Matrix Computations

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.

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

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.

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!

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.