# METT VIII - 8th Workshop on Matrix Equations and Tensor Techniques

Nov 6 – 8, 2019
MPI Magdeburg
Europe/Berlin timezone

## Rational Krylov for Stieltjes matrix functions: convergence and pole selection

Nov 8, 2019, 11:10 AM
25m
Prigogine (MPI Magdeburg)

### Prigogine

#### MPI Magdeburg

MPI for Dynamics of Complex Technical Systems Sandtorstr. 1 39106 Magdeburg
Talk Talks

### Speaker

Stefano Massei (EPF Lausanne)

### 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.

### Primary authors

Stefano Massei (EPF Lausanne) Leonardo Robol (University of Pisa)

### Presentation materials

There are no materials yet.