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

Abstract

AbstractThe Fréchet derivative of the matrix function 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 when the direction term 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.