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arXiv · 2008.12926

Computing low-rank approximations of the Fréchet derivative of a matrix function using Krylov subspace methods

Peter Kandolf·Antti Koskela·Samuel D. Relton·Marcel Schweitzer
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Abstract

The Fréchet derivative $L_f(A,E)$ of the matrix function $f(A)$ plays an important role in many different applications, including condition number estimation and network analysis. We present several different Krylov subspace methods for computing low-rank approximations of $L_f(A,E)$ when the direction term $E$ is of rank one (which can easily be extended to general low-rank). We analyze the convergence of the resulting method for the important special case that $A$ is Hermitian and $f$ is either the exponential, the logarithm or a Stieltjes function. In a number of numerical tests, both including matrices from benchmark collections and from real-world applications, we demonstrate and compare the accuracy and efficiency of the proposed methods.

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Peter Kandolf, Antti Koskela, Samuel D. Relton, Marcel Schweitzer. 2020-08-29. Computing low-rank approximations of the Fréchet derivative of a matrix function using Krylov subspace methods. https://arxiv.org/abs/2008.12926

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