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

The Frechet derivative Lf(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 Lf(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 methods both in the Hermitian and non-Hermitian case. 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.