, Gautschi has produced a very attractive textbook. Just exactly who would use it and how is another matter. The first chapter is especially impressive. Students may find the business of machine arithmetic rather dull but it is truly fundamental to the whole business of numerical analysis (and frequently the source of otherwise mysteriously bad behavior of algorithms). The author does a very nice job of homing in on the real difficulties. This is as good a treatment as I have seen, but it is pitched at a fairly sophisticated level. The other three chapters in the “introductory” first half of the book are equally well written discussions of more or less standard material in approximation, interpolation, numerical differentiation and integration, and solution of nonlinear equations.
The “advanced” portion of the book consists of three chapters: two on numerical solutions of the initial value problem of ordinary differential equations and one on two-point boundary value problems. This is a sophisticated and rigorous treatment at least a notch above the level of the earlier chapters. Gautschi attends to both theoretical issues and practical questions of implementation. Throughout the book there is consistent evidence of the author’s intention to make the book teachable and student-friendly.
The text has a very complete bibliography as well as an early section pointing the student to a selection of general numerical analysis textbooks, more specialized books and monographs, and relevant journals. Solutions to selected exercises are provided (“to give students an idea of what is expected”) and a full solution manual is available to instructors. All chapters have dual sets of regular exercises and programming exercises. (Matlab is commonly used throughout the book.)
Where does this attractive book fit in the scheme of things? The latter chapters on numerical solution of ordinary differential equations would work splendidly for a second course in numerical analysis at the graduate level. The first chapters by themselves would probably not constitute a good introduction to numerical analysis by themselves because of the absence of numerical linear algebra (especially for student who would take only an introductory course). They would nonetheless make for excellent supplemental or reference material.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.