Computing low-rank approximations of the Fr\'echet derivative of a matrix function using Krylov subspace methods

TL;DR

This paper introduces Krylov subspace methods for efficiently computing low-rank approximations of the Fréchet derivative of matrix functions, with convergence analysis and numerical validation on Hermitian matrices.

Contribution

It develops new Krylov subspace algorithms for low-rank Fréchet derivative approximation, including convergence analysis for Hermitian matrices and specific functions.

Findings

Methods are accurate and efficient in numerical tests.

Convergence is established for Hermitian matrices and specific functions.

Algorithms outperform existing approaches in benchmark and real-world cases.

Abstract

The Fr\'echet 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.