Chebyshev Approximation by Exponential Expression with Relative Error
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CHEBYSHEV APPROXIMATION BY EXPONENTIAL EXPRESSION WITH RELATIVE ERROR P. S. Malachivskyy,a† Ya. V. Pizyur,b N. V. Danchak,a and E. Â. Orazovc
UDC 519.65
Abstract. The properties of the Chebyshev approximation by an exponential expression with the smallest relative error are investigated and the sufficient condition for its existence is established. A method to determine the parameters of such approximation is proposed and justified. The error of the Chebyshev approximation by an exponential expression is estimated. A numerical example confirming the theoretical results is presented. Keywords: Chebyshev (uniform) approximation, points of alternation, relative error, kernel of approximation error. INTRODUCTION Consider the problem of Chebyshev approximation by exponential expression æ n ö E n ( a ; x ) = A exp ç å a i x i + c x p ÷ , x ³ 0, c ¹ 0, p ¹ k ( k = 0, n ) , è i =1 ø
(1)
with respect to unknown parameters A , c, p, and a i ( i = 1, n ) . Approximation by exponential expression (1) is used to describe various physical dependences, in particular, for n = 0 and n = 1 to describe thermometric performances of a germanium sensor [1] and constants of the velocity of chemical reactions [2]. Expression (1) does not satisfy the Haar condition [3, 4]; therefore, it is necessary to investigate the existence of the Chebyshev approximation by such expression. EXISTENCE OF THE CHEBYSHEV APPROXIMATION BY EXPONENTIAL EXPRESSION WITH RELATIVE ERROR The theorem below establishes the class of functions f ( x ) for which there exists Chebyshev approximation by expression (1) with a relative error. THEOREM 1. A sufficient condition for the existence of Chebyshev approximation by expression (1) for a positive function f ( x ) ( f ( x ) Î C[ a, b], f ( x ) > 0) continuous on the interval [ a, b] , a ³ 0, with a relative error on [ a, b] , is the following inequalities: (2) W ( n ) > 0 , W ( n ) ¹ Wr( n ) , r = 0, n , where W (n ) =
Dn + 1 ( f ; z 2 , z 3 , K , z n + 4 ) ; D n + 1 ( f ; z1 , z 2 , K , z n + 3 )
a
(3)
Ya. S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine, †[email protected]. bNational University “Lviv Polytechnic,” Lviv, Ukraine, [email protected]. c Ukrainian Academy of Printing, Lviv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 2, March–April, 2015, pp. 145–150. Original article submitted February 13, 2014. 286
1060-0396/15/5102-0286
©
2015 Springer Science+Business Media New York
Dk (U ; z j , z j + 1 , K , z j + k + 1 ) =
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Dk -1 (U ; z j + 1 , z j + 2 , K , z j + k + 1 ) Dk -1 ( sk -1 ; z j + 1 , z j + 2 ,.. , z j + k + 1 )
Dk -1 (U ; z j , z j + 1 , K , z j + k ) D k -1 ( sk -1 ; z j , z j + 1 , K , z j + k )
, k = 2, 3,... , j = 1, n - k + 3 ;
(4)
D1 (U ; z j , z j + 2 ) = U ( z j + 2 ) -U ( z j ) , j = 1, n + 2 ;
(5)
if z1 = 0, ì0 ï Wr( n ) = í Dn + 1 ( lr ; z 2 , z 3 ,... , z n + 4 ) if z1 > 0; ïî Dn + 1 ( lr ; z1 , z 2 ,... , z n+ 3 )
(6)
sk ( x ) = x k ; lk ( x ) = x k ln
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