Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics). xix, 469 p. (2001).

This second volume in a series of 5 on matrix algorithms is devoted to eigenvalue-eigenvector computations. As in the first volume on basic decompositions (1998;

Zbl 0910.65012), the approach is very constructive, algorithmic, and self contained. The author has chosen not to heavily cross reference between the different volumes but to assume a general understanding of what is needed from the previous theory and algorithms. The first part concentrates on general dense matrices and the main issues are subsequently: the general theory, the QR algorithm, and the symmetric eigenvalue problem. Also singular values and the generalized eigenvalue problem are discussed here. In the second part large matrices are treated. Again the general theory is given and is followed by Krylov methods (in particular Arnoldi and Lanczos). The last chapter shifts to subspace iteration and the Jacobi-Davidson method.
The material for this second volume is less elementary than in the first volume. Yet the author succeeds again in picking up the essential stepping stones to allow the student or the practinoner, who might not be a specialist in numerical linear algebra, to safely embark the reading of this volume. It confirms the expectation that the whole series will grow out to be a reference work for the next generation.