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

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.