Lecture 37

# Lecture 37 - Gauss Quadratures change of variables so that...

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Gauss Quadratures change of variables so that the interval of integration is [ - 1,1] to simplify the math select functional values at non- uniformly distributed points to achieve higher accuracy

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Gauss Quadrature on [ a , b ] Coordinate transformation from [ a , b ] to [ - 1,1] t 2 t 1 a b - - = - + + - = 1 1 d d 1 1 d d b a dx x g dx 2 a b 2 b a x 2 a b f dx x f ) ( ) )( ( ) ( 2 2 1 1 d d d b a a b x x x x a x x b - + = + = - ⇒ = = = x
Gauss Quadrature on [ - 1, 1] Choose ( c 0 , c 1 , x 0 , x 1 ) such that the method 0 1 2 3 1 0 0 1 1 1 f ( )dx c (x ) c f (x ) x f - = + x 1 x 0 -1 1

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Gauss Quadrature on [ - 1, 1] Exact integral for f = x 0 , x 1 , x 2 , x 3 Four equations for four unknowns 1 0 0 1 1 1 f (x)dx c f (x ) c f (x ) - = + = - = = = + = = = + = = = + = = = + = = = - - - - 3 1 x 3 1 x 1 c 1 c x c x c 0 dx x x f x c x c 3 2 dx x x f x c x c 0 xdx x f c c 2 dx 1 1 f 1 0 1 0 3 1 1 3 0 1 1 0 3 3 2 1 1 2 0 1 1 0 2 2 1 1 0 1 1 0 1 1 1 0 ) 3 1 ( f ) 3 1 ( f dx ) x ( f I 1 1 + - = = -
Gauss Quadrature on [ - 1, 1] Choose ( c 0 , c 1 , c 2 , x 0 , x 1 , x 2

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