Implementing Clenshaw-Curtis Quadrature,
II Computing the Cosine Transformation
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 trouble some.  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.
CACM May, 1972
Gentleman, W. M.
