Accurate Approximation of the Matrix Hyperbolic Cosine Using Bernoulli Polynomials

José M. Alonso (), Javier Ibáñez, Emilio Defez and Fernando Alvarruiz
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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

Mathematics, 2023, vol. 11, issue 3, 1-22

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.

Keywords: Bernoulli matrix polynomials; matrix hyperbolic cosine; matrix functions approximation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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