How do you Calculate the Points in Gaussian Quadrature?

In summary, the Gaussian Quadrature method for numerically integrating a function requires the values of the function at specific points, which are calculated using the zeros of the Legendre polynomial of the same order. The Newton method can be used to find these points by iteratively narrowing down their location using a recursive relation.
  • #1
ecastro
254
8
How do you calculate the necessary points in a function to numerically integrate it using the Gaussian Quadrature?

If I were to evaluate a function using two points, the Gaussian Quadrature needs the value of the function at ##\displaystyle{\pm \sqrt{\frac{1}{3}}}## with weights of unity. How did they calculate the ##\displaystyle{\pm \sqrt{\frac{1}{3}}}##?

I have looked through an example of how the points are located, and in this example, the point is estimated by dividing the length of the function into the number of required points (if the function is to be integrated with two points, the whole function is divided into three parts), and somehow narrow down the location of the point by using this relation:
\begin{equation*}
x - \frac{P_n\left(x\right)}{P_n'\left(x\right)}
\end{equation*}
where ##x## is the estimated point and the functions on the second term are the Legendre Polynomials. The result of this difference will be the new location of the point. This process is repeated until the result is almost zero.

If the above equation is acceptable, what is the concept behind the second term (the ratio between the Legendre Polynomials)?
 
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  • #2
The abscissas (or points) used in a Gauss-Legendre quadrature of order ##n## are the zeros of the Legendre polynomial of the same order, i.e. ##P_n(x)##. So, for order 2 they are the two roots of ##P_2(x)##. As you noted they are ##\pm \sqrt{1/3}##. The formula you are giving is the Newton method. That is, you can find the root of a function f(x) by using the recursive relation
##x_{i+1}=x_i-\frac{f(x_i)}{f'(x_i)}##,
starting from a guess ##x_0##. The method is derived by doing a Taylor expansion of ##f(x)## around ##x=x_i##.
 
  • #3
I see. Now it makes sense. Thank you for your help!
 

Related to How do you Calculate the Points in Gaussian Quadrature?

1. What is Gaussian Quadrature and why is it used?

Gaussian Quadrature is a numerical integration technique used to approximate the definite integral of a function. It is used because it provides a more accurate result compared to other numerical integration methods, especially for functions with highly oscillatory behavior.

2. How do I choose the number of points in Gaussian Quadrature?

The number of points in Gaussian Quadrature is determined by the degree of the polynomial that the method can exactly integrate. For example, if you want to integrate a polynomial of degree n, you will need n+1 points in Gaussian Quadrature.

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

Gaussian Quadrature uses specific weights and points to approximate the integral, whereas other methods use uniform intervals. This allows for a more accurate approximation, especially for functions with complex behavior.

4. How do you calculate the weights and points in Gaussian Quadrature?

The weights and points in Gaussian Quadrature are calculated using a mathematical formula involving the orthogonal polynomials associated with the integral. These polynomials can be found in tables or calculated using recurrence relations.

5. Can Gaussian Quadrature be used for integrals with infinite limits?

Yes, Gaussian Quadrature can be used for integrals with infinite limits by transforming the integral into one with finite limits. This can be done using a change of variables or by applying a transformation formula.

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