= Two-node Gaussian quadrature with quadratic weight
{title2=$\int_{-1}^1(1-x^2)f(x)dx\simeq\frac23[f(-1/\sqrt5)+f(1/\sqrt5)]$}
The even weight $1-x^2$ has moments $m_0=4/3$ and $m_2=4/15$. Its monic quadratic orthogonal polynomial is $x^2-1/5$, so the two <Gaussian quadrature> nodes are $\pm1/\sqrt5$. Symmetry and constant exactness give weights $2/3$ each. These nodes and weights integrate every polynomial of degree at most three exactly. Degree four fails: the true integral is $4/35$, whereas the rule gives $4/75$.
Back to article page
