Chebyshev–Gauss quadrature ($x_k=\cos ((2k-1)\pi /(2n)),a_k=\pi /n$)

The first-kind rule approximates ${\int }_{-1}^{1}f(x)(1-x^2{)}^{-1/2}dx$ at the zeros of the Chebyshev polynomial $T_n$. It integrates all polynomials of degree at most $2n-1$ exactly. The weights follow by integrating the Lagrange interpolation polynomial at those nodes. Orthogonality kills the quotient when a degree-$2n-1$ polynomial is divided by $T_n$; its remainder is recovered by interpolation. No $n$-node rule can integrate every degree-$2n$ polynomial, since the squared nodal polynomial has zero quadrature sum and strictly positive weighted integral.

// Target: probability-and-statistics.bigb