Abstract
The method of constructing the Chebyshev approximation by a rational expression for functions of many variables is proposed. The idea of the method is based on constructing the boundary mean-power approximation in Ep norm as p→∞. The least squares method with two variable weight functions is used to construct this approximation. One weight function ensures the construction of mean-power approximation, and the other one refines parameters of the rational expression by linearization scheme. The convergence of the method is provided by the original method of sequentially refining the values of the weight functions. Algorithms for calculating the parameters of the Chebyshev approximation of functions of many variables by a rational expression with absolute and relative errors is described.
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References
L. Collatz and W. Krabs, Approximationstheorie: Tschebyscheffsche Approximation mit Anwendungen (Teubner Studienbucher Mathematik) Vieweg+Teubner Verlag (1973).
T. Azizov, O. Melnyk, O. Orlova, A. Kalenchuk-Porkhanova, and L. Vakal, “Calculation of reinforced concrete ceilings with normal cracks accounting the Chebyshev approximation,” in: Proc. 6th Intern. Sci. Conf. “Reliability and Durability of Railway Transport Engineering Structures and Buildings” Transbud-2017 (April 19–21, 2017, Kharkiv, Ukraine), Kharkiv (2017), pp. 1–7.
V. Peiris, N. Sharon, N. Sukhorukova, and J. Ugon, “Rational approximation and its application to improving deep learning classifiers,” arXiv:2002.11330v1 [math.OC] 26 Feb 2020.
A. A. Kalenchuk-Porkhanova, “Best Chebyshev approximation of functions of one and many variables,” Cybern. Syst. Analysis, Vol. 45, No. 6, 988-996 (2009).
Y. Nakatsukasa, O. Sete, and L. N. Trefethen, “The AAA algorithm for rational approximation,” SIAM J. Sci. Comput., Vol. 40, No. 3, A1494–A1522 (2018).
S.-I. Filip, Y. Nakatsukasa, L. N. Trefethen, and B. Beckermann, “Rational minimax approximation via adaptive barycentric representations,” SIAM J. Sci. Comput., Vol. 40, No. 4, A2427–A2455 (2018).
L. V. Petrak, “Approximation of functions of many variables by rational fractions,” Tr. IMM UNTs AN USSR, Issue 6: Optimization Programs (Approximation of Functions), 130–144 (1975).
P. S. Malachivskyy, Y. N. Matviychuk, Y. V. Pizyur, and R. P. Malachivskyi, “Uniform approximation of functions of two variables,” Cybern. Syst. Analysis, Vol. 53, No. 3, 426–431 (2017).
P. S. Malachivskyy, Y. V. Pizyur, R. P. Malachivskyi, and O. M. Ukhanska, “Chebyshev approximation of functions of several variables,” Cybern. Syst. Analysis, Vol. 56, No. 1, 76–86 (2020).
N. N. Kalitkin, Numerical Methods [in Russian], Nauka, Moscow (1978).
P. S. Malachivskyy and Y. V. Pizyur, Solving Problems in the Maple Environment [in Ukrainian], RASTR–7, Lviv (2016).
P. S. Malachivskyy, Y. V. Pizyur, and R. P. Malachivskyi, “Uniform approximation by a rational expression,” Komp. Tekhnologii Drukarstva, No. 1 (39), 54–59 (2018).
P. S. Malachivskyy, B. R. Montsibovych, Y. V. Pizyur, and R. P. Malachivskyi, “Chebyshev approximation of functions of two variables by a rational expression,” Matematychne ta Komp. Modelyuvannya, Ser. Tekhnichni Nauky, Issue 19, 75–81 (2019).
E. Ya. Remez, Fundamentals of Numerical Methods of Chebyshev Approximation [in Russian], Naukova Dumka, Kyiv (1969).
P. S. Malachivskyy and V. V. Skopetskii, Continuous and Smooth Minimax Spline Approximation [in Ukrainian], Naukova Dumka, Kyiv (2013).
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Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2020, pp. 146–156.
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Malachivskyy, P.S., Pizyur, Y.V. & Malachivsky, R.P. Chebyshev Approximation by a Rational Expression for Functions of Many Variables. Cybern Syst Anal 56, 811–819 (2020). https://doi.org/10.1007/s10559-020-00302-0
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DOI: https://doi.org/10.1007/s10559-020-00302-0