Categoría: Applications

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.

Advances in the Approximation of the Matrix Hyperbolic Tangent

Advances in the Approximation of the Matrix Hyperbolic Tangent (License CC BY 4.0), J. Ibáñez, J.M. Alonso, J. Sastre, E. Defez, P.A. Alonso, Mathematics 2021, 9(11), 1219; https://doi.org/10.3390/math9111219, Researchgate Link.

In this paper, we introduce two approaches to compute the matrix hyperbolic tangent. While one of them is based on its own definition and uses the matrix exponential, the other one is focused on the expansion of its Taylor series. For this second approximation, we analyse two different alternatives to evaluate the corresponding matrix polynomials. This resulted in three stable and accurate codes, which we implemented in MATLAB and numerically and computationally compared by means of a battery of tests composed of distinct state-of-the-art matrices.

Our results show that the Taylor series-based methods were more accurate, although somewhat more computationally expensive, compared with the approach based on the exponential matrix. To avoid this drawback, we propose the use of a set of formulas that allows us to evaluate polynomials in a more efficient way compared with that of the traditional Paterson–Stockmeyer method, thus, substantially reducing the number of matrix products (practically equal in number to the approach based on the matrix exponential), without penalising the accuracy of the result.

Simulation of harmonic oscillators on the lattice

Simulation of harmonic oscillators on the lattice, M.Tung, J. Ibáñez, E. Defez, J. Sastre, Mathematical Methods in the Applied Sciences 43(14), May 2020. Researchgate Link

This work deals with the simulation of a two‐dimensional ideal lattice having simple tetragonal geometry. The harmonic character of the oscillators give rise to a system of second‐order linear differential equations, which can be recast into matrix form. The explicit solutions which govern the dynamics of this system can be expressed in terms of matrix trigonometric functions.

For the derivation we employ the Lagrangian formalism to determine the correct solutions, which extremize the underlying action of the system. In the numerical evaluation we develop diverse state‐of‐the‐art algorithms which efficiently tackle equations with matrix sine and cosine functions. For this purpose, we introduce two special series related to trigonometric functions. They provide approximate solutions of the system through a suitable combination.

For the final computation an algorithm based on Taylor expansion with forward and backward error analysis for computing those series had to be devised. We also implement several MATLAB programs which simulate and visualize the two‐dimensional lattice and check its energy conservation.

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.

Modelling acoustics on the Poincaré half-plane

Modelling acoustics on the Poincaré half-plane. Michael M. Tung. Journal of Computational and Applied Mathematics DOI10.1016/j.cam.2017.10.037

Novel advances in the field of metamaterial research have permitted the engineering of devices with extraordinary characteristics. Here, we explore the possibilities in transformation acoustics to implement a model for the simulation of acoustic wave propagation on the Poincaré half-plane-the simplest model possessing hyperbolic geometry and also of considerable historical interest. We start off from a variational principle on the given spacetime manifold to find the design description of the model in the laboratory. After examining some significant geometrical and physical properties of the Poincaré half-plane model, we derive a general formal solution for its acoustic wave propagation. A numerical example for the evolution of the acoustic potential on a rectangular region of the Poincaré half-plane concludes this discussion.