Implementing Clenshaw-Curtis quadrature, I methodology and experience

Abstract

Clenshaw-Curtis quadrature is a particularly important automatic quadrature scheme for a variety of reasons, especially the high accuracy obtained from relatively few integrand values. However, it has received little use because it requires the computation of a cosine transformation, and the arithmetic cost of this has been prohibitive. This paper is in two parts; a companion paper, “II Computing the Cosine Transformation,” shows that this objection can be overcome by computing the cosine transformation by a modification of the fast Fourier transform algorithm. This first part discusses the strategy and various error estimates, and summarizes experience with a particular implementation of the scheme.

Summation Paths in Clenshaw-Curtis Quadrature

“…Efficient algorithms for particular n values have been reported long time ago [2,5] by taking advantage of the fast Fourier transform (FFT) approach. The analysis done in section 3 unveils the occurrence of a general mathematical structure consisting of families of coefficients ordered into complete binary trees.…”

Disentangling Complexity in Bayesian Automatic Adaptive Quadrature

“…In the frame of the BAAQ, the selection of the CC quadrature sums for the approximation of the Riemann integrals (1) over macroscopic integration ranges is motivated by several features of the spanning Chebyshev polynomials: closeness to the polynomials of the best approximation, symmetry properties, easiness of the analytic computation of the quadrature weights for different weight functions [4,9,[11][12][13].…”

Fast Construction of the Fejér and Clenshaw–Curtis Quadrature Rules

“…In fact, already in 1972 W.M. Gentleman [6] implemented the Clenshaw-Curtis rule with n + 1 nodes by means of a discrete cosine transformation, which has to be carried out anew at every instance of quadrature, however. Recently, a direct computation (once for all) of the Clenshaw-Curtis weights by means of DFTs of order 2n was submitted to The Mathworks Central File Exchange by G. von Winckel [13].…”