### Speaker

### Description

Evaluating the action of a matrix function on a vector, that is $x=f(\mathcal M)v$, is an ubiquitous task in applications. When the matrix $\mathcal M$ is large, subspace projection method, such as the rational Krylov method, are usually employed.

In this work, we provide a quasi-optimal pole choice for rational Krylov methods applied to this task when $f(z)$ is either Cauchy-Stieltjes or Laplace-Stieltjes (or, which is equivalent, completely monotonic) for a positive definite matrix $\mathcal M$.

Then, we consider the case when the argument $\mathcal M$ has the Kronecker structure $\mathcal M=I \otimes A - B^T \otimes I$, and is applied to a vector obtained vectorizing a low-rank matrix. This finds application, for instance, in solving fractional diffusion equation on rectangular domains.

We introduce an error analysis for the numerical approximation of $x$. Pole choices and explicit convergence bounds are given also in this case.