On Fast Implementation of Clenshaw-Curtis and Fej\'{e}r-type Quadrature Rules

TL;DR

This paper introduces fast algorithms for Clenshaw-Curtis and Fejér quadrature rules using FFT-based coefficient computation and efficient moment evaluation, demonstrating comparable accuracy to Gauss-Jacobi quadrature with improved efficiency.

Contribution

It presents novel fast implementation methods for Clenshaw-Curtis and Fejér quadrature rules using FFT and recursive algorithms, with accompanying Matlab codes.

Findings

Quadrature rules achieve similar convergence rates as Gauss-Jacobi quadrature.

Algorithms are stable and accurate for functions with finite regularity.

Methods are more CPU-efficient than traditional Gauss quadrature computations.

Abstract

Based upon the fast computation of the coefficients of the interpolation polynomials at Chebyshev-type points by FFT, DCT and IDST, respectively, together with the efficient evaluation of the modified moments by forwards recursions or by the Oliver's algorithm, this paper presents interpolating integration algorithms, by using the coefficients and modified moments, for Clenshaw-Curtis, Fej\'er's first and second-type rules for Jacobi or Jacobi weights multiplied by a logarithmic function. The corresponding {\sc Matlab} codes are included. Numerical examples illustrate the stability, accuracy of the Clenshaw-Curtis, Fej\'{e}r's first and second rules, and show that the three quadratures have nearly the same convergence rates as Gauss-Jacobi quadrature for functions of finite regularities for Jacobi weights, and are more efficient upon the cpu time than the Gauss evaluated by fast computation of the weights and nodes by {\sc Chebfun}.