Categoría: Polynomial Families

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.

Beyond Paterson–Stockmeyer: Advancing Matrix Polynomial Computation

For over fifty years, the Paterson–Stockmeyer method has been considered the benchmark for efficient matrix polynomial evaluation. In our recent open access article, we provide a summary of recent advances in this area and present a constructive scheme that evaluates a degree‑20 matrix polynomial using only 5 matrix multiplications—two fewer than Paterson–Stockmeyer.

We also show how the coefficients of this scheme can be derived from the solutions of a single equation involving one coefficient, and we include the full process in our supplementary materials.

Publication Details

  • Title: Beyond Paterson–Stockmeyer: Advancing Matrix Polynomial Computation
  • Authors: J. Sastre, J. Ibáñez, J. M. Alonso, E. Defez
  • Journal: WSEAS Transactions on Mathematics, Vol. 24, pp. 684–693, 2025
  • Conference: 5th Int. Conf. on Applied Mathematics, Computational Science and Systems Engineering (AMCSE), Paris, France, April 14–16, 2025
  • Open Access: https://doi.org/10.37394/23206.2025.24.68
  • Supplementary Material:

Main Contributions

  • Survey of recent advances in matrix polynomial evaluation.
  • Constructive result: A method to compute a degree‑20 matrix polynomial with just 5 matrix multiplications, improving efficiency over Paterson–Stockmeyer (needing 7 matrix products).
  • Coefficient derivation: All coefficients can be obtained by solving an equation in one unknown, documented step by step in the .txt file.
  • Generalization: We propose a framework for evaluation formulas of the type yk2(A)y_{k2}(A), see with Ck2C_k^2​ available variables, and set two conjectures for future research.

Why This Matters

Reducing matrix multiplications significantly lowers computational cost, which is crucial for:

  • Large-scale scientific computing
  • Numerical linear algebra
  • AI and machine learning models involving matrix functions

Access and Resources

Next Steps

If you work with matrix functions or large-scale computations:

  • Try the 5-multiplication scheme for degree‑20 polynomials.
  • Benchmark against Paterson–Stockmeyer.
  • Explore adapting the rational-coefficient approach to other degrees.

We welcome collaboration on proving the conjectures and extending these ideas to broader polynomial families.

Accurate Approximation of the Matrix Hyperbolic Cosine Using Bernoulli Polynomials

José M. Alonso, Javier Ibáñez, Emilio Defez and Fernando Alvarruiz. Mathematics, vol. 11, 520, 2022. https://doi.org/10.3390/math11030520.

This paper presents three different alternatives to evaluate the matrix hyperbolic cosine using Bernoulli matrix polynomials, comparing them from the point of view of accuracy and computational complexity. The first two alternatives are derived from two different Bernoulli series expansions of the matrix hyperbolic cosine, while the third one is based on the approximation of the matrix exponential by means of Bernoulli matrix polynomials. We carry out an analysis of the absolute and relative forward errors incurred in the approximations, deriving corresponding suitable values for the matrix polynomial degree and the scaling factor to be used. Finally, we use a comprehensive matrix testbed to perform a thorough comparison of the alternative approximations, also taking into account other current state-of-the-art approaches. The most accurate and efficient options are identified as results.

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.

Accurate approximation of the hyperbolic matrix cosine using Bernouilli matrix Polynomials

Accurate approximation of the hyperbolic matrix cosine using Bernouilli matrix Polynomials, E. Defez, J. Ibáñez, J.M. Alonso, J.Peinado and J. Sastre, in the International Conference Mathematical Modeling in Engineering & Human Behaviour 2021,  Mathematical Modelling Conference Series at the Institute for Multidisciplinary Mathematics, Universitat Politècnica de València,  July, 14-16, 2021, Valencia (Spain).