- Journal Home
- Volume 43 - 2025
- Volume 42 - 2024
- Volume 41 - 2023
- Volume 40 - 2022
- Volume 39 - 2021
- Volume 38 - 2020
- Volume 37 - 2019
- Volume 36 - 2018
- Volume 35 - 2017
- Volume 34 - 2016
- Volume 33 - 2015
- Volume 32 - 2014
- Volume 31 - 2013
- Volume 30 - 2012
- Volume 29 - 2011
- Volume 28 - 2010
- Volume 27 - 2009
- Volume 26 - 2008
- Volume 25 - 2007
- Volume 24 - 2006
- Volume 23 - 2005
- Volume 22 - 2004
- Volume 21 - 2003
- Volume 20 - 2002
- Volume 19 - 2001
- Volume 18 - 2000
- Volume 17 - 1999
- Volume 16 - 1998
- Volume 15 - 1997
- Volume 14 - 1996
- Volume 13 - 1995
- Volume 12 - 1994
- Volume 11 - 1993
- Volume 10 - 1992
- Volume 9 - 1991
- Volume 8 - 1990
- Volume 7 - 1989
- Volume 6 - 1988
- Volume 5 - 1987
- Volume 4 - 1986
- Volume 3 - 1985
- Volume 2 - 1984
- Volume 1 - 1983
Cited by
- BibTex
- RIS
- TXT
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 Arnoldi 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 better 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 acceleration 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} }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 Arnoldi 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 better 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 acceleration are often considerably more efficient than the standard $(m+1)$-step restarted ones.