An Improved Taylor Algorithm for Computing the Matrix Logarithm

Javier Ibáñez, Jorge Sastre, Pedro Ruiz, José M. Alonso and Emilio Defez
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Javier Ibáñez: Instituto de Instrumentación para Imagen Molecular, Universitat Politècnica de València, Av. dels Tarongers, 14, 46011 Valencia, Spain
Jorge Sastre: Instituto de Telecomunicaciones y Aplicaciones Multimedia, Universitat Politècnica de València, Ed. 8G, Camino de Vera s/n, 46022 Valencia, Spain
Pedro Ruiz: Instituto de Instrumentación para Imagen Molecular, Universitat Politècnica de València, Av. dels Tarongers, 14, 46011 Valencia, Spain
José M. Alonso: Instituto de Instrumentación para Imagen Molecular, Universitat Politècnica de València, Av. dels Tarongers, 14, 46011 Valencia, Spain
Emilio Defez: Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Ed. 8G, Camino de Vera s/n, 46022 Valencia, Spain

Mathematics, 2021, vol. 9, issue 17, 1-19

Abstract: 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.

Keywords: matrix logarithm; matrix square root; inverse scaling and squaring method; Taylor series; Paterson–Stockmeyer method; matrix polynomial evaluation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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