1R3. Matrix Algorithms, Volume II: Eigensystems. - GW Stewart (Dept of Comput Sci, Univ of Maryland, College Park MD). SIAM, Philadelphia. 2001. 469 pp. (Softcover). ISBN 0-89871-503-2. $50.00.

Reviewed by A Mahajan (Dept of Mech Eng and Energy Processes, S Illinois Univ, Carbondale IL 62901).

This book contains a comprehensive presentation of computations involving the eigenvalues and eigenvectors of a matrix. It is the second volume in a projected five-volume series on matrix algorithms. The first volume was about basic decompositions. The second volume (this book) presents eigensystems. The next three books are projected to treat iterative methods for linear systems, sparse direct methods, and special topics, including fast algorithms for structured matrices. The author has managed to keep this volume fairly independent of the first volume, hence a basic knowledge of linear algebra is sufficient to understand most topics in the book is necessary. The book is really intended to be a reference book for scientists and engineers who need tools to solve problems in eigensystems. According to the author, the intended audience is the nonspecialist whose needs cannot be satisfied by black boxes. The book does live up to the author’s stated aim. The author gives detailed derivations that will help anyone who wants to adapt the methods to particular problems. The book really is not intended to be a textbook due to the lack of extensive solved and unsolved problems, but can be used as a self study by graduate students or even undergraduate students in honors programs.

The book is divided into two parts: Dense Eigenproblems and Large Eigenproblems. The book has six chapters, a reference list of over 300 references, and an excellent subject index. The first chapter presents the underlying theory for eigensystems. The second chapter describes the widely used QR algorithm. Chapter 3 deals with symmetric matrices and the singular value decomposition. The fourth chapter deals with large matrices for which the computation of a complete eigensystem is not possible, hence an algebraic and analytic theory of eigensystems is presented. The fifth chapter deals with the Krylov sequence methods, in particular the Arnoldi and Lanczos methods. The sixth chapter presents alternative methods such as subspace iteration and the Jacobi-Davidson method.

The book provides pseudo codes for many algorithms. The author does admit that the pseudo codes are for illustration only and should not be regarded as finished implementations, and it is difficult to verify their correctness. He also mentions that whereever possible he has checked the algorithms against MATLAB implementations. If there is a weakness in the book, it is right here, for the author could have provided these MATLAB programs on disk, or at the very least, made them available on a website. The reviewer did implement some of the algorithms in MATLAB, and it did take some time to develop the implementation. Of course, this in no way impacts the significance of the material to the serious reader who is looking for a detailed treatise on the subject, but does impact a student or practicing engineer who could be looking for a quick implementation of a particular algorithm. A recommendation to the author and publisher would be to make these MATLAB programs available on a website so that they can be easily downloaded.

Matrix Algorithms, Volume II: Eigensystems is an excellent treatise on eigensystems. It should be purchased by all libraries and any individuals who make extensive use of computations in eigensystems.