QUADRATURE_RULES_CLENSHAW_CURTIS
Rules for Approximate Integration

QUADRATURE_RULES_CLENSHAW_CURTIS is a dataset directory which contains examples of Clenshaw Curtis quadrature rules for the interval [-1,+1]. A quadrature rule is a set of n points x and associated weights w so that the integral of a function f(x) over some particular region can be approximated by:

Integral f(x) dx = Sum ( 1 <= i <= n ) w(i) * f(x(i))

For this directory, a quadrature rule is stored as three files, containing the weights, the points, and a file containing two points defining the corners of the rectangular region. The dimension of the region is deduced implicitly from the dimension of the points.

Example:

Here are the files for a Clenshaw Curtis rule of order 9:

Here is the text of an "W" file storing the weights of such a rule

  0.1587301587301588E-01
  0.1462186492160182
  0.2793650793650794
  0.3617178587204897
  0.3936507936507936
  0.3617178587204898
  0.2793650793650794
  0.1462186492160182
  0.1587301587301588E-01
      

Here is the text of a "X" file storing the abscissas of such a rule:

 -1.0000000000000000
 -0.9238795325112867
 -0.7071067811865475
 -0.3826834323650897
   0.000000000000000
  0.3826834323650898
  0.7071067811865475
  0.9238795325112867
  1.0000000000000000
      

Here is the text of an "R" file storing the lower and upper limits of the region, needed to determine the integration region:

 -1.0000000000000000
  1.0000000000000000
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Related Data and Programs:

CLENSHAW_CURTIS_RULE, a C++ program which defines a Clenshaw Curtis quadrature rule.

SANDIA_RULES, a C++ library which produces 1D quadrature rules of Chebyshev, Clenshaw Curtis, Fejer 2, Gegenbauer, generalized Hermite, generalized Laguerre, Hermite, Jacobi, Laguerre, Legendre and Patterson types.

SPARSE_GRID_CC, a dataset directory which contains multidimensional Smolyak sparse grids based on the 1D Clenshaw Curtis rule;

TOMS424, a FORTRAN77 library which estimates the integral of a function using Clenshaw-Curtis quadrature.

Sample Files:

CC_O1, Clenshaw Curtis rule of order 1:

• cc_o1_x.txt, the abscissas.
• cc_o1_w.txt, the weights.
• cc_o1_r.txt, the range.

CC_O2, Clenshaw Curtis rule of order 2:

• cc_o2_x.txt, the abscissas.
• cc_o2_w.txt, the weights.
• cc_o2_r.txt, the range.

CC_O3, Clenshaw Curtis rule of order 3:

• cc_o3_x.txt, the abscissas.
• cc_o3_w.txt, the weights.
• cc_o3_r.txt, the range.

CC_O4, Clenshaw Curtis rule of order 4:

• cc_o4_x.txt, the abscissas.
• cc_o4_w.txt, the weights.
• cc_o4_r.txt, the range.

CC_O5, Clenshaw Curtis rule of order 5:

• cc_o5_x.txt, the abscissas.
• cc_o5_w.txt, the weights.
• cc_o5_r.txt, the range.

CC_O9, Clenshaw Curtis rule of order 9:

• cc_o9_x.txt, the abscissas.
• cc_o9_w.txt, the weights.
• cc_o9_r.txt, the range.

CC_O17, Clenshaw Curtis rule of order 17:

• cc_o17_x.txt, the abscissas.
• cc_o17_w.txt, the weights.
• cc_o17_r.txt, the range.

CC_O33, Clenshaw Curtis rule of order 33:

• cc_o33_x.txt, the abscissas.
• cc_o33_w.txt, the weights.
• cc_o33_r.txt, the range.

CC_O65, Clenshaw Curtis rule of order 65:

• cc_o65_x.txt, the abscissas.
• cc_o65_w.txt, the weights.
• cc_o65_r.txt, the range.

You can go up one level to the DATASETS page.