Higher Order Frechet Derivatives of Matrix Functions and the Level-2 Condition Number

The Fr\'echet derivative $L_f$ of a matrix function $f \colon \mathbb{C}^{n\times n} \mapsto \mathbb{C}^{n\times n}$ controls the sensitivity of the function to small perturbations in the matrix. While much is known about the properties of $L_f$ and how to compute it, little attention has been given to higher order Fr\'echet derivatives. We derive sufficient conditions for the $k$th Fr\'echet derivative to exist and be continuous in its arguments and we develop algorithms for computing the $k$th 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 Fr\'echet 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