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An Improved Taylor Algorithm for Computing the Matrix Logarithm

Author

Listed:
  • 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)

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.

Suggested Citation

  • Javier Ibáñez & Jorge Sastre & Pedro Ruiz & José M. Alonso & Emilio Defez, 2021. "An Improved Taylor Algorithm for Computing the Matrix Logarithm," Mathematics, MDPI, vol. 9(17), pages 1-19, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:17:p:2018-:d:620346
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    References listed on IDEAS

    as
    1. Yu, Philip L.H. & Wang, Xiaohang & Zhu, Yuanyuan, 2017. "High dimensional covariance matrix estimation by penalizing the matrix-logarithm transformed likelihood," Computational Statistics & Data Analysis, Elsevier, vol. 114(C), pages 12-25.
    2. Sastre, J. & Ibáñez, J. & Defez, E., 2019. "Boosting the computation of the matrix exponential," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 206-220.
    3. Robert B. Israel & Jeffrey S. Rosenthal & Jason Z. Wei, 2001. "Finding Generators for Markov Chains via Empirical Transition Matrices, with Applications to Credit Ratings," Mathematical Finance, Wiley Blackwell, vol. 11(2), pages 245-265, April.
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