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.

Roger Dannenberg, co-creator of the Nyquist language, explores the application of generative AI to music at the UPV

Roger Dannenberg, co-creator of the Nyquist language, explores the application of generative AI to music at the UPV.

The pioneer in computer music and professor of Art, Music, and Computer Science at Carnegie Mellon University in the United States, Roger B. Dannenberg, visited Valencia from June 27 to July 4 to participate in academic, cultural, and research activities organized by the Polytechnic University of Valencia (UPV), its High-Performance Scientific Computing (HiPerSC) group, and American Space Valencia.

Among these activities, Dannenberg delivered a lecture titled “Music in the Age of Artificial Intelligence (AI) and Machine Learning.” The session included live demonstrations of improvised melodies generated by AI, developed as part of a collaboration funded by the Valencian Regional Government through project CIAICO/2023/275.

25/06/2026 – Roger Dannenberg visits Valencia to present advances from an iTEAM project on generative AI for music

30/06/2026 – Roger Dannenberg, en Valencia. El cocreador del lenguaje Nyquist explora en la UPV la aplicación a la música de la IA generativa

01/07/2026 – Talk video: “Music in the Age of Artificial Intelligence (AI) and Machine Learning.”

01/07/2026 – Music in the Age of AI and Machine Learning with Prof. Roger Dannenberg

01/07/2026 – Agenda UPV – Charla »Music in the Age of AI and Machine Learning – Invitation to attend a talk by Prof. Roger Dannenberg»

01/07/2026 – “UPV AL DIA” students – Conferència de Roger Dannenberg: descobreix com la IA transforma la música amb el cocreador d’Audacity.

01/07/2026 – “UPV AL DIA” – Conferència de Roger Dannenberg: descobreix com la IA transforma la música amb el cocreador d’Audacity.

01/07/2026 – Telegram UPV – Conferència de Roger Dannenberg: descobreix com la IA transforma la música amb el cocreador d’Audacity.

01/07/2026 – Whatsapp UPV – Conferència de Roger Dannenberg: descobreix com la IA transforma la música amb el cocreador d’Audacity.

Roger Dannenberg, co-creator of Audacity, visits the UPV to collaborate on AI projects

Roger Dannenberg, pioneer of computational music, visits UPV within the CIAICO/2023/275 project led by Jorge Sastre and Nuria Lloret.

Below you can access press coverage of Roger Dannenberg’s visit to the UPV, where he gave a live presentation of his virtual musical interpreter system.

The visit is part of the Valencian Regional Government’s Ciaico/2023/275 project, which focuses on artificial intelligence applications and is co-directed by UPV professors Nuria Lloret and Jorge Sastre.

07/11/2025 – Roger Dannenberg, cocreador de Audacity, visita la UPV para hablar de composición musical con IA

18/11/2025 – Roger Dannenberg, el creador del Audacity, en La Tarde Con Marina

07/11/2025 – Roger Dannenberg, cocreador de Audacity, visita la UPV para colaborar en proyectos de IA

20/11/2025 – La inteligencia artificial ya puede improvisar música en un grupo: «Nos lleva a un nuevo territorio musical»

07/11/2025 – Roger Dannenberg, cocreador de Audacity, visita la UPV para colaborar en proyectos de IA

18/11/2025 – Roger Dannenberg, el creador del Audacity, en La Tarde Con Marina

07/11/2025 – Roger Dannenberg, cocreador de Audacity, visita la UPV para colaborar en proyectos de IA

18/11/2025 –
Roger Dannenberg, el creador del Audacity, en La Tarde Con Marina

07/11/2025 – Roger Dannenberg, cocreador de Audacity, visita la UPV para colaborar en proyectos de IA

17/11/2025 – Roger Dannenberg, co-creator of Audacity, visits UPV

10/11/2025 – ¿Realmente se puede componer música con inteligencia artificial? Roger Dannenberg, pionero mundial en música computacional, reflexionará sobre ello los días 12 y 13 de noviembre en la Universitat Politècnica de València

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.

Polynomial approximations for the matrix logarithm with computation graphs

Polynomial approximations for the matrix logarithm with computation graphs, E. Jarlebring, J. Sastre, J. Ibáñez, Linear Algebra Applications, in Press (open access), 2024. https://doi.org/10.1016/j.laa.2024.10.024, https://arxiv.org/abs/2401.10089, code.

In this article the matrix logarithm is computed by using matrix polynomial approximations evaluated by using matrix polynomial multiplications and additions. The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. The main computational effort lies in matrix-matrix multiplications and left matrix division. In this work we illustrate that the number of such operations can be substantially reduced, by using a graph based representation of an efficient polynomial evaluation scheme. A technique to analyze the rounding error is proposed, and backward error analysis is adapted. We provide substantial simulations illustrating competitiveness both in terms of computation time and rounding errors.

