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Scientific Computing

Michael T. Heath
Publisher: 
SIAM
Publication Date: 
2018
Number of Pages: 
567
Format: 
Paperback
Edition: 
2nd revised edition
Series: 
Classics in Applied Mathematics
Price: 
94.00
ISBN: 
978-1-611975-57-4
Category: 
Textbook
[Reviewed by
Bill Satzer
, on
08/4/2019
]
This book is a newer entry in the Classics In Applied Mathematics series from SIAM. It is an updated republication of the second edition of a book first published in 1997. Its subject is really numerical analysis. The author suggests that “scientific computing” and “numerical analysis” are synonymous, but the term “scientific computing” - always a little fuzzy - has taken on a broader meaning over the last decade or so and now often focuses more on aspects of computer implementation and delves more deeply into applications. 
 
The author takes a nontraditional approach that emphasizes the motivations for and ideas underlying numerical methods. The book is intended for graduate students and advanced undergraduates in computer science, mathematics, and engineering. The prerequisites are multivariate calculus, linear algebra and basic acquaintance with differential equations. Because there are computer exercises, access to and facility with numerical software packages are expected. Pointers to readily available software are given throughout, but no particular software or programming language is required.
 
The topics that are treated here are fairly standard. The first half of the book discusses primarily algebraic problems, while the second half focuses on problems involving derivatives and integrals. Numerical linear algebra gets more emphasis than usual. There is more material here than could be handled in a single semester, but not enough for two. The more advanced topics – ones likely not likely to be included in a single semester course – are basic treatments of boundary value problems for ordinary differential equations, partial differential equations, the fast Fourier transform, and stochastic simulation.
 
This book is more challenging in several respects than a typical introduction to numerical analysis. The treatment of linear least squares, for example, goes into some detail on Householder transformations and Givens rotations as part of the author’s discussion of orthogonalization methods. More commonly these details are tucked away – often unmentioned – in expositions of higher-level algorithms such as singular value decomposition. Existence, uniqueness, and conditioning of solutions are also treated more extensively than in comparable introductory texts. The author says that he “adopts a fairly sophisticated perspective”, and expects a “reasonable level of maturity” from students. 
 
The problems and exercises also demand more from students. Not many are straightforward. Some students without prior experience with some kind of mathematical software would find the computer exercises quite challenging. While many examples are provided, they tend to be brief and relatively simple compared with what the author expects students to be capable of in the problems and exercises. 
 
All the same, this is an attractive presentation of several aspects of numerical analysis. The sections on numerical linear algebra and linear least squares are particularly well done. The author’s treatment of optimization is also noteworthy: clear, thorough and concise.
 
This would be an excellent choice for a text with a class of more mathematically mature students. With an average class seeing numerical analysis for the first time, it might not work very well.

 

Bill Satzer (bsatzer@gmail.com), 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.

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