Customer Reviews

Matrix Computations (Johns Hopkins Studies in the Mathematical Sciences)

5 star:		(20)
4 star:		(6)
3 star:		(5)
2 star:		(0)
1 star:		(1)

 

 

This is not a complete review. I just wanted to say something important about the book. I'm a second year computer science PhD student, comfortable with linear algebra. I have been using this book for a couple of months to implement SVD (singular value decomposition) and unfortunately the book turned out to introduce some difficulties.

First of all, it's annoyingly terse! You must be quite comfortable with matrices and all the manipulations etc. to "grasp" the main idea behind an algorithm. I'm talking about truly understanding, not implementing line by line. Most of the times, you will need a paper and pencil to understand what's going on during the execution of an algorithm.

Yet, there's one more important thing: There are typos, and worse: there are mistakes. A specific example:

page 456, Algorithm 8.6.2 The SVD Algorithm

It doesn't talk about how to extract U and V in the decomposition A = U^T*D*V and the last line is incorrect.
diag(I_p, U, I_{q+m-n}) is not an n*n matrix, so you cannot multiply B with this matrix from left. Maybe, there's something I couldn't catch, but this is book's deficiency again.

page 252 Example 5.4.2 about Householder Bidiagonalization

The given matrices do not constitute a correct bidiagonalization, I checked them with Matlab.

and a typo: page 216 5.1.9 Applying Givens rotations, the 4th and 5th line of the algorithm is incorrectly written.

A(1, j) = ...
A(2, j) = ...

should be

A(i, j) = ...
A(k, j) = ...

So, these are the ones I encountered. This book is unmatched in its category in terms of depth and coverage, but it definitely needs a new edition with a more careful treatment.

This is one of the definitive texts on computational linear algebra, or more specifically, on matrix computations. The term "matrix computations" is actually the more apt name because the book focuses on computational issues involving matrices,the currency of linear algebra, rather than on linear algebra in the abstract. As an example of this distinction, the authors develop both "saxpy" (scalar "a" times vector "x" plus vector "y") based algorithms and "gaxpy" (generalized saxpy, where "a" is a matrix) based algorithms, which are organized to exploit very efficient low-level matrix computations. This is an important organizing concept that can lead to more efficient matrix algorithms.

For each important algorithm discussed, the authors provide a concise and rigorous mathematical development followed by crystal clear pseudo-code. The pseudo-code has a Pascal-like syntax, but with embedded Matlab abbreviations that make common low-level matrix operations extremely easy to express. The authors also use indentation rather than tedious BEGIN-END notation, another convention that makes the pseudo-code crisp and easy to understand. I have found it quite easy to code up various algorithms from the pseudo-code descriptions given in this book. The authors cover most of the traditional topics such as Gaussian elimination, matrix factorizations (LU, QR, and SVD), eigenvalue problems (symmetric and unsymmetric), iterative methods, Lanczos method, othogonalization and least squares (both constrained and unconstrained), as well as basic linear algebra and error analysis.

I've use this book extensively during the past ten years. It's an invaluable resource for teaching numerical analysis (which invariably includes matrix computations), and for virtually any research that involves computational linear algebra. If you've got matrices, chances are you will appreciate having this book around.

First, this isn't Numerical Recipes. If you're looking for cut&paste code, you're just looking in the wrong place. This is for people who need a deep understanding of the computational issues, and are going to put a lot of time into an implementation. It's for people who are completely at ease with linear algebra, standard matrix-oriented problems, and dense mathematical notation.

Despite its demand for a reader well versed in theory, this really is about practice. It's about the nasty effects of finite-precision arithmetic, about specific ways of minimizing the harm they cause. These techniques take full advantage of any special features in the problem, including banding and symmetry. This also deals briefly with caching issues, which are even more important now than when this book was written. Cache data can get to the processor in 1-10 cycles, in a modern workstation processor, but main memory access costs 100-1000 cycles. TLB misses can cost many thousands of cycles, even when data is already in memory. Clearly, good data structures and well-orgnized access patterns can make a huge difference, but one that is mentioned only briefly. The section on parallel computation is brief and helpful, but overdue for review. The authors could never have foreseen today's multi-(thread, core, processor) systems, Blue Gene, or clusters.

