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Numerical Integration: Gaussian Quadrature

  1. Sep 11, 2009 #1
    [tex]\int^{1}_{-1}f(x)dx = \sum^{n}_{j=-n}a_{j}f(x_{j}) [/tex]

    Why does [tex]\sum_{j}a_{j} = 2 [/tex] ?

    I know that the aj's are weights, and in the case of [-1,1], they are calculated using the roots of the Legendre polynomial, but I don't understand why they all add up to 2.
  2. jcsd
  3. Sep 11, 2009 #2
    I believe that they add up to 2 because they are symmetric about the midpoint of the integration range of [-1,1]. For instance if you used a sampling of 10 points (on the same interval of integration) 5 would be positive and 5 would be negative. If you summed up the 5 positive points you would you arrive at 2.0

    Last edited: Sep 11, 2009
  4. Sep 11, 2009 #3
    yeah when graphed out weights vs. abscissa, it looks like the attached photo, which is symmetric.

    Attached Files:

  5. Sep 12, 2009 #4
    For more information on Gaussian Quadrature see,

    "Gaussian Quadrature Formulas" by Stroud and Secrest.

    This was printed in 1966 but it is still accurate for today.

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