Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    Thanks
    Matt
     
    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.

    Thanks
    Matt
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Numerical Integration: Gaussian Quadrature
  1. Gaussian Quadrature` (Replies: 3)

Loading...