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ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis, pp. 381-404, 2024/07/01
While preconditioning is a long-standing concept to accelerate iterative methods for linear systems, generalizations to matrix functions are still in their infancy. We go a further step in this direction, introducing polynomial preconditioning for Krylov subspace methods that approximate the action of the matrix square root and inverse square root on a vector. Preconditioning reduces the subspace size and therefore avoids the storage problem together with—for non-Hermitian matrices—the increased computational cost per iteration that arises in the unpreconditioned case. Polynomial preconditioning is an attractive alternative to current restarting or sketching approaches since it is simpler and computationally more efficient. We demonstrate this for several numerical examples.
Keywords: polynomial preconditioning, matrix square root, inverse square root, Krylov space, matrix functions