Three point Gaussian Quadrature formula derivation

Question

For 3-point Gaussian quadrature, I'm not sure how the $5/9$ and $8/9$ coefficients are found. I am able to derive $x0, x1, x2$ in $g(x0), g(x1), g(x2)$ but I'm not sure how to get the rest.

Answer 1

Let $a,b,c$ be the weights at $-\sqrt{3/5},0,\sqrt{3/5}$ respectively. These should be chosen so that $x^n$ is integrated exactly for $n=0,1,2,\dots$ as far as possible. So:

• $n=0$ yields $a+b+c=2$
• $n=1$ yields $a=c$; this also takes care of all other odd $n$
• $n=2$ yields $2a(3/5)=2/3$, hence $a=5/9$.

This determines the weights. After that, the orthogonality of Legendre polynomials kicks in to deliver exact integration of $x^4$ at no additional charge.