Customer Reviews

Introduction to Numerical Analysis

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One of the best numerical analysis books I ever came across. This describes the theory behind numerical analysis, so if you expect to find a lot of numerical examples and written algorithms, this is NOT the book you're looking for.
Though there are some examples and algorithm, this is a math book, not a computer science oriented book. So buy this book if you are interested in the mathematical theory and ideas behind numerical analysis. Algorithms come and go, but the theory is always the same.
In my work as a computational physicist I use this book extensively and find it invaluable.
It takes some time to get used to, but little effort in understanding math never killed anyone.

Numerical analysis is the study and art of determining how to get high quality answers out of computers with finite precision: in other words, all of them. This may not sound like a big issue - you can always use double precision, right? Well, no. Binary computers can't even represent 0.1 exactly. The numbers are wrong from the start, and go downhill fast. This book addresses the twin questions: how fast, and how to preserve as much accuracy as possible.

Atkinson gives a clear, readable exposition. Chapters cover all the classic topics: error analysis, solutions of nonlinear systems, and issues in vector and matrix manipulations. Matrix analysis skips discussion of sparse systems, though, and omits the different kinds of decompositions available for matrices in special form. He also presents chapters on integration and solution of differential equations, also staples of scientific computing, though maybe not quite as common as the other topics. Some of the best material, though, comes in sections on interpolation and function approximation, something that came up in my own work recently. A typical engineer equates polynomial approximation with truncated Taylor series, but that's a real mistake. Atkinson describes techniques based on sets of orthogonal polynomials. For an approximation of given polynomial degree, my application showed an order of magnitude reduction in error when we stopped using Taylor series. Your milage may vary, but orthogonal polynomials never give worse results. Also note that they don't affect how the approximation polynomial is used - just the way you pick the coefficients.

I fault this book only for minor points. First, discussion early on predates general acceptance of IEEE 754 - with denorms and other weirdness, problems are slightly different than before, but wide availability means that almost everyone has the same problems (early Java implementations notwithstanding). Second, it refers to "stable" problems as "well posed." Many problems, molecular dynamics among them, have inherently chaotic features no matter how they're phrased. The problem is what it is, and calling it "badly posed" suggest that beating it into shape will somehow "pose" it better - directing attention away from dealing with its true nature. Despite a few pickable nits, this is an outstanding introduction for a diligent reader, and should be on the shelves of any programmer involved in scientific computing.

//wiredweird

Out of some 7 or 8 numerical analysis texts from which I have learned or taught, this is easily the best. Its organization is standard, its exposition is excellent, it is comprehensive in its coverage of introductory topics, it has a very good bibliography, and its problems are very good. It is a good introduction for graduate students; it is a little advanced for most undergraduates, though strong undergraduates would benefit from its use. No computer coding is supplied though coding from the book's explanations is straightforward.

The material is all mostly valid, and the topics presented are treated in a sophisticated manner. This is not meant as an elementary text in numerical analysis.

One unfortunate distraction, that appears on every page, is the obsolete font. By comparison with fonts in more recently written texts, including those books by the same publisher, the printed text of this book appears smudged. Despite the author's claim in the preface to the second edition, it is not true that "all sections have been rewritten". But maths books are notoriously expensive to retype, because of the intricate equations that appear. I suspect what happened here is that the publisher largely went the easy route of re-using the older camera-ready files.

Another backwardness is the reference in the above mentioned preface, written in 1987, to software packages by IMSL and NAG. These certainly still exist. But by now, packages by Mathematica, Maple and Matlab are more prevalent, at least for undergraduate students. Though for readers experienced in this subject or in programming, they should be able to write code implementing the algorithms.

In my opinion, Atkinson is the best writer of numerical analysis textbooks. I learned numerical analysis from him and have used his books every time I have taught the course. The algorithms are written in a combination of pseudocode and Pascal-like language, as in the := being used for assignment. I disagree with the other reviewers who criticize the text for being unreadable. It is completely understandable, provided you have the mathematical background one would expect of a numerical analysis student. Three semesters of calculus are considered the minimum background needed for a numerical analysis class. If you have taken and understood them, then this reading is not that difficult.
I have used this text as a reference in my earlier work and would still be using it if the third edition had not come out. I will continue to use Atkinson's fine texts in my numerical analysis classes and would not hesitate to use this edition if for some reason I could not obtain the later one.

This is a standard textbook by a leading authority. There is little hand-waiving here. However, this is hardly a book to learn by.

The typesetting could have been a bit better. I wish the proofs had been set off from the examples and the text a little more. There is also too much referencing to earlier equations. Rather than referring me over and over to equatin (6.2.1), just re-write the equation.

Also, this book is starting to show its age. It is now 11 years old, so its bibliography is a bit outdated, as are references to computer programs.

My most severe criticism of this book is that it is sorely lacking in explanations. There is little intuition provided here. Definately not an undergrad book. A much better text to learn from--but not as useful as a reference as this book is--is Burden and Faires. B&F make lots of use of pseudo-code and I applaud them for it. It helps detangle some of the math.