A Matrix Spline Method for a Class of Fourth-Order Ordinary Differential Problems

A Matrix Spline Method for a Class of Fourth-Order Ordinary Differential Problems, M.M. Tung, E. Defez, J. Ibáñez, J.M. Alonso, J.I. Real-Herráiz, Mathematics 2022, 10(16), 2826, https://www.doi.org/10.3390/math10162826

Differential matrix models provide an elementary blueprint for the adequate and efficient treatment of many important applications in science and engineering. In the present work, we suggest a procedure, extending our previous research results, to represent the solutions of nonlinear matrix differential problems of fourth order given in the form ?(4)(?)=?(?,?(?)) in terms of higher-order matrix splines. The corresponding algorithm is explained, and some numerical examples for the illustration of the method are included.

On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem

On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem, E. Defez, J. Ibáñez, J.M. Alonso, M.M. Tung, T.P. Real-Herraiz, Teresa Pilar. Mathematics, 2021, 9(18), 2262, https://doi.org/10.3390/math9182262

Matrix differential equations are at the heart of many science and engineering problems. In this paper, a procedure based on higher-order matrix splines is proposed to provide the approximated numerical solution of special nonlinear third-order matrix differential equations, having the form ?(3)(?)=?(?,?(?)). Some numerical test problems are also included, whose solutions are computed by our method.

Press: UPV Researchers ‘Revolutionize’ Matrix Calculation with a ‘Faster and More Accurate’ Method

UPV Researchers ‘Revolutionize’ Matrix Calculation with a ‘Faster and More Accurate’ Method

Below you can access the press coverage of our discovery of a new, faster and more accurate method for calculating matrix functions:

29/08/2023 – Un método matemático valenciano más rápido y preciso jubila el cálculo tradicional de funciones de matrices.

29/08/2023 – Jorge Sastre (Investigador UPV): «El descubrimiento supone una revolución en el campo de las matemáticas»

29/08/2023 – Revolución en las matemáticas: investigadores de la UPV descubren un método más rápido y preciso para calcular funciones de matrices

31/08/2023 – Nuestro descubrimiento en ​⁠Radio Nacional España

31/08/2023 – Interview Onda Cero radio on our last improvements on Efficient Evaluation of Matrix Polynomials

29/08/2023 – Investigadores de la UPV «revolucionan» el cálculo de matrices con un método «más rápido y preciso»

29/08/2023 – Un método valenciano más rápido jubila el cálculo tradicional de funciones de matrices

29/08/2023 – Investigadores de la UPV descubren un método más rápido para calcular matrices

27/03/2023 – Un grupo de matemáticos en Valencia mejora un sistema de cálculo «imbatible» durante 50 años

29/08/2023 – Un método valenciano más rápido jubila el cálculo tradicional de funciones de matrices

29/08/2023 – Un método valenciano más rápido jubila el cálculo tradicional de funciones de matrices

29/08/2023 – Un método valenciano más rápido jubila el cálculo tradicional de funciones de matrices

29/08/2023 – Investigadores de la UPV «revolucionan» el cálculo de matrices con un método «más rápido y preciso»

30/08/2023 – Investigadores de la Politècnica revolucionan un método de cálculo usado desde hace 50 años

29/08/2023 – Un mètode valencià més ràpid deixa arrere el càlcul tradicional de funcions de matrius

04/09/2023 – Entrevistem Jorge Sastre, catedràtic i matemàtic de la UPV i descobridor d’un nou càlcul de matrius revolucionari

29/08/2023 – Un método valenciano más rápido jubila el cálculo tradicional de funciones de matrices

03/09/2023 – Revolución matemática en València: investigadores de la Universitat Politécnica desarrollan un método “más rápido y preciso” en el cálculo de matrices

29/08/2023 – Investigadores de la UPV «revolucionan» el cálculo de matrices con un método «más rápido y preciso»

29/08/2023 – Un método matemático valenciano más rápido y preciso jubila el cálculo tradicional de funciones de matrices

07/09/2023 – Acabamos de revolucionar la forma de calcular matrices. Se lo debemos a unos investigadores de Valencia

07/09/2023 – Investigadores de la UPV descubren un método más rápido y preciso para calcular funciones de matrices que desplaza un sistema con medio siglo de primacía

27/03/2023 – Un grupo de matemáticos en Valencia mejora un sistema de cálculo «imbatible» durante 50 años

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.