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Reviewer: Ian Gladwell

This excellent text is carefully organized as a reference. Mainly, it deals with iterative solution techniques for solving most types of algebraic eigenvalue problems which are of interest and for which there exist reliable algorithms. There are a large number of distinguished authors including the editors. The author(s) are identified at the beginning of each major section. After a brief introductory chapter explaining the use of templates and the overall organization of the text, Chapter 2 is devoted to a description of the six types of eigenvalue problems to be considered, each of which is discussed in one of Chapters 4-9. The first five types are: Hermitian eigenproblems, solving $Ax=\lambda x$ with $A$ Hermitian; generalized Hermitian eigenproblems, solving $Ax=\lambda Bx$ with $A$ and $B$ Hermitian; the singular value decomposition, computing the positive square roots of the eigenvalues of $A^*A$; non-Hermitian eigenproblems, solving $Ax=\lambda x$ without constraints on $A$; and generalized non-Hermitian eigenproblems, solving $Ax=\lambda Bx$. The chapters on and the discussion of each of these problems are sufficiently similar that the authors adopt a uniform structure which, though repetitive, makes the algorithms for specific problems simple to reference. In Chapter 2, for each of these problems, the authors define the problem, equivalencies such as similarity and congruence relations, conditioning, alternative specifications which are important for the efficiency and even the viability of the algorithms (for example, the aim may be to compute a small specified number of the largest eigenvalues, or to compute all the positive eigenvalues, which may be a small but unknown number, or to compute all the eigenvalues, which is usually a large and difficult problem). In each case, they also discuss some related eigenproblems and an example of how each type of eigenproblem can arise in practice, using a simple mass-spring problem for explanation. In Chapter 3, iterative projection methods are introduced as they are the major tool for computing extremal eigenvalues of large sparse eigenproblems employing rapidly convergent iterations. Particular examples are the power and Rayleigh-Ritz iterations but also a number of more refined methods are described. The iterative projection methods compute certain well-separated extreme eigenvalues quickly but other eigenvalues slowly, or effectively not at all. So, the remainder of this chapter is devoted to spectral transformations aimed at transforming the eigenvalue problem so that the eigenvalues of interest may be computed from well separated extreme eigenvalues of the transformed problem. In each of Chapters 4--8, a specific problem is discussed. In each case there is a brief discussion of direct methods, that is methods such as the {\it QR} algorithm which are iterative in nature but which are almost guaranteed to converge in a fixed number of iterations. Here, references are made to widely available software such as that in {\it LAPACK}. Then, each chapter has a number of sections devoted to iterative methods, particularly subspace iterations, Lanczos iterations and variants, Jacobi-Davidson iterations, and, in the case of non-Hermitian problems, Arnoldi iterations. In many cases, a well-structured pseudocode is given and where appropriate there is a discussion of available software. Most of these chapters finish with a section on assessing stability and accuracy. In Chapter 9, the authors discuss nonlinear eigenvalue problems. The major emphasis is on two problems. The first, the quadratic eigenvalue problem, has received significant attention in the literature. For the quadratic eigenvalue problem, $(\lambda^2 M+\lambda C + K)x=0$, the concentration is on transformations to a linear eigenproblem. The second part of the chapter considers nonlinear eigenvalue problems with orthogonal constraints. Here, the emphasis is far more experimental than in the earlier parts of the text where there are more established and well-developed algorithms. The software is written as \textsc{Matlab} templates which are to be made available on a web site associated with the text. The final two chapters are concerned with matters that are of common interest. In Chapter 10, there are brief discussions of compressed storage mechanisms, of structured and fast matrix multiplication, of direct solvers for sparse linear systems (including a table of the available software), of iterative solvers for sparse linear systems, and of important considerations for parallelism. In Chapter 11, a variety of methods for preconditioning eigenvalue and generalized eigenvalue problems are discussed. This chapter also contains a number of pseudocodes. Finally, an appendix discusses very briefly a number of special eigenvalue problems not treated in the text and gives references to the literature for further discussion and algorithms. The text has a bibliography of nearly 500 items and a rather short index. In Chapter 1, the intended audience is identified as students and teachers who want simple but effective algorithms, general scientists and engineers who need easy-to-use reliable software, and the high performance computing community who need access to the details of algorithms for fine tuning for performance. In my view, the text should be useful in all these contexts. Certainly, I would envisage using it as a practical companion and source of advice when I next teach a course on the algebraic eigenproblem.