On the inverse of the Caputo matrix exponential. (E. Defez, M. Tung, B. Chen-Charpentier and J.M. Alonso) Mathematics (MDPI, ISSN 2227-7390). Vol. 7 (12), 2019, pp. 1137. doi:10.3390/math7121137

# Category: Matrix Exponential

## 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.

## 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**.

Abstract: 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.