References

The theory summarised in Numerical Quadrature and the derivations in Quadrature Rules follow the sources listed below.

For the Chebyshev-based rules, Trefethen [1] is the natural starting point: it settles the question of how Clenshaw-Curtis compares with Gauss quadrature in practice and gives the taxonomy of the three Chebyshev variants. The construction of the Clenshaw-Curtis weights used in this package follows Reid [15]; Waldvogel [18] gives the fast $O(n \log n)$ alternative. Davis and Rabinowitz [19] is the standard monograph on numerical integration, and Trefethen [20] covers the approximation theory that the convergence results rest on.

The tanh-sinh rule goes back to Takahasi and Mori [4], who introduced the double-exponential transformation; Mori and Sugihara [5] surveys it and its relatives, and Mori [6] tells the story of its discovery. Trefethen and Weideman [3] explains why the trapezoidal rule underlying it is spectrally accurate on the real line, and Bailey et al. [7] documents its role as the standard scheme for high-precision quadrature.