Algorithms for sparse matrix eigenvalue problems. [DBLKLN, block Lanczos algorithm with local reorthogonalization strategy]
Eigenvalue problems for an n by n matrix A where n is large and A is sparse are considered. A is assumed to be unstructured: it cannot be reordered to have narrow bandwidth, nor can linear systems of the form Ax = b be solved by special techniques. The eigenvalue problem for such a matrix is well-solved only when A is symmetric and the eigenvalues to be computed are among the largest or smallest eigenvalues of A. The Lanczos algorithm offers a very good solution in this case. Algorithms are proposed for three difficult problems: computing one (or a few) eigenvalue(s) of a symmetric matrix where the eigenvalue(s) are not near the end of the spectrum; computing a number of interior eigenvalues of a symmetrix matrix; computing one (or a few) eigenvalues(s) of a non-symmetric matrix. Two algorithms for finding a single interior eigenvalue are investigated. An algorithm using the Lanczos algorithm to solve the linear systems which arise in the Rayleigh quotient iteration is much superior. The algorithm for finding many eigenvalues is built directly upon the Lanczos algorithm without reorthogonalization. It is found that the Lanczos algorithm is suitable for very large problems only if a limited reorthogonalization is included. A block generalization of the unreorthogonalized Lanczos algorithm is developed, and necessary conditions for its stability are given. The investigations and analyses are largely empirical. The third algorithm reduces the non-symmetric eigenproblem to a series of symmetric eigenvalue problems. It is a Rayleigh quotient-like iteration using singular vectors. A theoretical analysis of the local convergence properties of the algorithm is given. There are two obvious computational approaches, one by way of a poorly conditioned extreme eigenvalue problem, the other by way of an interior eigenvalue problem. The latter approach is superior. 3 figures, 25 tables.
- Research Organization:
- Stanford Univ., CA (USA). Dept. of Computer Science
- DOE Contract Number:
- EY-76-S-03-0326-030
- OSTI ID:
- 7254102
- Report Number(s):
- SU-326P30-52; STAN-CS-77-595
- Resource Relation:
- Other Information: Thesis
- Country of Publication:
- United States
- Language:
- English
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