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Unformatted text preview: Gaussian Quadrature James Keesling 1 Quadrature Using Points with Unequal Spacing In Newton-Cotes 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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