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Article

Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method

1
Instituto de Telecomunicaciones y Aplicaciones Multimedia, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
2
Instituto de Instrumentación para Imagen Molecular, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
*
Author to whom correspondence should be addressed.
Academic Editor: Luca Gemignani
Mathematics 2021, 9(14), 1600; https://doi.org/10.3390/math9141600
Received: 14 May 2021 / Revised: 27 June 2021 / Accepted: 1 July 2021 / Published: 7 July 2021
(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)
Recently, two general methods for evaluating matrix polynomials requiring one matrix product less than the Paterson–Stockmeyer method were proposed, where the cost of evaluating a matrix polynomial is given asymptotically by the total number of matrix product evaluations. An analysis of the stability of those methods was given and the methods have been applied to Taylor-based implementations for computing the exponential, the cosine and the hyperbolic tangent matrix functions. Moreover, a particular example for the evaluation of the matrix exponential Taylor approximation of degree 15 requiring four matrix products was given, whereas the maximum polynomial degree available using Paterson–Stockmeyer method with four matrix products is 9. Based on this example, a new family of methods for evaluating matrix polynomials more efficiently than the Paterson–Stockmeyer method was proposed, having the potential to achieve a much higher efficiency, i.e., requiring less matrix products for evaluating a matrix polynomial of certain degree, or increasing the available degree for the same cost. However, the difficulty of these family of methods lies in the calculation of the coefficients involved for the evaluation of general matrix polynomials and approximations. In this paper, we provide a general matrix polynomial evaluation method for evaluating matrix polynomials requiring two matrix products less than the Paterson-Stockmeyer method for degrees higher than 30. Moreover, we provide general methods for evaluating matrix polynomial approximations of degrees 15 and 21 with four and five matrix product evaluations, respectively, whereas the maximum available degrees for the same cost with the Paterson–Stockmeyer method are 9 and 12, respectively. Finally, practical examples for evaluating Taylor approximations of the matrix cosine and the matrix logarithm accurately and efficiently with these new methods are given. View Full-Text
Keywords: efficient; matrix polynomial evaluation; matrix function; Taylor approximation; cosine; logarithm efficient; matrix polynomial evaluation; matrix function; Taylor approximation; cosine; logarithm
MDPI and ACS Style

Sastre, J.; Ibáñez, J. Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method. Mathematics 2021, 9, 1600. https://doi.org/10.3390/math9141600

AMA Style

Sastre J, Ibáñez J. Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method. Mathematics. 2021; 9(14):1600. https://doi.org/10.3390/math9141600

Chicago/Turabian Style

Sastre, Jorge, and Javier Ibáñez. 2021. "Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method" Mathematics 9, no. 14: 1600. https://doi.org/10.3390/math9141600

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