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* **2021**, *9*(11), 1219; https://doi.org/10.3390/math9111219, Researchgate 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.

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