Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Volume I: Theory

First published in 1985, this book presents background material, descriptions, and supporting theory relating to practical numerical algorithms for the solution of huge eigenvalue problems. This book deals with 'symmetric' problems. However, in this book, 'symmetric' also encompasses numerical procedures for computing singular values and vectors of real rectangular matrices and numerical procedures for computing eigenelements of nondefective complex symmetric matrices. Although preserving orthogonality has been the golden rule in linear algebra, most of the algorithms in this book conform to that rule only locally, resulting in markedly reduced memory requirements. Additionally, most of the algorithms discussed separate the eigenvalue (singular value) computations from the corresponding eigenvector (singular vector) computations. This separation prevents losses in accuracy that can occur in methods which, in order to be able to compute further into the spectrum, use successive implicit deflation by computed eigenvector or singular vector approximations.

Dr. Jane K. Cullum is a Technical Staff Member in the Computer and Computational Sciences Division at the Los Alamos National Laboratory. Prior to coming to Los Alamos in late 1998, she was a Researcher and a Senior Manager in the Mathematical Sciences Department at the IBM Research Center in Yorktown Heights, NY. She is active in the Society for Industrial and Applied Mathematics (SIAM) and she is a Fellow of the IEEE and a recipient of the IEEE Third Millennium Medal in 2000. Her research contributions focus on practical numerical algorithms for large-scale or difficult problems. She has published contributions to problem partitioning, preconditioning, algebraic multigrid, model reduction, eigenvalue problems and modal analysis, system stability, data analysis, optimization, and control theory, and on understanding the behavior of and relationships between different types of numerical algorithms.

Dr. Ralph A. Willoughby (1923-2001) was a member of the Research Staff in the Mathematical Sciences Department in IBM Research for 34 years (1957-1991). He was a primary catalyst in the development of sparse matrix theory and its applications and in 1968 organized the first Symposium on Sparse Matrices. He has been called the father of Sparse Matrix Theory. In addition to his work on direct methods for sparse matrices and his work with Dr. Cullum on eigenvalue algorithms, he is also well known for his contributions to the solution of stiff systems of differential equations.