New matrix series expansions for the matrix cosine approximations

New matrix series expansions for the matrix cosine approximations, E. Defez, J. Ibáñez, P. Alonso, J.M. Alonso J. Peinado, and P. Alonso-Jordá, in the International Conference Mathematical Modelling in Engineering and Human Behaviour 2019,  Mathematical Modelling Conference Series at the Institute for Multidisciplinary Mathematics, Universitat Politècnica de València,  July, 10-12, 2019, Valencia (Spain).

Fast Taylor polynomial evaluation for the matrix cosine

Fast Taylor polynomial evaluation for the matrix cosine, J. Sastre, J. Ibañez, P. Alonso, J. Peinado and E. Defez, 18th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2018, July 2018, Rota (Cadiz)-SPAIN.

In this work we introduce a new method to compute the matrix cosine. It is based on recent new matrix polynomial evaluation methods for the Taylor approximation and forward and backward error analysis. The matrix polynomial evaluation methods allow to evaluate the Taylor polynomial approximation of the cosine function more efficiently than using Paterson-Stockmeyer method. A MATLAB implementation of the new algorithm is provided, giving better efficiency and accuracy than state-of-the-art algorithms.

 

 

An efficient and accurate algorithm for computing the matrix cosine based on New Hermite approximations

An efficient and accurate algorithm for computing the matrix cosine based on New Hermite approximations, E. Defez, J. Ibáñez, J. Peinado, J. Sastre and P. Alonso, Journal of Computational and Applied Mathematics, Volume 348, pp. 1-13. March 2019. https://doi.org/10.1016/j.cam.2018.08.047, Preprint, Matlab code cosmtayher.m.

In this work we introduce new rational-polynomial Hermite matrix expansions which allow us to obtain a new accurate and efficient method for computing the matrix cosine. This method is compared with other state-of-the-art methods for computing the matrix cosine, including a method based on Padé approximants, showing a far superior efficiency, and higher accuracy. The algorithm implemented on the basis of this method can also be executed either in one or two NVIDIA GPUs, which demonstrates its great computational capacity.

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.

A new efficient and accurate spline algorithm for the matrix exponential computation

A new efficient and accurate spline algorithm for the matrix exponential computation, Emilio Defez, Javier Ibáñez, Jorge Sastre, Jesús Peinado, Pedro Alonso. Journal of Computational and Applied Mathematics, Volume 337, pp. 354-365. August 2018. Preprint, Matlab code expmspl.m.

Abstract: In this work an accurate and efficient method based on matrix splines for computing matrix exponential is given. An algorithm and a MATLAB implementation have been developed and compared with the state-of-the-art algorithms for computing the matrix exponential. We also developed a parallel implementation for large scale problems. This implementation allowed us to get a much better performance when working with this kind of problems.

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

Abstract: 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.