Clenshaw–Curtis quadrature

Summary

Clenshaw–Curtis quadrature integrates a function on $[-1, 1]$ by interpolating it at the points $\cos(k\pi/N)$ and integrating the interpolating polynomial exactly. The substitution $x = \cos\theta$ turns the problem into a cosine series, so the weights can be computed fast with the discrete cosine transform. Its nodes for $N$ are contained in its nodes for $2N$, so doubling $N$ reuses all earlier function values. For many smooth functions it is about as accurate as Gauss–Legendre quadrature with the same number of points.