1: Style of presentation is confusing; reads like a reference, not a learning book
This book appears to be very comprehensive, encompassing topics not usually found in the same volume, including numerical linear algebra, nonlinear optimization, numerical integration, and numerical solutions to ODE's and PDE's. The authors claim that this book is intended as a textbook for undergraduates wishing to engage in scientific computing, but the book's style of presentation is more appropriate for an advanced reference than a learning text. At the same time, the book is not thorough enough to be used as a reference.
In many chapters, the authors begin a topic in a general abstract framework, introduce concepts such as stability or convergence, and only later present specific methods and algorithms. The presentation and explanation of certain topics is very clear, but one has to skip around and sort through a lot of confusing nonsense to find the worthwhile parts of this text. Many ideas that could be expressed clearly in words are instead put in equation form where they are cryptic and unintuitive. The algorithms and diagrams are a little more useful and easier to understand, but often the book ignores discussion of the practicality of the various algorithms, and sometimes it ignores simple enhancements that have become common practice, instead only presenting algorithms in forms that are impractical and seldom used.
There is a wealth of other books covering much of the other material in this text. Where the material overlaps with Golub's "Matrix Computations", Golub's book is far superior, starting from a more basic level, covering the material in more depth, exploring more possibilities, and providing a much healthier dose of explicit algorithms. Trefethen's textbook "Numerical Linear Algebra" covers similar material in a much better fashion. The material on numerical optimization is covered in a much more readable fashion in the book by Nocedal and Wright, although that book definitely moves much slower. I can't say about the material on numerical methods for differential equations because it is an area I have not worked with.
This book might make a good reference for people who are already well-versed with many of the methods in this book and who wish to look up one topic here or there. However, I can say that I did not find it a useful text to learn the material from, and I doubt anyone would.
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