Abstract
In this paper we propose a stable variant of Simpler GMRES. It is based on the adaptive choice of the Krylov subspace basis at a given iteration step using the intermediate residual norm decrease criterion. The new direction vector is chosen as in the original implementation of Simpler GMRES or it is equal to the normalized residual vector as in the GCR method. We show that such an adaptive strategy leads to a well-conditioned basis of the Krylov subspace and we support our theoretical results with illustrative numerical examples.
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The work of the first author was supported by the grant No. 201/09/P464 of the Grant Agency of the Czech Republic.
The work of the second author was supported by the project IAA100300802 of the Grant Agency of the Academy of Sciences of the Czech Republic and by the Institutional Research Plan AV0Z10300504 “Computer Science for the Information Society: Models, Algorithms, Applications”.
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Jiránek, P., Rozložník, M. Adaptive version of Simpler GMRES. Numer Algor 53, 93–112 (2010). https://doi.org/10.1007/s11075-009-9311-2
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DOI: https://doi.org/10.1007/s11075-009-9311-2
Keywords
- Nonsymmetric linear systems
- Krylov subspace methods
- Minimum residual methods
- Numerical stability
- Rounding errors