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## 2013.58: Higher Order Frechet Derivatives of Matrix Functions and the Level-2 Condition Number

2013.58: Nicholas J. Higham and Samuel D. Relton (2014) Higher Order Frechet Derivatives of Matrix Functions and the Level-2 Condition Number. SIAM Journal on Matrix Analysis and Applications, 35 (3). pp. 1019-1037. ISSN 1095-7162

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## Abstract

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.

Item Type: Article matrix function, Frechet derivative, Gateaux derivative, higher order derivative, matrix exponential, matrix logarithm, matrix square root, matrix inverse, matrix calculus, partial derivative, Kronecker form, level-2 condition number, expm, logm, sqrtm, MATLAB MSC 2000 > 15 Linear and multilinear algebra; matrix theoryMSC 2000 > 65 Numerical analysis 2013.58 Nick Higham 16 April 2015