Higher Order Fréchet Derivatives of Matrix Functions and the Level-2 Condition Number

Abstract

Abstract. The Fréchet derivative L f of a matrix function f : C n×n → C n×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 Fréchet derivatives. We derive sufficient conditions for the kth Fréchet derivative to exist and be continuous in its arguments and we develop algorithms for computing the kth 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 Fréchet 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.Key words. matrix function, Fréchet derivative, Gâteaux 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 AMS subject classifications. 65F30, 65F60DOI. 10.1137/130945259 Introduction. Matrix functions f : Cn×n → C n×n such as the matrix exponential, the matrix logarithm, and matrix powers A t for t ∈ R are being used within a growing number of applications including model reduction [5] The Fréchet derivative of f at A ∈ C n×n is the unique function L f (A, · ) that is linear in its second argument and for all E ∈ C n×n satisfies

Krylov Approximation of Linear ODEs with Polynomial Parameterization

“…From one viewpoint, our result is related to recent work on computations and theory for Fréchet derivatives of matrix functions, e.g., [11,20,19]. As an illustration of a relation, consider the special case N = 1.…”

Testing Matrix Function Algorithms Using Identities

“…The overall cost of computing the backward error estimate is O(n 6 ) flops if we explicitly form the Kronecker matrices and then find the minimum 2-norm solution by QR factorization. This clearly limits n. The Kronecker matrices are highly structured [Higham and Relton 2013a], [Higham and Relton 2013b], but it is not clear how to take advantage of this structure in using a direct method to solve the problem.…”

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

“…To investigate the condition number of the Fréchet derivative we will need higher order Fréchet derivatives of matrix functions, which were recently investigated by Higham and Relton [19]. We now summarize the key results that we will need from that work.…”

“…Higham and Relton show that a sufficient condition for the second Fréchet derivative L (2) f (A, ·, ·) to exist is that f is 4p − 1 times continuously differentiable on an open set containing the eigenvalues of A [19,Thm. 3.5], where p is the size of the largest Jordan block of A.…”

“…Higham and Relton [19,Thm. 3.5] show that the second Fréchet derivative can be calculated in a similar way, as the top right-hand block of a 4n × 4n matrix:…”