Estimating the Condition Number of the Frechet Derivative of a Matrix Function

Higham, Nicholas J. and Relton, Samuel D. (2013) Estimating the Condition Number of the Frechet Derivative of a Matrix Function. [MIMS Preprint]

The Fr\'{e}chet derivative $L_f$ of a matrix function $f \colon \mathbb{C}^{n\times n} \to \mathbb{C}^{n\times n}$ is used in a variety of applications and several algorithms are available for computing it. We define a condition number for the Fr\'{e}chet 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.
Item Type: MIMS Preprint matrix function, condition number, Frechet derivative, Kronecker form, matrix exponential, matrix logarithm, matrix powers, matrix $p$th root, MATLAB, expm, logm, sqrtm MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Nick Higham 20 Dec 2013 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/2086