Numerical Integration: Gaussian Quadrature

In summary, the conversation is discussing the reason why the sum of aj's is equal to 2. The speaker explains that the aj's are weights calculated using the roots of the Legendre polynomial and that they add up to 2 because they are symmetric about the midpoint of the integration range. The conversation also mentions the use of Gaussian Quadrature formulas to calculate these weights.
  • #1
Somefantastik
230
0
[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.
 
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  • #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:
  • #3
yeah when graphed out weights vs. abscissa, it looks like the attached photo, which is symmetric.
 

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  • Gaussian Weights 300.jpg
    Gaussian Weights 300.jpg
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  • #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
 

Related to Numerical Integration: Gaussian Quadrature

1. What is numerical integration and why is it used?

Numerical integration involves approximating the value of an integral by using numerical methods. It is used when the integral cannot be solved analytically or when the function is too complex to integrate by hand.

2. What is Gaussian Quadrature and how does it work?

Gaussian Quadrature is a numerical integration method that uses a weighted sum of function values at specific points to approximate the integral. The points and weights are chosen in a way that minimizes the error of the approximation.

3. What is the difference between Gaussian Quadrature and other numerical integration methods?

Gaussian Quadrature is more accurate than other numerical integration methods, such as the Trapezoidal rule or Simpson's rule. This is because it uses a higher number of function evaluations and carefully chosen points and weights to minimize the error.

4. How do you determine the number of points to use in Gaussian Quadrature?

The number of points used in Gaussian Quadrature is determined by the degree of the polynomial that the method can exactly integrate. For example, a Gaussian Quadrature with n points can exactly integrate polynomials of degree 2n-1.

5. Can Gaussian Quadrature be used for any type of integral?

No, Gaussian Quadrature is most effective for integrals where the integrand is smooth and well-behaved. It may not work well for integrals with singularities or oscillatory behavior. In these cases, other numerical integration methods may be more suitable.

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