Volume 18, Issue 3
Improving Eigenvectors in Arnoldi's Method

Zhong Xiao Jia & Ludwig Elsner

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J. Comp. Math., 18 (2000), pp. 265-276

Published online: 2000-06

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  • Abstract

The Ritz vectors obtained by Arnoldi's method may not be good approximations and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the efficiency of Arnolid type algorithms, we propose a strategy that uses Ritz values obtained from an m-dimensional Krylov subspace but chooses modified approximate eigenvectors in an (m+1)-dimensional Krylov subspace. Residual norm of each new approximate eigenpair is minimal over the span of the Ritz vector and the (m+1)th basis vector, which is available when the m-step Arnoldi process is run. The resulting modified m-step Arnoldi method is bettern than the standard m-step one in theory and cheaper than the standard (m+1)-step one. Based on this strategy, we present a modified m-step restarted Arnoldi algorithm.Numerical examples show that the modified m-step restarted algorithm and its version with Chebyshev accelration are often considerably more efficient than the standard (m+1)-step restarted ones.

  • Keywords

Large unsymmetric The m-step Arnoldi process The m-step Arnoldi method Eigenvalue Rit

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@Article{JCM-18-265, author = {}, title = {Improving Eigenvectors in Arnoldi's Method}, journal = {Journal of Computational Mathematics}, year = {2000}, volume = {18}, number = {3}, pages = {265--276}, abstract = { The Ritz vectors obtained by Arnoldi's method may not be good approximations and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the efficiency of Arnolid type algorithms, we propose a strategy that uses Ritz values obtained from an m-dimensional Krylov subspace but chooses modified approximate eigenvectors in an (m+1)-dimensional Krylov subspace. Residual norm of each new approximate eigenpair is minimal over the span of the Ritz vector and the (m+1)th basis vector, which is available when the m-step Arnoldi process is run. The resulting modified m-step Arnoldi method is bettern than the standard m-step one in theory and cheaper than the standard (m+1)-step one. Based on this strategy, we present a modified m-step restarted Arnoldi algorithm.Numerical examples show that the modified m-step restarted algorithm and its version with Chebyshev accelration are often considerably more efficient than the standard (m+1)-step restarted ones. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9040.html} }
TY - JOUR T1 - Improving Eigenvectors in Arnoldi's Method JO - Journal of Computational Mathematics VL - 3 SP - 265 EP - 276 PY - 2000 DA - 2000/06 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9040.html KW - Large unsymmetric KW - The m-step Arnoldi process KW - The m-step Arnoldi method KW - Eigenvalue KW - Rit AB - The Ritz vectors obtained by Arnoldi's method may not be good approximations and even may not converge even if the corresponding Ritz values do. In order to improve the quality of Ritz vectors and enhance the efficiency of Arnolid type algorithms, we propose a strategy that uses Ritz values obtained from an m-dimensional Krylov subspace but chooses modified approximate eigenvectors in an (m+1)-dimensional Krylov subspace. Residual norm of each new approximate eigenpair is minimal over the span of the Ritz vector and the (m+1)th basis vector, which is available when the m-step Arnoldi process is run. The resulting modified m-step Arnoldi method is bettern than the standard m-step one in theory and cheaper than the standard (m+1)-step one. Based on this strategy, we present a modified m-step restarted Arnoldi algorithm.Numerical examples show that the modified m-step restarted algorithm and its version with Chebyshev accelration are often considerably more efficient than the standard (m+1)-step restarted ones.
Zhong Xiao Jia & Ludwig Elsner. (1970). Improving Eigenvectors in Arnoldi's Method. Journal of Computational Mathematics. 18 (3). 265-276. doi:
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