Mes: noviembre 2026

New Results Of Harmonic Oscillators On The Lattice

Emilio Defez Candel, Michael M. Tung, and Màrius Josep Fullana i Alfonso present a contribution at MME&HB 2026.

The work builds on the group’s research into the simulation of harmonic oscillators on lattices, a problem with connections to both mathematical physics and numerical simulation models. The contribution is part of the Mathematics and Physics session of the conference.

Publication Details

  • Title: More about harmonic oscillators on the lattice
  • Authors: Emilio Defez Candel, Michael M. Tung, Màrius Josep Fullana i Alfonso
  • Conference: Mathematical Modelling in Engineering & Human Behaviour 2026 (MME&HB 2026), Valencia, July 12–15 of 2026
  • Proceedings: Mathematical Modelling Conference Series at the Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, pp. 383–389
  • ISBN: 978-84-09-89823-7
  • Enlace: https://imm.webs.upv.es/jornadas/2026/2026_Proceedings_MME_HB.pdf

Main Contributions

  • Extension of the study of harmonic oscillators on a lattice.
  • Connection with simulation models and mathematical physics.
  • Contribution to the Mathematics and Physics session of the conference.

Why This Matters

Harmonic oscillators on a lattice are a reference model in physics and numerical simulation. The results presented may be relevant to researchers working on network dynamics and numerical methods for Hamiltonian systems.

New Taylor method for the matrix sign function

Emilio Defez, José Miguel Alonso, Javier Ibáñez and Jorge Sastre present a new Taylor-based method for computing the matrix sign function at MME&HB 2026.

The matrix sign function is an important tool in numerical linear algebra, with applications to the solution of matrix equations and matrix decomposition problems. Its efficient computation remains an interesting challenge, particularly when accuracy and computational cost must be considered together.

Building on the research group’s experience in polynomial approximation and efficient matrix polynomial evaluation, the authors introduce a new approach within the family of Taylor methods for this function. The contribution extends their ongoing work on polynomial-based algorithms for matrix functions, exploring how Taylor approximations can provide an alternative framework for their numerical computation.

The proposed method adds to the group’s research into efficient algorithms for matrix functions, with a particular focus on developing numerical techniques that combine mathematical accuracy with practical computational efficiency.

Publication Details

  • Title: A new Taylor method for the matrix sign function
  • Authors: Emilio Defez, José Miguel Alonso, Javier Ibáñez, Jorge Sastre
  • Conference: Mathematical Modelling in Engineering & Human Behaviour 2026 (MME&HB 2026), Valencia, July 12–15 of 2026
  • Proceedings: Mathematical Modelling Conference Series at the Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, pp. 356–363
  • ISBN: 978-84-09-89823-7
  • Enlace: https://imm.webs.upv.es/jornadas/2026/2026_Proceedings_MME_HB.pdf

Main Contributions

  • A new method based on Taylor series for the matrix sign function.
  • An alternative within the set of polynomial techniques for this function.
  • A connection with the group’s research on efficient matrix polynomial evaluation.

Why This Matters

The matrix sign function arises in control problems, the solution of Riccati equations, and the computation of other matrix functions. An additional Taylor method broadens the range of available algorithmic approaches.

Optimal order-24 matrix exponential approximation with applications to generative AI flows

At MME&HB 2026, Jorge Sastre, José Miguel Alonso, Javier Ibáñez, Julio Andrés, Roger B. Dannenberg, Daniel Faronbi, and Nuria Lloret present a 24th-degree polynomial approximation to the matrix exponential, designed to meet the computational demands of generative AI workflows.

The work builds on Taylor-based methods and polynomial evaluation schemes that go beyond the limitations of conventional approaches, with the aim of reducing computational costs without compromising numerical stability. It also draws on the group’s recent advances in efficient matrix polynomial evaluation and error analysis for high-order approximations.

By bringing these developments together, the proposed approach addresses a relevant computational challenge in generative AI: making the evaluation of matrix functions more efficient while retaining the numerical reliability required by demanding computational workloads.

Publication Details

  • Title: Optimal order-24 matrix exponential approximation with applications to generative AI flows
  • Authors: Jorge Sastre, José Miguel Alonso, Javier Ibáñez, Julio Andrés, Roger B. Dannenberg, Daniel Faronbi, Nuria Lloret
  • Conference: Mathematical Modelling in Engineering & Human Behaviour 2026 (MME&HB 2026), Valencia, July 12–15 of 2026
  • Proceedings: Mathematical Modelling Conference Series at the Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, pp. 343–355
  • ISBN: 978-84-09-89823-7
  • Enlace: https://imm.webs.upv.es/jornadas/2026/2026_Proceedings_MME_HB.pdf

Main Contributions

  • 24th-degree approximation of the matrix exponential.
  • Direct application to generative AI workflows.
  • Integration with matrix polynomial evaluation schemes developed by the group.

Why This Matters

Generative AI workflows require high-performance matrix operations. A high-order approximation with controlled computational cost can help reduce computation time in these models.

A MATLAB Tool for the Stable Generation of Matrix Polynomial Evaluation Schemes with Two-Product Savings

The efficient evaluation of high-degree matrix polynomials is central to the numerical approximation of matrix functions. For decades, the Paterson–Stockmeyer (PS) method has provided the standard benchmark for reducing the number of matrix–matrix multiplications required by these computations. Recent theoretical advances, however, have shown that further savings are possible through evaluation schemes that eliminate two matrix products (2M) relative to the classical approach. Despite this progress, numerically stable implementations of such schemes have remained largely restricted to a few specific polynomial degrees and applications.

