UDC 519.65
1 Mathematical Modeling Center of the Ya.S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics,
National Academy of Sciences of Ukraine, Lviv, Ukraine
 Petro.Malachivskyy@gmail.com
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2 Mathematical Modeling Center of the Ya.S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics,
National Academy of Sciences of Ukraine, Lviv, Ukraine
levkom@gmail.com
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CHEBYSHEV APPROXIMATION OF THE MULTIVARIABLE FUNCTIONS
BY THE EXPONENTIAL EXPRESSION
Abstract. A method for constructing the Chebyshev approximation by the exponential expression of the multivariable functions with relative error is proposed. It generates an intermediate Chebyshev approximation by a polynomial of the values of the logarithm of a function with absolute error. An iterative scheme based on the least squares method with a variable weight function was used to construct a Chebyshev approximation of the multivariable functions by a generalized polynomial. The presented results of the solution of test examples confirm the fast convergence of the method when calculating the parameters of the Chebyshev approximation of table-defined continuous functions of one, two, and three variables.
Keywords: Chebyshev approximation by the exponential expression, multivariable functions, mean-power approximation, least squares method, variable weight function.
FULL TEXT
REFERENCES
- Cavoretto R. A numerical algorithm for multidimensional modeling of scattered data points. Comp. Appl. Math. 2015. Vol. 34. P. 65–80.
- Iske A. Approximation theory and algorithms for data analysis. New York: Springer, 2018.
- 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.
- Collatz L., Krabs V. Approximation theory. Chebyshev approximations and their applications [Russian translation]. Moscow: Nauka, 1978. 272 p.
- Dunham C.B. Approximation with one (or few) parameters nonlinear. Journal of Computational and Applied Mathematics. 1988. Vol. 21, N 1. P. 115–118.
- Braess D. Nonlinear approximation theory. Berlin; Heidelberg: Springer–Verlag, 1986.
- DeVore R. Nonlinear approximation. Acta Numerica. Cambridge University Press. 1998. Vol. 7. P. 51–150.
- DeVore R.A. , Kunoth A. Nonlinear approximation and its applications. In multscale, nonlinear and adaptive approximation. Berlin; Heidelberg: Springer–Verlag, 2009. P. 169–201.
- Temlyakov V. Nonlinear methods of approximation. J. of FOCM. 2003. Vol. 3. P. 33–107.
- Luke Yu. Special mathematical functions and their approximations [Russian translation]. Moscow: Mir, 1980. 608 p.
- Popov B.A., Tesler G.S. Calculation of functions on a computer. Directory [in Russian]. Kiev: Nauk. dumka, 1984. 599 p.
- Popov B.A., Tesler G.S. Approximation of functions for technical applications. Kiev: Nauk. dumka, 1980. 352 p.
- Dunham C.B. Difficulties in fitting scientific data. ACM SIGNUM Newsletter. 1990. Vol. 25, N 3. P. 15–20.
- Kalenchuk-Porkhanova A.O., Vakal L.P. Reproduction of functional dependences on the basis of nonlinear approximations of some types. Abstracts of International Conf. “Problems of decision making under uncertainties.” 21–25 May 2007. Chernivtsi, Ukraine, 2007. P. 135–137.
- Zuowei Shen, Haizhao Yang, and Shijun Zhang. Nonlinear approximation via compositions. Neural Networks. Vol. 119, Nov. 2019. P. 74–84.
- Malachivskyy P.S., Pizyur Ya.V., Danchak N.V., Orazov E.B. Chebyshev approximation by exponential-power expression. Cybernetics and Systems Analysis. 2013. Vol. 49, N 6. P. 877–881.
- Malachivskyy P.S., Pizyur Ya.V., Danchak N.V., Orazov E.B. Chebyshev approximation by exponential expression with relative error. Cybernetics and Systems Analysis. 2015. Vol. 51, N 2. P. 286–290.
- Malachivskyy P.S., Matviychuk Y.N., Pizyur Ya.V., Malachivskyi R.P. Uniform approximation of functions of two variables. Cybernetics and Systems Analysis. 2017. Vol. 53, N 3. P. 426–431.
- Malachivskyy P.S., Pizyur Ya.V., Malachivskyi R.P., and Ukhanska O.M. Chebyshev approximation of functions of several variables. Cybernetics and Systems Analysis. 2020. Vol. 56, N 1. P. 76–86.
- Remez E.Ya. Fundamentals of numerical methods for the Chebyshev approximation. Kiev: Nauk. dumka, 1969. 623 p.
- Malachivsky P.S., Skopetsky V.V. Continuous and smooth minimax spline approximation [in Ukrainian]. Kyiv: Nauk. dumka, 2013. 270 p.
- Malachivsky PS, Pizyur Ya.V. Solve problems in the Maple environment [in Ukrainian]. Lviv: “RASTR – 7”, 2016. 282 p.
