Higher order fréchet derivatives of matrix functions and the level-2 condition number

Abstract

The Fréchet derivative Lf of a matrix function f: ℂn×n → ℂn×n controls the sensitivity of the function to small perturbations in the matrix. While much is known about the properties of Lf and how to compute it, little attention has been given to higher order Fréchet derivatives. We derive sufficient conditions for the kth Fréchet derivative to exist and be continuous in its arguments and we develop algorithms for computing the kth derivative and its Kronecker form. We analyze the level-2 absolute condition number of a matrix function ("the condition number of the condition number") and bound it in terms of the second Fréchet derivative. For normal matrices and the exponential we show that in the 2-norm the level-1 and level-2 absolute condition numbers are equal and that the relative condition numbers are within a small constant factor of each other. We also obtain an exact relationship between the level-1 and level-2 absolute condition numbers for the matrix inverse and arbitrary nonsingular matrices, as well as a weaker connection for Hermitian matrices for a class of functions that includes the logarithm and square root. Finally, the relation between the level-1 and level-2 condition numbers is investigated more generally through numerical experiments.