## Accurate Approximation of the Matrix Hyperbolic Cosine Using Bernoulli Polynomials

José M. Alonso, Javier Ibáñez, Emilio Defez and Fernando Alvarruiz. Mathematics, vol. 11, 520, 2022. https://doi.org/10.3390/math11030520.

This paper presents three different alternatives to evaluate the matrix hyperbolic cosine using Bernoulli matrix polynomials, comparing them from the point of view of accuracy and computational complexity. The first two alternatives are derived from two different Bernoulli series expansions of the matrix hyperbolic cosine, while the third one is based on the approximation of the matrix exponential by means of Bernoulli matrix polynomials. We carry out an analysis of the absolute and relative forward errors incurred in the approximations, deriving corresponding suitable values for the matrix polynomial degree and the scaling factor to be used. Finally, we use a comprehensive matrix testbed to perform a thorough comparison of the alternative approximations, also taking into account other current state-of-the-art approaches. The most accurate and efficient options are identified as results.

## Euler polynomials for the matrix exponential approximation

José M. Alonso, Javier Ibáñez, Emilio Defez, Pedro Alonso-Jordá. Journal of Computational and Applied Mathematics, vol. 425, 115074, 2023. https://doi.org/10.1016/j.cam.2023.115074.

In this work, a new method to compute the matrix exponential function by using an approximation based on Euler polynomials is proposed. These polynomials are used in combination with the scaling and squaring technique, considering an absolute forward-type theoretical error. Its numerical and computational properties have been evaluated and compared with the most current and competitive codes dedicated to the computation of the matrix exponential. Under a heterogeneous test battery and a set of exhaustive experiments, it has been demonstrated that the new method offers performance in terms of accuracy and stability which is as good as or even better than those of the considered methods, with an intermediate computational cost among all of them. All of the above makes this a very competitive alternative that should be considered in the growing list of available numerical methods and implementations dedicated to the approximation of the matrix exponential.

## 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.

## Accurate approximation of the hyperbolic matrix cosine using Bernouilli matrix Polynomials

Accurate approximation of the hyperbolic matrix cosine using Bernouilli matrix Polynomials, E. Defez, J. Ibáñez, J.M. Alonso, J.Peinado and J. Sastre, in the International Conference Mathematical Modeling in Engineering & Human Behaviour 2021,  Mathematical Modelling Conference Series at the Institute for Multidisciplinary MathematicsUniversitat Politècnica de València,  July, 14-16, 2021, Valencia (Spain)

## 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.