On Fast and Stable Implementation of Clenshaw-Curtis and Fejér-Type Quadrature Rules

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, Fejér's first and second-type rules for Jacobi or Jacobi weights multiplied by a logarithmic function. The corresponding Matlab codes are included. Numerical examples illustrate the stability, accuracy of the Clenshaw-Curtis, Fejér'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 Chebfun.

A spectral method for nonlocal diffusion operators on the sphere

“…where 3 F 2 is a generalized hypergeometric function [33, §16.2.1] and B is the beta function [33, §5.12.1]. Using Sister Celine's technique [34, §127] or induction [35], a recurrence relation can be derived for the modified moments…”

On the Convergence Rate of Clenshaw–Curtis Quadrature for Jacobi Weight Applied to Functions with Algebraic Endpoint Singularities

“…For these two cases, the modified moments are stably computed by Oliver algorithm in [25] with one starting values and one end value to compute the modified moments. In this paper we consider…”

“…For the n-point Clenshaw-Curtis quadrature rule (6), the coefficients b k of the interpolant P n (x) is evaluated by FFT, and the moments M k , except in two cases (11) and (12), are computed by the forward recursion (9), which is perfectly numerically stable. For these two cases, the moments M k are calculated by the Oliver algorithm in [25]. While for the Gauss-Jacobi quadrature, we cite [x, w] = jacpts(n, α, β) in Chebfun [30], which costs O(n) operations for n-point Gauss-Jacobi quadrature.…”

On the use of Hahn’s asymptotic formula and stabilized recurrence for a fast, simple and stable Chebyshev–Jacobi transform

“…( 3.8) It is known that for α > β and α = − 1 2 + N 0 or for α < β and β = − 1 2 + N 0 , neither forward nor backward recurrence is stable. This has been addressed by Xiang et al (2014) by transforming the initial value problem into a boundary value problem with a sufficiently accurate asymptotic expansion for µ (α,β ) N and subsequent use of Oliver's algorithm (see Oliver (1968) where p N (x) is defined by (1.1), and where A α,β n is given by (3.2). If we let the vector [s 2N ] n = (A α,β n ) −1 for n = 0, .…”

“…where 3 F 2 is a generalized hypergeometric function (Olver et al, 2010, §16.2.1). Using Sister Celine's technique (Rainville, 1960, §127) or induction (see Xiang et al (2014)), a recurrence relation can be derived for the modified moments:…”