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! CC_ABSCISSAS computes the Clenshaw Curtis abscissas.
Discussion:
The interval is [ -1, 1 ].
The abscissas are the cosines of equally spaced angles between
180 and 0 degrees, including the endpoints.
X(I) = cos ( ( ORDER - I ) * PI / ( ORDER - 1 ) )
except for the basic case ORDER = 1, when
X(1) = 0.
If the value of ORDER is increased in a sensible way, then
the new set of abscissas will include the old ones. One such
sequence would be ORDER(K) = 2*K+1 for K = 0, 1, 2, ...
When doing interpolation with Lagrange polynomials, the Clenshaw Curtis
abscissas can be a better choice than regularly spaced abscissas.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
04 December 2007
Author:
John Burkardt
Reference:
Charles Clenshaw, Alan Curtis,
A Method for Numerical Integration on an Automatic Computer,
Numerische Mathematik,
Volume 2, Number 1, December 1960, pages 197-205.
Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
Joerg Waldvogel,
Fast Construction of the Fejer and Clenshaw-Curtis Quadrature Rules,
BIT Numerical Mathematics,
Volume 43, Number 1, 2003, pages 1-18.
Parameters:
Input, integer ( kind = 4 ) N, the order of the rule.
Output, real ( kind = 8 ) X(N), the abscissas.
| Type | Intent | Optional | Attributes | Name | ||
|---|---|---|---|---|---|---|
| integer(kind=4) | :: | n | ||||
| real(kind=8) | :: | x(n) |