The matrix exponential is one of the most frequently evaluated matrix functions, and small improvements in its numerical computation can have a significant impact on large-scale scientific workloads. In a paper published in the International Journal of Computer Mathematics, J. M. Alonso, E. Defez, J. Ibáñez, J. Sastre, and A. Ibáñez introduce a new method based on Charlier polynomials in combination with the scaling-and-squaring technique.

The approximating matrix polynomial is evaluated through an efficient polynomial evaluation scheme that keeps the computational cost competitive with existing state-of-the-art codes. The authors perform a comprehensive numerical comparison across different matrix types. In their testbed, the variant expm_charlier_1 consistently appears among the most accurate and stable methods, while other variants and a reference code (expm_new) show higher error levels in a substantial fraction of the cases.

A distinctive aspect of this work is the use of a discrete orthogonal polynomial family in the matrix-exponential context. To our knowledge, discrete orthogonal polynomials, like the case of Charlier polynomials, have not been previously employed to address the matrix exponential computation.The article thus contributes a new polynomial family to the toolbox of matrix-exponential algorithms and provides empirical evidence of its numerical behaviour, without relying on asymptotic arguments alone.

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Publication Details

• Title: Computing Matrix Exponentials via Charlier Method
• Authors: J. M. Alonso, E. Defez, J. Ibáñez, J. Sastre, A. Ibáñez
• Journal: International Journal of Computer Mathematics, 1-20, 2026
• DOI: 10.1080/00207160.2026.2697770

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Main Contributions

• A new algorithm for the matrix exponential based on Charlier matrix polynomialsand scaling-and-squaring.
• Efficient evaluation of the approximating matrix polynomial via the Paterson–Stockmeyer method.
• A numerical study showing that the expm_charlier_1 variant achieves competitive accuracy and stability across different matrix classes.
• The first use, to the authors’ knowledge, of discrete orthogonal polynomials (Charlier) for matrix-exponential computation.

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Why This Matters

Matrix exponentials appear in the solution of differential equations, in control theory, and in many scientific computing pipelines. Extending the set of reliable polynomial-based methods gives practitioners more options when selecting an algorithm for a specific matrix structure or accuracy requirement.