Categoría: Matrix Exponential

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.

Euler polynomials for the matrix exponential approximation

José M. Alonso, Javier Ibáñez, Emilio Defez, Pedro Alonso-Jordá. Journal of Computational and Applied Mathematics, vol. 425, 115074, 2023. https://doi.org/10.1016/j.cam.2023.115074.

In this work, a new method to compute the matrix exponential function by using an approximation based on Euler polynomials is proposed. These polynomials are used in combination with the scaling and squaring technique, considering an absolute forward-type theoretical error. Its numerical and computational properties have been evaluated and compared with the most current and competitive codes dedicated to the computation of the matrix exponential. Under a heterogeneous test battery and a set of exhaustive experiments, it has been demonstrated that the new method offers performance in terms of accuracy and stability which is as good as or even better than those of the considered methods, with an intermediate computational cost among all of them. All of the above makes this a very competitive alternative that should be considered in the growing list of available numerical methods and implementations dedicated to the approximation of the matrix exponential.

Boosting the computation of the matrix exponential

Boosting the computation of the matrix exponential, J. Sastre, J. Ibáñez, E. Defez, Appl. Math. Comput. in press, 2018, doi:10.1016/j.amc.2018.08.017, Preprint, Matlab code expmpol.m.

This paper presents new Taylor algorithms for the computation of the matrix exponential based on recent new matrix polynomial evaluation methods. Those methods are more efficient than the well known Paterson–Stockmeyer method. The cost of the proposed algorithms is reduced with respect to previous algorithms based on Taylor approximations. Tests have been performed to compare the MATLAB implementations of the new algorithms to a state-of-the-art Padé algorithm for the computation of the matrix exponential, providing higher accuracy and cost performances.

First article with an application of the new matrix polynomial evaluation methods from J. Sastre, Efficient evaluation of matrix polynomials, Linear Algebra Appl. 539, (2018) 229-250. With the new matrix polynomial evaluation methods, Taylor approximation methods are more efficient than Padé approximant based methods.

A new efficient and accurate spline algorithm for the matrix exponential computation

A new efficient and accurate spline algorithm for the matrix exponential computation, Emilio Defez, Javier Ibáñez, Jorge Sastre, Jesús Peinado, Pedro Alonso. Journal of Computational and Applied Mathematics, Volume 337, pp. 354-365. August 2018. Preprint, Matlab code expmspl.m.

In this work an accurate and efficient method based on matrix splines for computing matrix exponential is given. An algorithm and a MATLAB implementation have been developed and compared with the state-of-the-art algorithms for computing the matrix exponential. We also developed a parallel implementation for large scale problems. This implementation allowed us to get a much better performance when working with this kind of problems.

Efficient evaluation of matrix polynomials

Efficient evaluation of matrix polynomials, Jorge Sastre, Linear Algebra Applications, Vol. 539, pp. 229-250, Feb. 2018 (early version submitted on Feb. 18, 2016  in AMC-S-16-00951.pdf), submitted Oct. 2016, available online 2017, Preprint.

This paper presents a new family of methods for evaluating matrix polynomials more efficiently than the state-of-the-art Paterson–Stockmeyer method. Examples of the application of the methods to the Taylor polynomial approximation of matrix functions like the matrix exponential and matrix cosine are given. Their efficiency is compared with that of the best existing evaluation schemes for general polynomial and rational approximations, and also with a recent method based on mixed rational and polynomial approximants.

For many years, the Paterson–Stockmeyer method has been considered the most efficient general method for the evaluation of matrix polynomials. In this paper we show that this statement is no longer true. Moreover, for many years rational approximations have been considered more efficient than polynomial approximations, although recently it has been shown that often this is not the case in the computation of the matrix exponential and matrix cosine.

In this paper we show that in fact polynomial approximations provide a higher order of approximation than the state-of-the-art computational methods for rational approximations for the same cost in terms of matrix products. For an early unpublished version of this work submitted on Feb. 18, 2016 to Appl. Math. Comput. see AMC-S-16-00951.pdf.