An Improved Taylor Algorithm for Computing the Matrix Logarithm, J. Ibáñez, J. Sastre, P. Ruiz, J.M. Alonso, E. Defez ), Mathematics. Vol. 9 (17), 2021, pp. 2018.

The matrix logarithm is the used in many applications of science and engineering [2], such as machine learning [7,8,9,10], computer-aided design (CAD) [19], computer graphics [17], the analysis of the topological distances between networks [23], graph theory [11,12], quantum chemistry and mechanics [3,4], buckling simulation [5], biomolecular dynamics [6], the study of Markov chains [13], sociology [14], optics [15], mechanics [16], control theory [18], optimization [20], the study of viscoelastic fluids [21,22], the study of brain–machine interfaces [24], and also in statistics and data analysis [25], among other areas.

The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. In this work, we present a Taylor series algorithm, based on the free-transformation approach of the inverse scaling and squaring technique, that uses recent matrix polynomial formulas for evaluating the Taylor approximation of the matrix logarithm more efficiently than the Paterson–Stockmeyer method. Two MATLAB implementations of this algorithm, related to relative forward or backward error analysis, were developed and compared with different state-of-the art MATLAB functions. Numerical tests showed that the new implementations are generally more accurate than the previously available codes, with an intermediate execution time among all the codes in comparison.