Parallel matrix function evaluation via initial value ODE modeling

The purpose of this article is to propose ODE based approaches for the numerical evaluation of matrix functions f(A)f(A), a question of major interest in the numerical linear algebra. For that, we model f(A)f(A) as the solution at a finite time TT of a time dependent equation. We use parallel algorithms, such as the parareal method, on the time interval [0,T][0,T] in order to solve the obtained evolution equation. When f(A)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.