Estimating the Condition Number of the Fréchet Derivative of a Matrix Function
Nicholas John Higham, Samuel David Relton · SIAM Journal on Scientific Computing · 2014
The Fréchet derivative $L_f$ of a matrix function $f : \mathbb{C}^{n\times n} \to \mathbb{C}^{n\times n}$ is used in a variety of applications and several algorithms are available for computing it. We define a condition number for the Fréchet derivative and derive upper and lower bounds for it that differ by at most a factor $2$. For a wide class of functions we derive an algorithm for estimating the 1-norm condition number that requires $O(n^3)$ flops given $O(n^3)$ flops algorithms for evaluating $f$ and $L_f$; in practice it produces estimates correct to within a factor $6n$. Numerical experiments show the new algorithm to be much more reliable than a previous heuristic estimate of conditioning.