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N16 - ME 510 NUMERICAL METHODS FOR ME II Example on...

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ME – 510 NUMERICAL METHODS FOR ME II g51g85g82g73g17g3g39g85g17g3g41g68g85g88g78g3g36g85g213g81g111 Fall 2007 Example on Chebyshev Fitting Problem : Given the function f (x) = exp (- 2 x ) in the range [0 , 1], find the first-degree polynomial, P 1 (x), that fits f (x) according to Chebyshev (L g102 ) norm. Solution 1 : Using Orthogonality of Chebyshev Polynomial: Expansion of f (x) in Chebyshev orthogonal polynomials gives f (x) g167g3g51 1 (x) = a 0 T 0 (x) + a 1 T 1 (x) Where T 0 (x) = 1 and T 1 (x) = x ME – 510 NUMERICAL METHODS FOR ME II g51g85g82g73g17g3g39g85g17g3g41g68g85g88g78g3g36g85g213g81g111 Fall 2007 There are two ways to determine the coefficients a 0 , and a 1 . The exact procedure is to use the orthogonality property of the Chebyshev functions and evaluate the appropriate integrals. The first step is to re-define the independent variable, x, such that the range of interest is the range of orthogonality of the Chebyshev functions: 1 - x 2 0 - 1 0 - 1 - x 2 a - b a - b - x 2 z g32 g32 g32 2 1 z x g14 g32 e F(z) 1) (z - g14 g32 - 1 g116 z g116 + 1 The expansion becomes: F (z) = exp[- (z + 1)] g167g3g51 1 (z) = A 0 T 0 (z) + A 1 T 1 (z) Where T 0 (x) = 1 and T 1 (x) = x
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