Implementing Clenshaw-Curtis quadrature, II computing the cosine transformation

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Abstract
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.