Recent Theory and Applications on Numerical Algorithms and Special Functions
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On Fast and Stable Implementation of ClenshawCurtis and FejérType Quadrature Rules
Abstract
Based upon the fast computation of the coefficients of the interpolation polynomials at Chebyshevtype points by FFT, together with the efficient evaluation of the modified moments by forward recursions or by Oliver’s algorithms, this paper presents fast and stable interpolating integration algorithms, by using the coefficients and modified moments, for ClenshawCurtis, Fejér’s first and secondtype rules for Jacobi weights or Jacobi weights multiplied by a logarithmic function. Numerical examples illustrate the stability, efficiency, and accuracy of these quadratures.
1. Introduction
The interpolation quadrature of the ClenshawCurtis rule as well as Fejértype formulas for has been extensively studied since Fejér [1, 2] in 1933 and Clenshaw and Curtis [3] in 1960, where the nodes are of Chebyshev types while the weights are computed by sums of trigonometric functions. When (), this quadrature is called Fejér’s firsttype rule. This kind of points is called the first kind of Chebyshev points, while Fejér’s secondtype rule is corresponding to the Filippi points () and the ClenshawCurtistype quadrature to the ClenshawCurtis points (the second kind of Chebyshev points) (). For more details, see Davis and Robinowitz [4], Sloan and Smith [5, 6], Sommariva [7], Trefethen [8], Waldvogel [9], and so forth.
In the case , a connection between the Fejér, ClenshawCurtis quadrature rules, and discrete Fourier transforms (DFTs) was given by Gentleman [10, 11], where the ClenshawCurtis rule is implemented with nodes by means of a discrete cosine transformation (DCT). An independent approach along the same lines, unified algorithms based on DFTs of order for generating the weights of the two Fejér rules and of the ClenshawCurtis rule, was presented in Waldvogel [9]. A streamlined MATLAB code is given as well in [9]. In addition, Clenshaw and Curtis [3], O’Hara and Smith [12], Trefethen [8, 13], Xiang and Bornemann in [14], Xiang et al. [15–17], and so forth showed that the Gauss, ClenshawCurtis, and Fejér quadrature rules are about equally accurate.
In this paper, we focus the attention on the weight functions and For these two weight functions, the ClenshawCurtistype quadrature has been extensively studied in a series of papers of Piessens [18, 19] and Piessens and Branders [20–23], by using Chebyshev interpolant of at the ClenshawCurtis points together with the modified moments [24]: where is the Chebyshev polynomial of degree and can be efficiently computed by FFT [8, 10, 11] which is widely used for the approximation of highly oscillatory integrals such as [22, 25–30]. The modified moments satisfy the following recurrence formulas for Jacobi weights or Jacobi weights multiplied by [20].(i)For , by using Fasenmyer’s technique, the recurrence formula for the evaluation of the modified moments is with The forward recursion is numerically stable [20], except in two cases: (ii)For , for the recurrence formula [20] is with where is the Beta function and is the Psi function (see Abramowitz and Stegun [31]). The forward recursion is as numerically stable as (4) except for (6) or (7) [20].
Thus, the modified moments can be fast computed by the forward recursions (4) and (9) except the two cases (6) and (7), and the total costs for are operations.
However, in case (6) or (7), the accuracy of the forward recursion is catastrophic particularly when or and (see Table 1). In case (6) the relative errors of the computed values obtained by the forward recursion behave approximately as and in case (7) as
In this paper, we will consider interpolation approaches for ClenshawCurtis rules as well as Fejér’s first and secondtype formulas for with or , which can be efficiently calculated in operations. Computing the modified moments in cases (6) and (7) by Oliver’s algorithm [32] or Lozier’s algorithm [33] with one starting value and one end value, as well as offering the very short codes for the evaluation of the coefficients by FFT, is the topic of this paper.
This paper is organized as follows. In Section 2.1, we studied the asymptotic expansions of the modified moments . Based on the results of the asymptotic expansions, we discussed Oliver’s algorithm for the modified moments when the recursion is unstable in Section 2.2. Moreover, we gave some test to verify the accuracy and the CPU time of Oliver’s algorithm. In Section 2.3, the concise MATLAB codes for evaluation of the coefficients by FFT are presented. The efficiency and accuracy of the three quadratures are illustrated in Section 3.
2. Computation of the Modified Moments and the Coefficients of the Interpolation Polynomials
From (4) and (9), we see that if can be efficiently and stably computed then can be so. Note that the asymptotic behaviour of two linearly independent solutions of (4) satisfies (see Denef and Piessens [34] and Piessens and Branders [20]). Consequently, the forward recursions for (4) and (9) are perfectly stable except cases (6) and (7). In these two cases, both the forward recursions and backward recursions for (4) and (9) are numerically unstable. In the following, we will consider Oliver’s algorithms to evaluate modified moments based on their asymptotic formulas of the moments with one starting value and one end value for these two cases.
2.1. Asymptotic Expansions of the Modified Moments
Lemma 1 (Erdélyi [35]). If , , and is times continuously differentiable for , then where
Lemma 2. If , , and is times continuously differentiable for , then where
Proof. The proof can be directly derived from the proof of Lemma 1 and the proof of the Theorem 5 in Erdélyi [36].
Theorem 3. If , then where
Proof. For , taking in (3), we have
where
Consequently, the desired result can be derived by applying (16) to (22).
The above outcome can be extended to the case of , , since can be written as
where denotes the largest integer less than or equal to .
Theorem 4. If , then where
Proof. Letting in (8) and using , we have Applying Lemmas 1 and 2 to the two integrals on the right hand side of (27), respectively, leads to the desired result.
Remark 5. Piessens and Branders gave the first term of asymptotic expansion for and in [20] using the asymptotic theory of Fourier coefficients (Lighthill [37]). Furthermore, Piessens [18] presented the asymptotic expansion with explicit formulas for the first three terms for , which is equivalent to (20) except a minor clerical error. In [18], should be modified by . The first term of (25) is different from the first asymptotic term for proposed in [18]; that is, . In order to ensure the accuracy of the modified moments evaluated by Oliver’s method, the end value of modified moments must be computed very precisely. In this paper, we use the first fourterm truncation.
2.2. Oliver’s Algorithm
Letwhere “” denotes the transpose; then the modified moments can be solved by where is computed by hypergeometric function [20] when : If , is computed by using the asymptotic expression (20) with the first fourterm truncation. Particularly, Oliver’s algorithm can be fast implemented by applying LU factorization (chasing method) with operations.
In addition, for the weight in case of (6) or (7), the computation of the modified moments can be fixed up by Oliver’s algorithm similar to (29) with one starting and one end . Consequently, the modified moments can be solved by where The end value can be calculated by its asymptotic formula (25) with the first fourterm truncation.
Tables 2 and 3 show the accuracy of Oliver’s algorithm for and , and , respectively. Table 4 displays the accuracy of Oliver’s algorithm for and , and for . Table 5 displays the accuracy of Oliver’s algorithm for and , and for . Table 6 shows the CPU time for implementation of the two Oliver’s algorithms. From these tables, we see that Oliver’s algorithms have high efficiency and are precise. The MATLAB codes on Oliver’s algorithms and all the MATLAB codes in this paper can be downloaded from [38].



