Category: Hermite

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.

A new type of Hermite matrix polynomial series

A new type of Hermite matrix polynomial series (E. DEFEZ, M. TUNG), Quaestiones Mathematicae, 41(2), 2018, 205–212.

Abstract: Conventional Hermite polynomials emerge in a great diversity of applications
in mathematical physics, engineering, and related fields. However, in physical
systems with higher degrees of freedom it will be of practical interest to extend
the scalar Hermite functions to their matrix analogue. This work introduces various
new generating functions for Hermite matrix polynomials and examines existence and
convergence of their associated series expansion by using Mehlerís formula for the
general matrix case. Moreover, we derive interesting new relations for even- and
odd-power summation in the generating-function expansion containing Hermite matrix
polynomials. Some new results for the scalar case are also presented.

 

Two algorithms for computing the matrix cosine based on new Hermite approximations