Extend Fréchet derivative–based condition number estimation to Krylov subspace methods for f(A)b

Determine how to extend algorithms that simultaneously compute a matrix function f(A) and its Fréchet derivative L_f(A,·) for condition number estimation to Krylov subspace algorithms for computing the action f(A)b on one or more vectors.

Background

Condition number estimation for computing a matrix function f(A) often relies on access to the Fréchet derivative L_f(A,·). For certain functions, such as the exponential and logarithm, specialized algorithms can compute f(A) and L_f(A,·) simultaneously to deliver reliable condition estimates, albeit at significant computational overhead.

In contrast, when computing the action f(A)b via Krylov subspace algorithms, there is no straightforward analogue of these simultaneous computation frameworks, because computing f(A)b and computing L_f(A,·) share fewer computational components than in the f(A) case. The authors explicitly note that it is currently unclear how to adapt these approaches to Krylov methods for f(A)b, despite some recent progress on low-rank approximations of the Fréchet derivative.