An Improved Taylor Algorithm for Computing the Matrix Logarithm

Mathematics

Javier Ibáñez¹  Pedro Ruiz¹  José M. Alonso¹  Emilio Defez²  Jorge Sastre³ 

[1] Instituto de Instrumentación para Imagen Molecular, Universitat Politècnica de València, Av. dels Tarongers, 14, 46011 Valencia, Spain;Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Ed. 8G, Camino de Vera s/n, 46022 Valencia, Spain;Instituto de Telecomunicaciones y Aplicaciones Multimedia, Universitat Politècnica de València, Ed. 8G, Camino de Vera s/n, 46022 Valencia, Spain;

关键词: matrix logarithm;    matrix square root;    inverse scaling and squaring method;    Taylor series;    Paterson–Stockmeyer method;    matrix polynomial evaluation;   

DOI  :  10.3390/math9172018

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown