UDC 519.65
1 Ya.S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine
 Petro.Malachivskyy@gmail.com
|
|
CHEBYSHEV APPROXIMATION BY THE RATIONAL EXPRESSION OF FUNCTIONS
OF MANY VARIABLES
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 E p norm with 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 another one refines parameters of 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.
Keywords: Chebyshev approximation by rational expression, functions of many variables, mean-power approximation, least squares method.
FULL TEXT
REFERENCES
- Collatz L., Krabs V. Approximation theory. Chebyshev approximations and their applications [in Russian]. Moscow: Nauka, 1978. 272 p.
- Azizov T., Melnyk O., Orlova O., Kalenchuk-Porkhanova A., Vakal L. Calculation of reinforced concrete ceilings with normal cracks accounting the Chebyshev approximation. Proc. 6th International Scientific Conference “Reliability and Durability of Railway Transport Engineering Structures and Buildings” Transbud-2017 (19–21 April 2017, Kharkiv, Ukraine). Kharkiv, 2017. Р. 1–7.
- Peiris V., Sharon N., Sukhorukova N., Ugon J. Rational approximation and its application to improving deep learning classifiers. arXiv:2002.11330v1 [math.OC] 26 Feb 2020.
- Kalenchuk-Porkhanova A.A. Best Chebyshev approximation of functions of one and many variables. Cybernetics and Systems Analysis. 2009. Vol. 45, N 6. P. 988-996.
- Nakatsukasa Y., Sete O., Trefethen L.N. The AAA algorithm for rational approximation. SIAM J. Sci. Comput. 2018. Vol. 40, N 3. Р. A1494–A1522.
- Filip S.-I., Nakatsukasa Y., Trefethen L.N., Beckermann B. Rational minimax approximation via adaptive barycentric representations. SIAM J. Sci. Comput. 2018. Vol. 40, N 4. Р. A2427–A2455.
- Petrak L.V. Approximation of functions of several variables by rational fractions. Proceedings of the IMM UC AS USSR. 1975. Issue. 6: Optimization programs (approximation of functions). P. 130–144.
- Malachivskyy P.S., Matviychuk Y.N., Pizyur Y.V., Malachivskyi R.P. Uniform approximation of functions of two variables. Cybernetics and Systems Analysis. 2017. Vol. 53, N 3. Р. 426–431.
- Malachivskyy P.S., Pizyur Y.V., Malachivskyi R.P., Ukhanska O.M. Chebyshev approximation of functions of several variables. Cybernetics and Systems Analysis. 2020. Vol. 56, N 1. Р. 76–86.
- Kalitkin N.N. Numerical methods. Moscow: Nauka, 1978. 512 p.
- Malachivskyy P.S., Pizyur Ya.V. Solving problems in the Maple environment [in Ukrainian]. Lviv: RASTR–7, 2016. 282 p.
- Malachivskyy P.S., Pizyur Y.V., Malachivskyi R.P. Uniform approximation by rational expression. Computer printing technology. 2018. N 1 (39). P. 54–59.
- Malachivskyy P.S., Montsibovych B.R., Pizyur Y.V., Malachivsky R.P. Chebyshev approximation by rational expression of functions of two variables. Mathematical and computer modeling. Ser. Technical sciences. 2019. Iss. 19. P. 75–81.
- Remez E.Ya. Fundamentals of numerical methods of Chebyshev approximation. Kiev: Nauk. Dumka, 1969. 623 p.
- Malachivskyy P.S., Skopetsky V.V. Continuous and smooth minimax spline approximation. Kyiv: Nauk. Dumka, 2013. 270 p.
