Lanczos algorithms for large symmetric eigenvalue computations. Vol. I: Theory. Vol. II: Programs.

*(English)*Zbl 0574.65028
Progress in Scientific Computing, Vols. 3, 4. Boston-Basel-Stuttgart: BirkhĂ¤user. XIV, 273 p.; VII, 500 p. Set DM 178.00 (1985).

This is a research monograph presenting Lanczos procedures for the determination of eigenvalues and eigenvectors of large sparse matrices with special emphasis on computation. The focus is primarily on the authors’ research on single-vector Lanczos procedures with no reorthogonalization. The material is organized into two volumes. The first volume provides the theoretical background necessary for understanding the Lanczos procedures which are presented. The second volume contains FORTRAN programs for these procedures. The first three chapters of the first volume give an introduction to the Lanczos process in particular to computational aspects. This includes a brief review of basic results from matrix theory, a nice and understandable summary of Paige’s results on the behavior of the Lanczos procedure under finite precision arithmetic, a survey of the literature on the various types of Lanczos procedures and a collection of special properties of tridiagonal matrices. The proofs are mostly omitted and reference is given to the appropriate literature. The main part of the book starts with investigating the connection between the Lanczos procedure and the conjugate gradient method in exact arithmetic and under finite precision arithmetic. These relations are used to justify the development of a special Lanczos procedure with no reorthogonalization, in some cases carried on considerably beyond the dimension of the original matrix. In addition the connection with the conjugate gradient method justifies heuristically the identification test of this procedure, which is used to recognize the so-called spurious eigenvalues. These are eigenvalues of the Lanczos matrices, which cannot be considered as approximations to eigenvalues of the original matrix and which occur because of the loss of orthogonality of the Lanczos vectors. This Lanczos procedure is used to handle the real symmetric, Hermitian and the generalized symmetric, definite eigenvalue problem. It is adapted to compute singular values and singular vectors of rectangular matrices and to treat the eigenvalue problem for nondefective complex symmetric matrices. Finally a block Lanczos procedure is described and a hybrid method is given, which combines this block procedure with ideas from the single vector process. The construction and implementation of these procedure are explained in detail and numerous results of numerical experiments are presented. The corresponding FORTRAN programs are carefully documented, consistency checks are incorporated to verify the proper set up and samples of input/output are given. The whole presentation is detailed and understandable and should be accessible for readers with some knowledge of matrix eigenvalue problems on an introductory level.

Reviewer: A.Bunse-Gerstner

##### MSC:

65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

15-04 | Software, source code, etc. for problems pertaining to linear algebra |

15A18 | Eigenvalues, singular values, and eigenvectors |