Abstract
Rational Krylov methods are a powerful alternative for computing the product of a function of a large matrix times a given vector. However, the creation of the underlying rational subspaces requires solving sequences of large linear systems, a delicate task that can require intensive computational resources and should be monitored to avoid the creation of subspace different to those required whenever, e.g., the underlying matrices are ill-conditioned. We propose the use of robust preconditioned iterative techniques to speedup the underlying process. We also discuss briefly how the inexact solution of these linear systems can affect the computed subspace. A preliminary test approximating a fractional power of the Laplacian matrix is included.
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Acknowledgements
We wish to thank two anonymous referees for their comments which have improved the readability of the paper.
This project was partially supported by the Tor Vergata University project “MISSION: SUSTAINABILITY” “NUMnoSIDS”, CUP E86C18000530005, and by the INDAM-GNCS 2019 project “Tecniche innovative e parallele per sistemi lineari e non lineari di grandi dimensioni, funzioni ed equazioni matriciali ed applicazioni”. D. Bertaccini gratefully acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.
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Bertaccini, D., Durastante, F. (2021). Computing Function of Large Matrices by a Preconditioned Rational Krylov Method. In: Vermolen, F.J., Vuik, C. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2019. Lecture Notes in Computational Science and Engineering, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-030-55874-1_33
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DOI: https://doi.org/10.1007/978-3-030-55874-1_33
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