## On the backward and forward error of approximations of analytic functions and applications to the computation of matrix functions

Jorge Sastre, Javier Ibáñez, Journal of Computational and Applied Mathematics, Volume 419, 2003, 114706, https://doi.org/10.1016/j.cam.2022.114706

A new formula to write the forward error of Taylor approximations of analytical functions in terms of the backward error of those approximations is given, overcoming problems of the backward error analysis that use inverse functions. Examples for the backward error analysis of functions such as the matrix cosine cos(A) or cos(sqrt(A)) are given.

## Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method

Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method (License CC BY 4.0), J. Sastre, J. Ibáñez, Mathematics 20219(14), 1600; https://doi.org/10.3390/math9141600Researchgate link

Two general methods for evaluating matrix polynomials requiring one matrix product less than the Paterson–Stockmeyer method were proposed in J. Sastre, Efficient evaluation of Matrix Polynomials, Linear Algebra Applications, where the cost of evaluating a matrix polynomial is given asymptotically by the total number of matrix product evaluations. An analysis of the stability of those methods was given and the methods have been applied to Taylor-based implementations for computing the exponential, the cosine and the hyperbolic tangent matrix functions. Moreover, a particular example for the evaluation of the matrix exponential Taylor approximation of degree 15 requiring four matrix products was given, whereas the maximum polynomial degree available using Paterson–Stockmeyer method with four matrix products is 9. Based on this example, a new family of methods for evaluating matrix polynomials more efficiently than the Paterson–Stockmeyer method was proposed, having the potential to achieve a much higher efficiency, i.e., requiring less matrix products for evaluating a matrix polynomial of certain degree, or increasing the available degree for the same cost. However, the difficulty of these family of methods lies in the calculation of the coefficients involved for the evaluation of general matrix polynomials and approximations. In this paper, we provide a general matrix polynomial evaluation method for evaluating matrix polynomials requiring two matrix products less than the Paterson-Stockmeyer method for degrees higher than 30. Moreover, we provide general methods for evaluating matrix polynomial approximations of degrees 15 and 21 with four and five matrix product evaluations, respectively, whereas the maximum available degrees for the same cost with the Paterson–Stockmeyer method are 9 and 12, respectively. Finally, practical examples for evaluating Taylor approximations of the matrix cosine and the matrix logarithm accurately and efficiently with these new methods are given.

## Advances in the Approximation of the Matrix Hyperbolic Tangent

Advances in the Approximation of the Matrix Hyperbolic Tangent (License CC BY 4.0), J. Ibáñez, J.M. Alonso, J. Sastre, E. Defez, P.A. Alonso, Mathematics 20219(11), 1219; https://doi.org/10.3390/math9111219Researchgate Link.
In this paper, we introduce two approaches to compute the matrix hyperbolic tangent. While one of them is based on its own definition and uses the matrix exponential, the other one is focused on the expansion of its Taylor series. For this second approximation, we analyse two different alternatives to evaluate the corresponding matrix polynomials. This resulted in three stable and accurate codes, which we implemented in MATLAB and numerically and computationally compared by means of a battery of tests composed of distinct state-of-the-art matrices. Our results show that the Taylor series-based methods were more accurate, although somewhat more computationally expensive, compared with the approach based on the exponential matrix. To avoid this drawback, we propose the use of a set of formulas that allows us to evaluate polynomials in a more efficient way compared with that of the traditional Paterson–Stockmeyer method, thus, substantially reducing the number of matrix products (practically equal in number to the approach based on the matrix exponential), without penalising the accuracy of the result.

## Simulation of harmonic oscillators on the lattice

Simulation of harmonic oscillators on the lattice, M.Tung, J. Ibáñez, E. Defez, J. Sastre, Mathematical Methods in the Applied Sciences 43(14), May 2020. Researchgate Link

This work deals with the simulation of a two‐dimensional ideal lattice having simple tetragonal geometry. The harmonic character of the oscillators give rise to a system of second‐order linear differential equations, which can be recast into matrix form. The explicit solutions which govern the dynamics of this system can be expressed in terms of matrix trigonometric functions. For the derivation we employ the Lagrangian formalism to determine the correct solutions, which extremize the underlying action of the system. In the numerical evaluation we develop diverse state‐of‐the‐art algorithms which efficiently tackle equations with matrix sine and cosine functions. For this purpose, we introduce two special series related to trigonometric functions. They provide approximate solutions of the system through a suitable combination. For the final computation an algorithm based on Taylor expansion with forward and backward error analysis for computing those series had to be devised. We also implement several MATLAB programs which simulate and visualize the two‐dimensional lattice and check its energy conservation.

## 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.

## 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, PreprintMatlab 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.

## 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.