An Improved Taylor Algorithm for Computing the Matrix Logarithm

[EN] The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Pade 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.This research was funded by the European Regional Development Fund (ERDF) and the Spanish Ministerio de Economia y Competitividad Grant TIN2017-89314-P.Ibáñez González, JJ.; Sastre, J.; Ruíz Martínez, PA.; Alonso Abalos, JM.; Defez Candel, E. (2021). An Improved Taylor Algorithm for Computing the Matrix Logarithm. Mathematics. 9(17):1-19. https://doi.org/10.3390/math917201811991