A Process for Solving a Few Extreme Eigenpairs of Large Sparse Positive Definite Generalized Eigenvalue Problem
Year: 2000
Author: Chong-Hua Yu, O. Axelsson
Journal of Computational Mathematics, Vol. 18 (2000), Iss. 4 : pp. 387–402
Abstract
In this paper, an algorithm for computing some of the largest (smallest) generalized eigenvalues with corresponding eigenvectors of a sparse symmetric positive definite matrix pencil is presented. The algorithm uses an iteration function and inverse power iteration process to get the largest one first, then executes $m-1$ Lanczos-like steps to get initial approximations of the next $m-1$ ones, without computing any Ritz pair, for which a procedure combining Rayleigh quotient iteration with shifted inverse power iteration is used to obtain more accurate eigenvalues and eigenvectors. This algorithm keeps the advantages of preserving sparsity of the original matrices as in Lanczos method and RQI and converges with a higher rate than the method described in [12] and provides a simple technique to compute initial approximate pairs which are guaranteed to converge to the wanted $m$ largest eigenpairs using RQI. In addition, it avoids some of the disadvantages of Lanczos and RQI, for solving extreme eigenproblems. When symmetric positive definite linear systems must be solved in the process, an algebraic multilevel iteration method (AMLI) is applied. The algorithm is fully parallelizable.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2000-JCM-9051
Journal of Computational Mathematics, Vol. 18 (2000), Iss. 4 : pp. 387–402
Published online: 2000-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 16
Keywords: Eigenvalue sparse problem.