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Explorations in Numerical Analysis

James V. Lambers and Amber C. Sumner
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Satzer
, on
This appealing introduction to numerical analysis is unusual in the sense that it is truly introductory. It begins with a lengthy chapter that tells students what numerical analysis is about, what its major components are, and why understanding error is so important. The book is designed to support a two-semester course, but it is structured to be able to support alternative courses.
The authors do not explicitly identify prerequisites, but they effectively assume competence with multidimensional calculus and basic linear algebra, as well as acquaintance with differential equations. No programming background is expected, but the authors strongly promote a hands-on approach. They provide an extensive introduction to MATLAB and expect students to use it throughout.
All the usual topics of basic numerical analysis are considered here, but the treatment is different in several respects. The chapter called “Understanding Error” is unusually detailed. It goes from questions of error analysis, sources of error, and error measurement to discussions of conditioning, stability, and convergence. This is followed by a careful discussion of the computer arithmetic. The presentation is clear and carefully paced. There is a danger in putting a large amount of material like this early in the book before students have been exposed to any substantial numerical algorithms, but the authors make it work. Good examples and exercises help.
Numerical linear algebra is handled particularly well. For instructors who want to concentrate on it, there is a thorough treatment of direct and iterative methods, least squares and eigenvalue problems. This includes topics like singular value decomposition, QR factorization, Given rotations, Householder reflections, and Cholesky factorization. Options are also available for instructors looking for a lighter treatment of linear algebra. If the full treatment is too much and instructors would rather focus on function approximation, for example, the chapter devoted to that has sufficient material to allow skipping the earlier full treatment of least squares. 
The authors pace the exposition very carefully; they begin with direct solutions of linear equations, focus first on diagonal systems, and gradually work from there. This is characteristic of the book, and maybe a bit of a concern for strong students who may find the pace too slow in some sections. Nonetheless, there is plenty of material to challenge even strong students.
The book has mostly standard treatments of nonlinear equations, optimization and differential equations. The chapters on polynomial approximation and approximation of functions are more elaborate and inclusive than in comparable texts.
The variety of exercises in the book is also a strength. There are exercises that test understanding of basic concepts, explorations that ask to students to work hands-on with ideas just presented, and end-of-chapter exercises. The authors state a number of theorems but never prove them. Sometimes a proof is assigned as an exercise. This seems a bit odd and appears to assume that students have already had experience writing proofs.
The authors tell us that the book started as a collection of notes that grew and were revised over the course of several years as the text developed and was classroom-tested. This book is definitely worth considering for anyone looking for a good introductory text.


Bill Satzer (, now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition to optical films.  He did his PhD work in dynamical systems and celestial mechanics.

See the publisher's web page.