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N18 - ME 510 NUMERICAL METHODS FOR ME II Numerical...

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ME 510 NUMERICAL METHODS FOR ME II Prof. Dr. Faruk Arınç Fall 2010 Newton - Cotes Formulae Trapezoidal Rule Simpson's One-third Rule Simpson's Three-eights Rule g11 g12 n 0 x n 0 1 2 n-1 h f (x) d x f 2 f 2 f ... 2 f f 2 x g35 g14 g14 g14 g14 g14 g179 3 ' ' n 0 2 - (x - x ) E (x) f ( ) 12 n g91 g32 g11 g12 n 1 - n 2 - n 2 1 0 x f f 4 f 2 ... f 2 f 4 f 3 h x d (x) f n 0 g14 g14 g14 g14 g14 g14 g35 g179 x 5 (4) n 0 4 - (x - x ) E (x) f ( ) 180 n g91 g32 g11 g12 n 0 x n 0 1 2 3 n-3 n-2 n-1 3 h f (x) d x f 3 f 3 f 2 f ... 2 f 3 f 3 f f 8 x g35 g14 g14 g14 g14 g14 g14 g14 g14 g179 Numerical Integration ME 510 NUMERICAL METHODS FOR ME II Prof. Dr. Faruk Arınç Fall 2010 Choose x i ’s such that, the sum gives the integral Gauss - Legendre Quadrature Method b n i i 0 a I f (x) dx w f (x ) i g32 g32 g35 g166 g179 n + 1 values of w i n + 1 values of f (x i ) => There are 2 n + 2 parameters which define a polynomial of degree 2 n + 1 ) (x f w n 0 i i i g166 g32 b a f (x) dx g179 exactly when f (x) is a polynomial of degree 2 n + 1.
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ME 510 NUMERICAL METHODS FOR ME II Prof. Dr. Faruk Arınç Fall 2010 Replace f(x) by a polynomial of degree n b b b n a a a I
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