Kandolf, P and Relton, SD orcid.org/0000-0003-0634-4587 (2017) A Block Krylov Method to Compute the Action of the Fréchet Derivative of a Matrix Function on a Vector with Applications to Condition Number Estimation. SIAM Journal on Scientific Computing, 39 (4). A1416-A1434. ISSN 1064-8275
Abstract
We design a block Krylov method to compute the action of the Fréchet derivative of a matrix function on a vector using only matrix-vector products, i.e., the derivative of $f(A)b$ when $A$ is subject to a perturbation in the direction $E$. The algorithm we derive is especially effective when the direction matrix $E$ in the derivative is of low rank, while there are no such restrictions on $A$. Our results and experiments are focused mainly on Fréchet derivatives with rank 1 direction matrices. Our analysis applies to all functions with a power series expansion convergent on a subdomain of the complex plane which, in particular, includes the matrix exponential. We perform an a priori error analysis of our algorithm to obtain rigorous stopping criteria. Furthermore, we show how our algorithm can be used to estimate the 2-norm condition number of $f(A)b$ efficiently. Our numerical experiments show that our new algorithm for computing the action of a Fréchet derivative typically requires a small number of iterations to converge and (particularly for single and half precision accuracy) is significantly faster than alternative algorithms. When applied to condition number estimation, our experiments show that the resulting algorithm can detect ill-conditioned problems that are undetected by competing algorithms.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | (c) 2017 SIAM. Published by SIAM under the terms of the Creative Commons 4.0 license. (https://creativecommons.org/licenses/by/4.0/) |
Keywords: | matrix function; matrix exponential; Fr´echet derivative; condition number; Krylov subspace; block Krylov subspace |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Medicine and Health (Leeds) > School of Medicine (Leeds) > Leeds Institute of Health Sciences (Leeds) > Centre for Health Services Research (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 14 Dec 2018 10:19 |
Last Modified: | 14 Dec 2018 10:19 |
Status: | Published |
Publisher: | Society for Industrial and Applied Mathematics |
Identification Number: | 10.1137/16M1077969 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:139977 |