Higham, Nicholas J. and Relton, Samuel D. (2014) Higher Order Frechet Derivatives of Matrix Functions and the Level2 Condition Number. SIAM Journal on Matrix Analysis and Applications, 35 (3). pp. 10191037. ISSN 10957162
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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 level2 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 2norm the level1 and level2 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 level1 and level2 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 level1 and level2 condition numbers is investigated more generally through numerical experiments.
Item Type:  Article 

Uncontrolled Keywords:  matrix function, Frechet derivative, Gateaux derivative, higher order derivative, matrix exponential, matrix logarithm, matrix square root, matrix inverse, matrix calculus, partial derivative, Kronecker form, level2 condition number, expm, logm, sqrtm, MATLAB 
Subjects:  MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis 
Depositing User:  Nick Higham 
Date Deposited:  16 Apr 2015 
Last Modified:  20 Oct 2017 14:13 
URI:  https://eprints.maths.manchester.ac.uk/id/eprint/2160 
Available Versions of this Item

Higher Order Frechet Derivatives of Matrix Functions and the Level2 Condition Number. (deposited 13 Nov 2013)
 Higher Order Frechet Derivatives of Matrix Functions and the Level2 Condition Number. (deposited 16 Apr 2015) [Currently Displayed]
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