ClenshawCurtisi

Clenshaw–Curtis quadrature is a numerical integration method used to approximate definite integrals. On a standard interval [-1,1], it computes the integral of a function f by a weighted sum ∑_{j=0}^n w_j f(x_j), where the nodes are x_j = cos(j π / n) for j = 0,...,n. These x_j are Chebyshev–Lobatto points, and the method extends to a general interval [a,b] by transforming x = (b−a)/2 t + (a+b)/2 and applying the rule to t ∈ [−1,1].

Weights w_j can be determined efficiently via the discrete cosine transform, by deriving the cosine-series coefficients of a simple endpoint function. Modern implementations use fast Fourier transform techniques to achieve O(n log n) time. The weights are nonnegative, and when n is a power of two the node set is nested, a property that is advantageous for adaptive schemes and multi-resolution schemes.

In terms of accuracy, Clenshaw–Curtis quadrature exhibits spectral (exponential) convergence for analytic functions and algebraic convergence for smoother but non-analytic functions. Empirically, its accuracy is often comparable to Gauss–Legendre quadrature with roughly half as many nodes, and it can outperform Gauss rules for functions with endpoint behavior that aligns with Chebyshev points.

Extensions include tensor-product constructions for higher dimensions and variants that emphasize different endpoint distributions. Clenshaw–Curtis is widely used in spectral methods and high-accuracy numerical integration due to its balance between accuracy, efficiency, and ease of implementation.