# N13 - ME 510 NUMERICAL METHODS FOR ME II Chebyshev Fitting...

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ME – 510 NUMERICAL METHODS FOR ME II gG±²³´µG³´¶·G¸¹´ºG»¼½ Fall 2007 Chebyshev Fitting Given: f (x) in [a , b] Problem : Find the best fitting, single degree polynomial with small degree, P n (x), that represents f (x) in the whole range, [a , b]. Solution : Change variable from x in [a , b] to z in [1 , +1], and f (x) to F(z) : a - b b - a - x 2 = z Minimize E (z) : ) ( F ! 1) + (n ) z - (z ... ) z - (z ) z - (z = (z) P - (z) F = (z) E 1) + (n n 1 0 n g ME – 510 NUMERICAL METHODS FOR ME II gG±²³´µG³´¶·G¸¹´ºG»¼½ Fall 2007 ! 1) (n ) ( F ) z - (z (z) E n 0 i 1) (n i G ± ² ² g ± Minimize E(z) means to find z 0 , z 1 , …, z n , which are the cross over points between F(z) and P n (z) where the error is exactly zero, such that the product, (z - z 0 ) (z - z 1 ) … (z -z n ) is minimized. Chebyshev proved the following: The roots of the (n+1)st Chebyshev polynomial, T n+1 (z), gives the cross- over (z) points:

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N13 - ME 510 NUMERICAL METHODS FOR ME II Chebyshev Fitting...

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