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arXiv · 2203.03930

Integral representations for higher-order Fr\'echet derivatives of matrix functions: Quadrature algorithms and new results on the level-2 condition number

Marcel Schweitzer
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Abstract

We propose an integral representation for the higher-order Fr\'echet derivative of analytic matrix functions $f(A)$ which unifies known results for the first-order Fr\'echet derivative of general analytic matrix functions and for higher-order Fr\'echet derivatives of $A^{-1}$. We highlight two applications of this integral representation: On the one hand, it allows to find the exact value of the level-2 condition number (i.e., the condition number of the condition number) of $f(A)$ for a large class of functions $f$ when $A$ is Hermitian. On the other hand, it also allows to use numerical quadrature methods to approximate higher-order Fr\'echet derivatives. We demonstrate that in certain situations -- in particular when the derivative order $k$ is moderate and the direction terms in the derivative have low-rank structure -- the resulting algorithm can outperform established methods from the literature by a large margin.

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Marcel Schweitzer. 2022-03-08. Integral representations for higher-order Fr\'echet derivatives of matrix functions: Quadrature algorithms and new results on the level-2 condition number. https://arxiv.org/abs/2203.03930

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