Volume 65, pp. 110-122, 2026.

A novel Krylov subspace method for approximating Fréchet derivatives of large-scale matrix functions

Daniel Kressner and Peter Oehme

Abstract

We present a novel Krylov subspace method for approximating $L_f(A, E) \mathbf{b}$, the matrix-vector product of the Fréchet derivative $L_f(A, E)$ of a large-scale matrix function $f(A)$ in direction $E$, a task that arises naturally in the sensitivity analysis of quantities involving matrix functions such as centrality measures for networks. It also arises in the context of gradient-based methods for optimization problems that feature matrix functions, e.g., when fitting an evolution equation to an observed solution trajectory. In principle, the well-known identity \[ f\left( \begin{bmatrix} A & E \\ 0 & A \end{bmatrix} \right) \begin{bmatrix} 0 \\ \mathbf{b} \end{bmatrix} = \begin{bmatrix} L_f(A, E) \mathbf{b} \\ f(A) \mathbf{b} \end{bmatrix} \] allows one to directly apply any standard Krylov subspace method, such as the Arnoldi algorithm, to address this task. However, this comes with the major disadvantage that the involved block-triangular matrix has unfavorable spectral properties, which impede the convergence analysis and, to a certain extent, also the observed convergence. To avoid these difficulties, we propose a novel modification of the Arnoldi algorithm that aims at better preserving the block-triangular structure. In turn, this allows one to bound the convergence of the modified method by the best polynomial approximation of the derivative $f^\prime$ on the numerical range of $A$. Several numerical experiments illustrate our findings.

Full Text (PDF) [390 KB], BibTeX , DOI: 10.1553/etna_vol65s110

Key words

matrix function, Fréchet derivative, Krylov subspace method

AMS subject classifications

15A16, 65F50, 65F60