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We propose an integral representation for the higher-order Fr\'echetderivative of analytic matrix functions $f(A)$ which unifies known results forthe first-order Fr\'echet derivative of general analytic matrix functions andfor higher-order Fr\'echet derivatives of $A^{-1}$. We highlight twoapplications of this integral representation: On the one hand, it allows tofind the exact value of the level-2 condition number (i.e., the conditionnumber 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 numericalquadrature methods to approximate higher-order Fr\'echet derivatives. Wedemonstrate that in certain situations -- in particular when the derivativeorder $k$ is moderate and the direction terms in the derivative have low-rankstructure -- the resulting algorithm can outperform established methods fromthe literature by a large margin.

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