Chebyshev Approximation by a Rational Expression for Functions of Many Variables
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CHEBYSHEV APPROXIMATION BY A RATIONAL EXPRESSION FOR FUNCTIONS OF MANY VARIABLES P. S. Malachivskyy,1 Ya. V. Pizyur,2 and R. P. Malachivsky³3
UDC 519.65
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 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. Keywords: Chebyshev approximation by rational expression, functions of many variables, mean-power approximation, least squares method. INTRODUCTION Let a continuous real function f ( X ) of n variables X = ( x1 , x 2 , K , x n ) be defined on the point set W = { X j }sj =1 from the bounded domain D, W Ì D, where D Ì R n and R n is a n-dimensional vector space. Function f ( X ) needs to be approximated on the point set W by the irreducible rational expression: k
R k , l ( a, b; X ) =
å ai j i ( X )
l -1
i=0
,
(1)
å bi y i ( X ) + y l ( X )
i=0
where j i ( X ) , i = 0, k , and y i ( X ) , i = 0, l , are systems of linearly independent real functions continuous on D, and
a i , i = 0, k , and bi , i = 0, l - 1 , are unknown parameters: {a i }ki = 0 Î A , A Í R k +1 , {bi }li -=10 Î B , B Í R l .
To construct the Chebyshev approximation by the rational expression (1) for the function f ( X ) on the point set W means to calculate the values of parameters a * and b* for which the condition is satisfied:
max | f ( X ) - R k , l ( a * , b* ; X )| =
X ÎW
min
max | f ( X ) - R k , l ( a, b; X ) | .
aÎA , bÎB X ÎW
(2)
1
Center of Mathematical Modeling, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine, [email protected]. 2Lviv Polytechnic National University, Lviv, Ukraine, [email protected]. 3Lohika System, Lviv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2020, pp. 146–156. Original article submitted October 25, 2019. 1060-0396/20/5605-0811 ©2020 Springer Science+Business Media, LLC
811
In many cases, for the same number of parameters, approximation by a rational expression provides a better approximation accuracy as compared with polynomial approximation [1]. Chebyshev approximation by a rational expression is used to represent elementary and special mathematical functions [1], to approximate solutions of differential and integral equations [1, 2], in neural networks [3], etc. A number of metho
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