Still, this is an indispensable reference for someone in the thick of numerical computation. Most programmers would do better, in lots of ways, usingn the GNU Scientific Library or one of the other production-quality packages out there. They don't always do the job, though. Emerging architectures, include hardware threading and reconfigurable computing, need new implementations of even well-known algorithms. If you have big mathematical problems and machines too exotic for the standard tools, you're on your own. Numerical computing is such a large topic that no one book can possibly cover it all. In the end, though, many other problems reduce to linear systems, and that's where this comes in. It may not be theonly book you'll need, but you'll need it.

//wiredweird

In certain ways, this book has been both a bane and a boon to my career as a computational mathematician. Way back in 1989, I had the mixed experience of taking a course in Numerical Analysis from Brian Smith at the University of New Mexico. Prof. Smith taught that course exclusively from this book (actually, from the 2nd edition). As a college sophomore, I was terribly out of my depth, but I managed to do okay. Later, I had the opportunity to study under Gene Golub at Stanford, although I was certainly not one of his better students :) Naturally, Prof. Golub also taught pretty much exclusively from this book, by the way, he is a gifted mathematician and wonderful instructor, and a real gentleman. Between these experiences, I'd say I became extremely familar with the contents of this book.
Okay, back to the actual book. If you've got a numerical linear algebra problem to solve, and you don't know which NAG or Matlab routine to use, or simiarly can't figure out why your Numerical Recipes ripped-off code is blowing up on a certain matrix, well, you'll find the reason in this book. The main issue is that you've got to know what you're looking for in order to find it, and that's kind of the kernel of the problem. Some reviewers have stated that the writing is terse, that it is too rigorous, etc. I don't really agree with these reviews, but I agree that it is not for the casual reader who wants a quick answer to the question of "how do I invert this thing". The book spends a lot of time with subtle details such as convergence and stability, and in my experience, these excellent treatments are wasted on most would-be users who are really just looking for a numerical silver bullet, which, of course, just doesn't exist. I find that the book is an invaluable reference, when I have a problem like this "okay, I've got a small Vandermonde matrix that may or may not be singular. What's the quickest algorithm to get a stable result?" Usually what happens is I read this book to understand the method, and then go looking for an existing implementation, usually in Matlab or NAG. I guess the bottom line is that if you're looking for a "recipe" you won't find them here. If you're looking for insight and understanding into a numerical method, you will find it here, but you'll have to work to eek out an implementation if you really feel that you need to write one.

Great book on the computational aspects of matrix computations. Has much more detail than NRiC for matrix computations -- of course, NRiC covers more topics. One the few places you can actually find out how to code SVD. A steal at $30. Highly recommended!

Once you have a grounding in matrix analysis and linear algebra this book makes a good reference. His explanations tend to be terse (even exceptionally so)- more suited for reminding someone who already knows how the algorithm works or was derived and simply can't remember the details. It lost a star as I've found some annoying typos (for example, in the pseudocode for the GMRES algorithm).

This book is the standard reference for all numerical linear algebra. It is a graduate-level applied math textbook written by practicing professionals for practicing professionals. If you are new to the topic you would probably prefer something like James Demmel's Applied Numerical Linear Algebra.

If you are interested in implementing the algorithms in this book, stop right now and first make sure that you can't use MATLAB or LAPACK instead, or even ScaLAPACK if you need a parallel implementation. Getting these algorithms right is hard, and the hard work has probably already been done by somebody else. LAPACK contains the accumulated wisdom of over forty years of research in numerical linear algebra, and MATLAB contains LAPACK. Don't re-invent the wheel.

On the other hand, if you want to understand how LAPACK works, or if you need to understand its numerical accuracy and stability, then this is the book for you.

Another reviewer has mentioned that this book contains numerous errata in the formulas. This is still true as of the third edition. Usually it is possible to detect and correct these errors by reading and understanding the surrounding text, but beware.

This book should be placed alongside "Principles of Mathematical Analysis" by Walter Rudin and "Finite Dimensional Vector Spaces" by Paul Halmos as a classic text, one which students/professionals of mathematics will use for years to come. A solid book covering computational matrix theory. I myself used it as a tool to bridge the gap between my formal training in Mathematics and my serious interest in computers. Reader should have some knowledge of basic linear algebra(ie understanding of vector spaces, L2 norms, etc..) before attempting this book. Excercises could be better. A good purchase for those with a more than passing interest.

I recently bought this book and am amazed at how detailed the information is presented. This a great book for anyone doing numerical analysis on the computer. The details on how to work around ill-conditioned matrices is great.

This book is an invaluable reference for anyone working in matrix computations or linear algebra. I have been using it for years and found it to be clear and comprehensive