Javier Ibáñez, Jorge Sastre, José M. Alonso, and Emilio Defez address this limitation by developing a software-assisted procedure that extends the 2M reduction to polynomials of degrees m∈{18,21,24,26,27,28} and to every degree m≥30, provided that the leading coefficient is nonzero. The main numerical challenge lies in determining the coefficients that define the evaluation schemes. These coefficients are obtained by solving systems of nonlinear equations, whose solutions may differ substantially in their numerical stability. The authors therefore introduce an automated verification procedure to identify and validate coefficient sets suitable for floating-point computations.

The central contribution is a MATLAB implementation that combines symbolic computation with variable-precision arithmetic to construct the reduced nonlinear systems, identify suitable solutions, assess their numerical stability, and estimate the precision required for reliable evaluation. Beyond the 2M cases, the framework also identifies polynomial degrees for which a one-product reduction is attainable, thereby extending the range of available evaluation strategies.

Numerical experiments involving several matrix functions show that the resulting schemes maintain accuracy comparable to that of the classical Paterson–Stockmeyer method and, in some cases, improve upon it, while consistently achieving the predicted reduction of two matrix–matrix multiplications. These results demonstrate that the proposed software provides a practical route from theoretical evaluation formulas to numerically reliable implementations, making the computational savings accessible to a substantially broader class of matrix polynomials.

Publication details

Main Contributions

  • Software procedure extending the two-matrix-product reduction to a broad range of polynomial degrees.
  • Automated verification routine for selecting numerically stable coefficient sets.
  • MATLAB implementation using variable-precision arithmetic and numerical validation on matrix functions.

Why This Matters

The reduction of two matrix–matrix multiplications represents both a theoretical and a practical improvement in the computational cost of evaluating high-degree polynomials. The developed tool makes this reduction reliably accessible across a broad range of polynomial degrees, potentially benefiting matrix function algorithms that rely on polynomial approximations.

A Systematic Framework for the Stable Evaluation of Matrix Polynomials with Product Savings

Evaluating high-degree matrix polynomials is a common task in matrix function approximation. A recently proposed evaluation scheme reduces the computational cost by one matrix multiplication (1M) compared with the Paterson–Stockmeyer (PS) method, offering a direct reduction in execution time for large matrices. The main difficulty, however, lies in determining suitable coefficients for these optimized formulas. Since they are obtained by solving nonlinear polynomial systems, several possible solutions may arise, with substantially different numerical stability properties. Selecting an unsuitable solution can lead to significant accuracy loss in floating-point arithmetic.

J. M. Alonso, J. Sastre, J. Ibáñez, and E. Defez address this issue by developing a systematic framework and its MATLAB implementation, MatrixPolEval1, for computing and validating stable coefficient sets for matrix polynomials of degrees m∈{8,10,12} and beyond. The framework automates coefficient selection using variable precision arithmetic (VPA) and assesses numerical stability through coefficient reconstruction errors, providing a reliability indicator for the expected accuracy. It also incorporates alternative evaluation structures to handle cases in which the original formulation fails to provide a robust solution, including degenerate configurations that require separate treatment.

Numerical experiments involving matrix polynomials arising in applications such as the matrix exponential and geometric series demonstrate that the proposed framework retains the one-multiplication advantage over the PS method while achieving comparable numerical accuracy. These results provide a practical basis for applying multiplication-efficient evaluation schemes without compromising numerical reliability.

Publication Details

Main Contributions

  • A systematic framework for identifying stable coefficient sets in evaluation schemes that save one matrix multiplication.
  • MATLAB implementation (MatrixPolEval1), featuring variable-precision verification and a reliability indicator.
  • Numerical validation for polynomials of degrees 8, 10, 12, and higher, including applications to the matrix exponential and geometric series.

Why This Matters

Saving one matrix multiplication can directly reduce execution time in large-scale problems. The proposed framework facilitates the reliable use of these optimized schemes without compromising numerical stability.

An Optimized Taylor Algorithm for the Matrix Exponential in Generative AI Workflows

The matrix exponential is a key operation in scientific computing and system simulation, with applications ranging from control theory and quantum mechanics to modern generative machine learning. Although Padé approximants combined with scaling and squaring have traditionally been the standard approach, recent advances in Taylor-based methods have shown that more efficient polynomial evaluation schemes can outperform the classical Paterson–Stockmeyer algorithm, both in accuracy and computational cost.

In this work, J. Sastre, D. Faronbi, J. M. Alonso, P. Traver, J. Ibáñez, and N. Lloret introduce an optimized Taylor-based algorithm for computing the matrix exponential, specifically developed to address the computational demands of generative AI flows. The method combines advanced matrix polynomial evaluation techniques with a rigorous error analysis and an adaptive strategy for selecting the Taylor approximation order and scaling factor. This allows the algorithm to adjust its computational effort to the properties of each input matrix while meeting a prescribed accuracy tolerance.

Beyond the numerical formulation itself, the main contribution lies in translating these algorithmic improvements into measurable gains in generative model training and inference. By reducing the number of matrix multiplications required without compromising numerical accuracy, the proposed approach addresses a significant computational bottleneck in flow-based architectures. Experiments on standard image datasets report training speedups ranging from 3.91× to 9.74× compared with the original implementation, while inference on batches of 128 samples is approximately 1.95× faster. These results illustrate how improvements in numerical algorithms can directly contribute to the efficiency and scalability of modern generative models.

Publication Details

Main Countributions

  • Optimized Taylor algorithm for the matrix exponential, designed for generative AI flows.
  • Error analysis and an adaptive strategy for selecting the Taylor approximation order and scaling factor.
  • Significant speedups while maintaining high numerical stability compared with reference implementations.

Why This Matters

Generative AI flows rely on computationally intensive matrix operations. A faster and more stable method for computing the matrix exponential can reduce the computational cost of these models without compromising accuracy.