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.

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

Two algorithms for computing the matrix cosine based on new Hermite approximations, E. Defez, J. Ibáñez, J. Sastre, J. Peinado, P. Alonso, in the International Conference Mathematical Modelling in Engineering and Human Behaviour 2017,  Mathematical Modelling Conference Series at the Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, Valencia, July. 19-21, 2017.

 

Efficient evaluation of matrix polynomials

Efficient evaluation of matrix polynomials, Jorge Sastre, Linear Algebra Applications, Vol. 539, pp. 229-250, Feb. 2018 (early version submitted on Feb. 18, 2016  in AMC-S-16-00951.pdf), submitted Oct. 2016, available online 2017, Preprint.

This paper presents a new family of methods for evaluating matrix polynomials more efficiently than the state-of-the-art Paterson–Stockmeyer method. Examples of the application of the methods to the Taylor polynomial approximation of matrix functions like the matrix exponential and matrix cosine are given. Their efficiency is compared with that of the best existing evaluation schemes for general polynomial and rational approximations, and also with a recent method based on mixed rational and polynomial approximants.

For many years, the Paterson–Stockmeyer method has been considered the most efficient general method for the evaluation of matrix polynomials. In this paper we show that this statement is no longer true. Moreover, for many years rational approximations have been considered more efficient than polynomial approximations, although recently it has been shown that often this is not the case in the computation of the matrix exponential and matrix cosine.

In this paper we show that in fact polynomial approximations provide a higher order of approximation than the state-of-the-art computational methods for rational approximations for the same cost in terms of matrix products. For an early unpublished version of this work submitted on Feb. 18, 2016 to Appl. Math. Comput. see AMC-S-16-00951.pdf.