# Rational Krylov methods for functions of matrices with applications to fractional partial differential equations

## TL;DR

This paper introduces a new pole selection strategy for rational Krylov methods to efficiently approximate functions of positive definite matrices, especially fractional powers, with promising numerical results for fractional PDEs.

## Contribution

It proposes a novel pole choice for rational Krylov methods tailored for fractional matrix functions, improving reliability and efficiency in solving fractional PDEs.

## Key findings

- Numerical experiments confirm the effectiveness of the new pole selection.
- The approach is promising for solving fractional partial differential equations.
- The method improves approximation accuracy for fractional matrix functions.

## Abstract

In this paper, we propose a new choice of poles to define reliable rational Krylov methods. These methods are used for approximating function of positive definite matrices. In particular, the fractional power and the fractional resolvent are considered because of their importance in the numerical solution of fractional partial differential equations. The results of the numerical experiments we have carried out on some fractional models confirm that the proposed approach is promising.