Computing the action of a matrix exponential on an interval via the $\star$-product approach

We present a new method for computing the action of the matrix exponential on a vector, $e^{At}v$. The proposed approach efficiently evaluates the solution for all $t$ within a prescribed bounded interval by expanding it into an orthogonal polynomial series. This method is derived from a new representation of the matrix exponential in the so-called $\star$-algebra, an algebra of bivariate distributions. The resulting formulation leads to a linear system equivalent to a matrix equation of Stein type, which can be solved by either direct or Krylov subspace methods. Numerical experiments demonstrate the accuracy and efficiency of the proposed approach in comparison to state-of-the-art techniques.