Mes: octubre 2026

A Charlier-Polynomial Approach to the Matrix Exponential

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.

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

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.

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.

New Generating Functions for Chebyshev, Gegenbauer, and Jacobi Matrix Polynomials

Generating functions are a fundamental tool in the analysis of orthogonal polynomials, but their extension to the matrix setting has remained fragmented across different polynomial families. In a new article published in Mediterranean Journal of Mathematics, Emilio Defez and Michael M. Tung present a unified approach to deriving generating functions for the matrix analogues of the Chebyshev, Gegenbauer, and Jacobi polynomials.

The work does not stop at the matrix level. Alongside the matrix generating functions, the authors establish several explicit identities for the corresponding scalar polynomial families. These results provide new analytical tools for a class of polynomials that play a growing role in the numerical solution of matrix differential equations, the approximation of inverse Laplace transforms, and the high-accuracy computation of matrix functions such as the exponential.

By bringing three classical families under a single derivation framework, the article offers a compact reference for researchers working with matrix orthogonal polynomials and opens the door to analogous results for other polynomial sequences.

Publication Details

  • Title: New Generating Functions for Chebyshev, Gegenbauer, and Jacobi Matrix Polynomials
  • Authors: E. Defez, M. M. Tung
  • Journal: Mediterranean Journal of Mathematics, Vol. 23, 2026, Article 118
  • DOI: 10.1007/s00009-026-03119-x

Main Contributions

  • A unified derivation of generating functions for Chebyshev, Gegenbauer, and Jacobi matrix polynomials.
  • New explicit identities for the corresponding scalar polynomial families.
  • A consistent analytical framework that connects three classical polynomial sequences in the matrix setting.

Why This Matters

Matrix orthogonal polynomials are increasingly used in numerical linear algebra and in the approximation of matrix functions. Having a unified source of generating functions simplifies the derivation of new series expansions and can reduce the analytical effort required when adapting classical polynomial techniques to the matrix case.