GaussianQuadrature

GaussianQuadrature - Gaussian Quadrature James Keesling 1...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 5

GaussianQuadrature - Gaussian Quadrature James Keesling 1...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online