= Chebyshev–Gauss quadrature
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{title2=$x_k=\cos((2k-1)\pi/(2n)),\quad a_k=\pi/n$}
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= Gauss–Chebyshev quadrature
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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
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