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$ {\rm SOR}$-methods for the eigenvalue problem with large sparse matrices

Author: Axel Ruhe
Journal: Math. Comp. 28 (1974), 695-710
MSC: Primary 65F15
MathSciNet review: 0378378
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Abstract: The eigenvalue problem $ Ax = \lambda Bx$, where A and B are large and sparse symmetric matrices, is considered. An iterative algorithm for computing the smallest eigenvalue and its corresponding eigenvector, based on the successive overrelaxation splitting of the matrices, is developed, and its global convergence is proved. An expression for the optimal overrelaxation factor is found in the case where A and B are two-cyclic (property A). Further, it is shown that this SOR algorithm is the first order approximation to the coordinate relaxation algorithm, which implies that the same overrelaxation can be applied to this latter algorithm. Several numerical tests are reported. It is found that the SOR method is more effective than coordinate relaxation. If the separation of the eigenvalues is not too bad, the SOR algorithm has a fast rate of convergence, while, for problems with more severe clustering, the c-g or Lanczos algorithms should be preferred.

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Keywords: Eigenvalues, sparse matrices, overrelaxation
Article copyright: © Copyright 1974 American Mathematical Society