  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.
