This article addresses the application of Charlier polynomials and their parameter dependent variants for the approximation of the matrix exponential function. While standard rational methods like Padé require matrix inversions that can create computational bottlenecks in large-scale or GPU-accelerated applications, polynomial approaches provide inversion-free alternatives based entirely on matrix multiplications. However, the efficiency of these polynomial methods relies heavily on balancing the polynomial degree and the scaling parameter. To our knowledge, discrete orthogonal polynomials have not been previously employed to address the matrix exponential computation. In this work, Charlier polynomials are used in conjunction with a novel procedure for determining both parameters, leading to a new numerical method. This method has been implemented and compared with state-of-the-art algorithms. Based on an extensive and heterogeneous battery of tests, the proposed method is equal to or slightly more accurate than existing techniques, with competitive computational cost and execution time.
Alonso et al. (Wed,) studied this question.