Accumulation of Round-Off Error in Fast Fourier Transforms
Journal of the ACM (JACM)
Algorithm 424: Clenshaw-Curtis quadrature [D1]
Communications of the ACM
Implementing Clenshaw-Curtis quadrature, I methodology and experience
Communications of the ACM
Certification of Algorithm 279: Chebyshev quadrature
Communications of the ACM
Numerical Analysis: A fast fourier transform algorithm for real-valued series
Communications of the ACM
Algorithm 279: Chebyshev quadrature
Communications of the ACM
The fast staggered transform, composite symmetries, and compact symmetric algorithms
IBM Journal of Research and Development
Algorithm 424: Clenshaw-Curtis quadrature [D1]
Communications of the ACM
Implementing Clenshaw-Curtis quadrature, I methodology and experience
Communications of the ACM
Uniform approximations to finite Hilbert transform and its derivative
Journal of Computational and Applied Mathematics - Special issue on proceedings of the international symposium on computational mathematics and applications
Quadrature rule for indefinite integral of algebraic-logarithmic singular integrands
Journal of Computational and Applied Mathematics
Quadrature rule for Abel's equations: Uniformly approximating fractional derivatives
Journal of Computational and Applied Mathematics
On the Check of Accuracy of the Coefficients of Formal Power Series
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Uniform approximation to fractional derivatives of functions of algebraic singularity
Journal of Computational and Applied Mathematics
Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants
ACM Transactions on Mathematical Software (TOMS)
An approximation method for high-order fractional derivatives of algebraically singular functions
Computers & Mathematics with Applications
Simulation-based optimal Bayesian experimental design for nonlinear systems
Journal of Computational Physics
Fast construction of Fejér and Clenshaw-Curtis rules for general weight functions
Computers & Mathematics with Applications
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In a companion paper to this, “I Methodology and Experiences,” the automatic Clenshaw-Curtis quadrature scheme was described and how each quadrature formula used in the scheme requires a cosine transformation of the integrand values was shown. The high cost of these cosine transformations has been a serious drawback in using Clenshaw-Curtis quadrature. Two other problems related to the cosine transformation have also been troublesome. First, the conventional computation of the cosine transformation by recurrence relation is numerically unstable, particularly at the low frequencies which have the largest effect upon the integral. Second, in case the automatic scheme should require refinement of the sampling, storage is required to save the integrand values after the cosine transformation is computed.This second part of the paper shows how the cosine transformation can be computed by a modification of the fast Fourier transform and all three problems overcome. The modification is also applicable in other circumstances requiring cosine or sine transformations, such as polynomial interpolation through the Chebyshev points.