Computing the action of a matrix exponential on an interval via the $\star$-product approach
This work addresses the need for efficient computation of matrix exponential actions over intervals, which is relevant for applications like differential equations, but the improvements are incremental over existing methods.
The paper introduces a new method to compute the action of a matrix exponential on a vector for all times within a bounded interval, using an orthogonal polynomial series derived from a \(\star\)-algebra representation. Numerical experiments show accuracy and efficiency compared to existing techniques.
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.