Author: joalab

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 𝑌(4)(𝑥)=𝑓(𝑥,𝑌(𝑥)) 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 𝑌(3)(𝑥)=𝑓(𝑥,𝑌(𝑥)). Some numerical test problems are also included, whose solutions are computed by our method.

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.