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Unformatted text preview: Gaussian Quadrature James Keesling 1 Quadrature Using Points with Unequal Spacing In NewtonCotes Integration we used points that were equally spaced. However, there was no need for the points to have any special spacing. If we wish to estimate the integral Z b a f ( x ) dx and have any set of points { x ,x 1 ,...,x n } , then we can estimate the integral by the formula Z b a f ( x ) dx A f ( x ) + A 1 f ( x 1 ) + + A n f ( x n ) . We can solve for the constants { A ,A 1 ,...,A n } by making the formula exact for the functions f ( x ) = 1 ,x,x 2 ,x 3 ,...,x n . This will give us n + 1 equations that we can use to solve for the constants { A ,A 1 ,A 2 ,...,A n } . In Gaussian Quadrature we use the interval [ 1 , 1] as the standard and the points { x ,x 1 ,...,x n } will all be contained in this interval. There is a matrix equation for the normalized constants { a ,a 1 ,...,a n } . M = Vandermonde([ x ,x 1 ,...,x n ]) = 1 x x 2 x n 1 x 1 x 2 1 x n 1 1 x 2 x 2 2 x n 2 . . . . . . . . . . . . 1 x n x 2 n x n n Let M T be the transpose of M . Let A be the column vector with entries a i . Let B be a column vector with entries b i = Z 1 1 x i dx = 1 ( 1) i +1 i + 1 . Then we get M T A = B and solving for A we get the following. A = ( M T ) 1 B 1 2 Choosing the Points We now have flexibility to choose the points { x ,x 1 ,...,x n } in a way that will make the estimate of the integral even more accurate. The theory behind the choice of points involvesestimate of the integral even more accurate....
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 Spring '07
 JURY
 Calculus

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