Gaussian quadrature

A quadrature rule that chooses BOTH where to sample and how much weight to give each sample, to maximize accuracy for a fixed number of evaluations -- unlike a composite rule's fixed, equally spaced grid.

The $n$ -point Gauss-Legendre quadrature rule samples $f$ at the $n$ roots of the degree- $n$ Legendre polynomial on $[-1,1]$ , each weighted by $w_i=\dfrac{2}{(1-x_i^2)[P_n'(x_i)]^2}$ , reaching degree of precision $2n-1$ -- the maximum possible for $n$ evaluations, and higher than any equally-spaced $n$ -point rule achieves. A general interval $[a,b]$ maps onto $[-1,1]$ by $x=\frac{b-a}{2}t+\frac{a+b}{2}$ .

Examples

2-point Gauss-Legendre quadrature samples at $\pm1/\sqrt3$ , reaching degree of precision 3 -- matching Simpson's rule's own degree of precision with one fewer function evaluation.