Abstract
Rational Krylov subspaces have become a fundamental ingredient in numerical linear algebra methods associated with reduction strategies. Nonetheless, many structural properties of the reduced matrices in these subspaces are not fully understood. We advance in this analysis by deriving bounds on the entries of rational Krylov reduced matrices and of their functions, that ensure an a-priori decay of their entries as we move away from the main diagonal. As opposed to other decay pattern results in the literature, these properties hold in spite of the lack of any banded structure in the considered matrices. Numerical experiments illustrate the quality of our results.
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Notes
Note that Dzhrbashyan is often also transliterated as Djrbashian. We chose to use Dzhrbashyan in accordance with [44].
The index \(j=3,4,\dots \) parametrizes the family of sequences of FD rational functions \(\{M_t^{(j)}\}_{t=0,\,\dots }\). All sequences in such family share the initial (finite) shifts \(\phi (\sigma _{s+1})\) up to \(j-3\). This means that given the indexes \(i<j\), it holds that \(M_t^{(i)} = M_t^{(j)}\) for \(t=0,\dots ,i-3\).
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Acknowledgements
The authors would like to thank the referees for the helpful comments and improvements suggested. The authors are members of Indam-GNCS, which support is gratefully acknowledged. This work has also been supported by Charles University Research Program No. UNCE/SCI/023.
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Pozza, S., Simoncini, V. Functions of rational Krylov space matrices and their decay properties. Numer. Math. 148, 99–126 (2021). https://doi.org/10.1007/s00211-021-01198-4
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DOI: https://doi.org/10.1007/s00211-021-01198-4