An Introduction to Numerical Analysis

good and fast

This was the textbook we used for my Numerical Analysis course. To be perfectly honest, I don't know if it's the the way the author presents the material or the incredibly dry and (to me at least) uninteresting subject matter or some superposition of the two, but I found it hard to learn from this book. It was the only math course in college (I took 17 total) that I finished without retaining any of the information. It's entirely possible that all current books on this subject are just as awful to read, so I hesitate to blame the author.

I took 3 courses from Professor Atkinson, and the knowledge gained from this book really impacted my future research. Yes, the software references are old-school, but the core ideas are lovely and clear. The knowledge I gained from this book helped me through a PhD at Berkeley. I still refer to this book often.

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

I give this a 4 out of 5 due to the errors I found in the book (minor, mostly in the exaple problems). In addition to vague number scheming in functions or algorithms for for solving functions.Otherwise a great book. Lots of material.

many typos

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.

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.