Gaussian quadrature

An $n$-node Gaussian quadrature rule uses orthogonal-polynomial roots and integrates every polynomial of degree at most $2n-1$ exactly.

For weight $w(x)=x$ on $[0,1]$, the two-node Gaussian quadrature nodes are roots of $x^2-6x/5+3/10$, the monic quadratic orthogonal to all linear polynomials. Its weights come from integrating the Lagrange interpolation cardinal functions. Orthogonality gives exactness on cubic polynomials and positive weights.

For integration against $\sin x$ on $[0,\pi ]$, the monic quadratic orthogonal polynomial is $x^2-\pi x+2$. Its two roots are the Gaussian quadrature nodes $c_{1,2}=\pi /2\mp \sqrt{{\pi }^2/4-2}$, with weights one. The rule integrates every cubic exactly: divide it by the orthogonal quadratic and integrate the degree-one remainder.

For an $n$-node Gaussian quadrature rule exact through degree $2n-1$, the cardinal Lagrange interpolation polynomial ${\ell }_j$ has square of degree $2n-2$. Exactness gives $A_j=\int {\ell }_j^2>0$ for a positive integration weight. This proof turns polynomial exactness directly into positivity.

Gauss-Hermite quadrature uses the roots of a degree-$N$ Hermite polynomial as nodes for an integral with Gaussian weight. Its positive interpolation weights make it exact for every polynomial of degree at most $2N-1$.