 Moment matching

For a measure $\mu$ with moments $m_j=\int x^{j}d\mu (x)$, a quadrature rule $Q(f)={\sum }_i{w}_{i}f(x_i)$ is exact through polynomial degree $d$ precisely whenNecessity follows by testing each monomial; sufficiency follows by linearity for any polynomial. These equations turn an integration requirement into algebraic constraints on the quadrature nodes and quadrature weights.

For ordinary integration on $[-1,1]$, the first moments are $m_0=2$, $m_1=0$, $m_2=2/3$ and $m_3=0$. Symmetric nodes $\pm a$ with equal quadrature weights $w$ therefore require $2w=2$ and $2w{a}^2=2/3$, giving $w=1$ and $a=1/\sqrt{3}$. Symmetry supplies the odd moments, producing the two-node Gaussian quadrature rule exact through degree three.