Generalizations and modifications of the GMRES iterative method |
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Authors: | Jen-Yuan Chen David R Kincaid David M Young |
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Institution: | (1) Department of Applied Mathematics, I-Shou University, Ta-Hsu, Kaohsiung, 840, Taiwan;(2) Center for Numerical Analysis, University of Texas at Austin, Austin, TX 78713-8510, USA |
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Abstract: | For solving nonsymmetric linear systems, the well-known GMRES method is considered to be a stable method; however, the work
per iteration increases as the number of iterations increases. We consider two new iterative methods GGMRES and MGMRES, which
are a generalization and a modification of the GMRES method, respectively. Instead of using a minimization condition as in
the derivation of GGMRES, we use a Galerkin condition to derive the MGMRES method. We also introduce another new iterative
method, LAN/MGMRES, which is designed to combine the reliability of GMRES with the reduced work of a Lanczos-type method.
A computer program has been written based on the use of the LAN/MGMRES algorithm for solving nonsymmetric linear systems arising
from certain elliptic problems. Numerical tests are presented comparing this algorithm with some other commonly used iterative
algorithms. These preliminary tests of the LAN/MGMRES algorithm show that it is comparable in terms of both the approximate
number of iterations and the overall convergence behavior.
This revised version was published online in June 2006 with corrections to the Cover Date. |
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Keywords: | iterative methods Generalized Minimum Residual (GMRES) method minimization condition Galerkin condition Lanczos-type methods 65F10 65H10 65N20 65N30 |
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