Abstract
Important matrix-valued functions f (A) are, e.g., the inverse A −1, the square root \({\sqrt{A}}\) and the sign function. Their evaluation for large matrices arising from pdes is not an easy task and needs techniques exploiting appropriate structures of the matrices A and f (A) (often f (A) possesses this structure only approximately). However, intermediate matrices arising during the evaluation may lose the structure of the initial matrix. This would make the computations inefficient and even infeasible. However, the main result of this paper is that an iterative fixed-point like process for the evaluation of f (A) can be transformed, under certain general assumptions, into another process which preserves the convergence rate and benefits from the underlying structure. It is shown how this result applies to matrices in a tensor format with a bounded tensor rank and to the structure of the hierarchical matrix technique. We demonstrate our results by verifying all requirements in the case of the iterative computation of A −1 and \({\sqrt{A}}\).
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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
This work was performed during the stay of the third author at the Max-Planck-Institute for Mathematics in the Sciences (Leipzig) and also supported by the Russian Fund of Basic Research (grants 05-01-00721, 04-07-90336) and a Priority Research Grant of the Department of Mathematical Sciences of the Russian Academy of Sciences.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Hackbusch, W., Khoromskij, B.N. & Tyrtyshnikov, E.E. Approximate iterations for structured matrices. Numer. Math. 109, 365–383 (2008). https://doi.org/10.1007/s00211-008-0143-0
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DOI: https://doi.org/10.1007/s00211-008-0143-0