ClenshawCurtisQuadratur

Clenshaw Curtis quadrature is a class of numerical integration methods used to approximate definite integrals. These methods rely on interpolating the integrand with Chebyshev polynomials and then integrating the interpolating polynomial exactly. The core idea is to express the function $f(x)$ as a sum of Chebyshev polynomials: $f(x) \approx \sum_{k=0}^n c_k T_k(x)$, where $T_k(x)$ are the Chebyshev polynomials of the first kind and $c_k$ are the coefficients. These coefficients are determined by evaluating the function at specific points, often the Chebyshev nodes. Once the interpolating polynomial is found, its integral is computed analytically. The error in Clenshaw Curtis quadrature is typically of a lower order than Newton-Cotes methods for the same number of function evaluations, especially for smooth integrands. These methods are particularly well-suited for integrands that are analytic or have smooth behavior over the integration interval. Variations exist, including adaptive versions that adjust the number of points based on the function's behavior. The accuracy of the method depends on the degree of the Chebyshev polynomial used for interpolation and the smoothness of the function being integrated.