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

TL;DR

This paper introduces an integral representation for higher-order Fréchet derivatives of matrix functions, enabling exact condition number computation and efficient numerical approximation, especially for Hermitian matrices and low-rank derivatives.

Contribution

It provides a unified integral framework for higher-order derivatives, facilitating exact condition number calculation and improved quadrature-based approximation methods.

Findings

Exact level-2 condition number for Hermitian matrices derived

Quadrature algorithms outperform existing methods in certain cases

Efficient approximation for derivatives with low-rank structure

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.