Volume 18, Issue 4
A Process for Solving a Few Extreme Eigenpairs of Large Sparse Positive Definite Generalized Eigenvalue Problem

Chong Hua Yu & O. Axelsson

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J. Comp. Math., 18 (2000), pp. 387-402

Published online: 2000-08

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  • 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 exectues 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 presevering 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 eigenproglems.When symmetric positive definite linear sysmtes must be solved in the process,an algebraic multilevel iteration method (AMLI) is applied. The algorithm is fully parallelizable.

  • Keywords

Eigenvalue sparse problem

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@Article{JCM-18-387, author = {}, title = {A Process for Solving a Few Extreme Eigenpairs of Large Sparse Positive Definite Generalized Eigenvalue Problem}, journal = {Journal of Computational Mathematics}, year = {2000}, volume = {18}, number = {4}, pages = {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 exectues 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 presevering 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 eigenproglems.When symmetric positive definite linear sysmtes must be solved in the process,an algebraic multilevel iteration method (AMLI) is applied. The algorithm is fully parallelizable. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9051.html} }
TY - JOUR T1 - A Process for Solving a Few Extreme Eigenpairs of Large Sparse Positive Definite Generalized Eigenvalue Problem JO - Journal of Computational Mathematics VL - 4 SP - 387 EP - 402 PY - 2000 DA - 2000/08 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9051.html KW - Eigenvalue KW - sparse problem AB - 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 exectues 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 presevering 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 eigenproglems.When symmetric positive definite linear sysmtes must be solved in the process,an algebraic multilevel iteration method (AMLI) is applied. The algorithm is fully parallelizable.
Chong Hua Yu & O. Axelsson. (1970). A Process for Solving a Few Extreme Eigenpairs of Large Sparse Positive Definite Generalized Eigenvalue Problem. Journal of Computational Mathematics. 18 (4). 387-402. doi:
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