Parallel Matrix Function Evaluation via Initial value ODE modelling

The purpose of this article is to propose ODE based approaches for the numerical evaluation of matrix functions $f(A)$, a question of major interest in the numerical linear algebra. To this end, we model $f(A)$ as the solution at a finite time $T$ of a time dependent equation. We use parallel algorithms, such as the parareal method, on the time interval $[0, T]$ in order to solve the evolution equation obtained. When $f(A)$ is reached as a stable steady state, it can be computed by combining parareal algorithms and optimal control techniques. Numerical illustrations are given.