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Accurate Approximation of the Matrix Hyperbolic Cosine Using Bernoulli Polynomials

Author

Listed:
  • José M. Alonso

    (Instituto de Instrumentación para Imagen Molecular, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain)

  • Javier Ibáñez

    (Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain)

  • Emilio Defez

    (Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain)

  • Fernando Alvarruiz

    (Departamento de Sistemas Informáticos y Computación, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain)

Abstract

This paper presents three different alternatives to evaluate the matrix hyperbolic cosine using Bernoulli matrix polynomials, comparing them from the point of view of accuracy and computational complexity. The first two alternatives are derived from two different Bernoulli series expansions of the matrix hyperbolic cosine, while the third one is based on the approximation of the matrix exponential by means of Bernoulli matrix polynomials. We carry out an analysis of the absolute and relative forward errors incurred in the approximations, deriving corresponding suitable values for the matrix polynomial degree and the scaling factor to be used. Finally, we use a comprehensive matrix testbed to perform a thorough comparison of the alternative approximations, also taking into account other current state-of-the-art approaches. The most accurate and efficient options are identified as results.

Suggested Citation

  • José M. Alonso & Javier Ibáñez & Emilio Defez & Fernando Alvarruiz, 2023. "Accurate Approximation of the Matrix Hyperbolic Cosine Using Bernoulli Polynomials," Mathematics, MDPI, vol. 11(3), pages 1-22, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:520-:d:1039860
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    References listed on IDEAS

    as
    1. Jan L. Cieśliński & Artur Kobus, 2020. "Locally Exact Integrators for the Duffing Equation," Mathematics, MDPI, vol. 8(2), pages 1-13, February.
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