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

2022/03/08 by Schweitzer, Marcel
#15A16 #65D30 #65F35 #65F60 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2203.03930

Abstract

We propose an integral representation for the higher-order Fréchet derivative of analytic matrix functions f(A) which unifies known results for the first-order Fréchet derivative of general analytic matrix functions and for higher-order Fréchet 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 Fréchet 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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