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

08/29/2020

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by   Peter Kandolf, et al.

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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.