Abstract
The Frechet derivative $L_f$ of a matrix function $f \, {:} \mathbb{C}^{n\times n} \mapsto \mathbb{C}^{n\times n}$ 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 Frechet derivatives. We derive sufficient conditions for the $k$th Frechet 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 Frechet 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 numbe...
Highlights
Matrix functions f : Cn×n → Cn×n such as the matrix exponential, the matrix logarithm, and matrix powers At for t ∈ R are being used within a growing number of applications including model reduction [5], numerical solution of fractional partial differential equations [9], analysis of complex networks [13], and computer animation [35]
The Frechet derivative is required, with recent examples including computation of correlated choice probabilities [1], registration of MRI images [6], Markov models applied to cancer data [14], matrix geometric mean computation [24], and model reduction [33], [34]
One purpose of our work is to investigate the connection between the level-1 and level-2 condition numbers of general matrix functions
Summary
Matrix functions f : Cn×n → Cn×n such as the matrix exponential, the matrix logarithm, and matrix powers At for t ∈ R are being used within a growing number of applications including model reduction [5], numerical solution of fractional partial differential equations [9], analysis of complex networks [13], and computer animation [35]. By the same argument as in the proof of Lemma 4.1 this matrix is linear in each Ei and continuing this process we eventually arrive at Kf(k)(A) ∈ Cn2k×n2 , which we call the Kronecker form of the kth Frechet derivative. We begin this section by deriving a bound for the level-2 absolute condition number for general functions f in the Frobenius norm.
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