Quadrature Rules

These are some examples of quadrature rules. 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:

A 2D quadrature rule for the [-1,1] square can be formed by using the product rule approach, based on a 1D Gauss-Legendre formula of order 3. The resulting product rule is of order 9.

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

   0.3086419753086420    
   0.4938271604938272    
   0.3086419753086420    
   0.4938271604938272    
   0.7901234567901235    
   0.4938271604938272    
   0.3086419753086420    
   0.4938271604938272    
   0.3086419753086420  
      

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

  -0.7745966692414835      -0.7745966692414835    
  -0.7745966692414835       0.0000000000000000    
  -0.7745966692414835       0.7745966692414835    
   0.0000000000000000      -0.7745966692414835    
   0.0000000000000000       0.0000000000000000    
   0.0000000000000000       0.7745966692414835    
   0.7745966692414835      -0.7745966692414835    
   0.7745966692414835       0.0000000000000000    
   0.7745966692414835       0.7745966692414835    
      

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    
     1.0000000000000000         1.0000000000000000  
      

Related Data and Programs:

• NINT_EXACTNESS is a FORTRAN90 program which can read a set of files defining a quadrature rule in 1D or multidimensions, and test it for exactness against monomial integrands.
• PRODUCT_FACTOR is a FORTRAN90 program which creates a multidimensional product rule whose factors are distinct 1D quadrature rules.
• PRODUCT_RULE is a FORTRAN90 program which creates a multidimensional product rule whose factors are identical 1D quadrature rules.
• QUADRATURE_RULES_HALTON is a collection of "quadrature rules" based on multidimensional Halton sequences.
• QUADRATURE_RULES_TRI is a collection of quadrature rules to be applied to triangular regions.
• QUADRATURE_TEST is an executable MATLAB program which reads the definition of a multidimensional quadrature rule from three files, applies the rule to a number of test integrals, and prints the results.
• QUADRULE is a FORTRAN90 library of routines which defines various quadrature rules.
• SPARSE_GRID_CC is a dataset collection of quadrature rules derived from the Clenshaw Curtis rule.
• SPARSE_GRID_F1 is a dataset collection of quadrature rules derived from the Fejer Type 1 rule.
• SPARSE_GRID_F2 is a dataset collection of quadrature rules derived from the Fejer Type 2 rule.
• SPARSE_GRID_GP is a dataset collection of quadrature rules derived from the Gauss-Patterson rule.
• SPARSE_GRID_NCC is a dataset collection of quadrature rules derived from the Newton Cotes Closed rule.
• SPARSE_GRID_NCO is a dataset collection of quadrature rules derived from the Newton Cotes Open rule.
• SPARSE_GRID_NCOH is a dataset collection of quadrature rules derived from the Newton Cotes Open Half rule.
• TABLE is the file format used to store this data;

Sample Files:

Clenshaw Curtis Quadrature Rules in 1D, defined on [-1,1]:

• cc_d1_o2_x.txt, the abscissas for the order 2 rule.
• cc_d1_o2_w.txt, the weights for the order 2 rule.
• cc_d1_o2_r.txt, the range of the integration region.
• cc_d1_o4_x.txt, the abscissas for the order 4 rule.
• cc_d1_o4_w.txt, the weights for the order 4 rule.
• cc_d1_o4_r.txt, the range of the integration region.

Clenshaw Curtis Product Quadrature Rules in 2D, defined on the [-1,1] square:

• cc_d2_o1x17_x.txt, the abscissas for the order 17 = (1x17) product rule.
• cc_d2_o1x17_w.txt, the weights for the order 17 = (1x17) product rule.
• cc_d2_o1x17_r.txt, the range of the integration region.
• cc_d2_o3x3_x.txt, the abscissas for the order 9 = (3x3) product rule.
• cc_d2_o3x3_w.txt, the weights for the order 9 = (3x3) product rule.
• cc_d2_o3x3_r.txt, the range of the integration region.
• cc_d2_o3x9_x.txt, the abscissas for the order 27 = (3x9) product rule.
• cc_d2_o3x9_w.txt, the weights for the order 27 = (3x9) product rule.
• cc_d2_o3x9_r.txt, the range of the integration region.
• cc_d2_o5x5_x.txt, the abscissas for the order 25 = (5x5) product rule.
• cc_d2_o5x5_w.txt, the weights for the order 25 = (5x5) product rule.
• cc_d2_o5x5_r.txt, the range of the integration region.
• cc_d2_o9x3_x.txt, the abscissas for the order 27 = (9x3) product rule.
• cc_d2_o9x3_w.txt, the weights for the order 27 = (9x3) product rule.
• cc_d2_o9x3_r.txt, the range of the integration region.
• cc_d2_o17x1_x.txt, the abscissas for the order 17 = (17x1) product rule.
• cc_d2_o17x1_w.txt, the weights for the order 17 = (17x1) product rule.
• cc_d2_o17x1_r.txt, the range of the integration region.

Clenshaw-Curtis Product Quadrature Rules in 3D, defined on the [-1,1] cube:

• cc_d3_o3x3x3_x.txt, the abscissas for the order 27 = (3x3x3) product rule.
• cc_d3_o3x3x3_w.txt, the weights for the order 27 = (3x3x3) product rule.
• cc_d3_o3x3x3_r.txt, the range of the integration region.

A product rule in 2D, defined by a Clenshaw Curtis rule of order 3 and a Gauss-Legendre rule of order 2, on the [-1,1] square:

• ccgl_d2_o3x2_x.txt, the abscissas for the 3x2 product rule.
• ccgl_d2_o3x2_w.txt, the weights for the 3x2 product rule.
• ccgl_d2_o3x2_r.txt, the range of the integration region.

Fejer Type 1 Sparse Quadrature Rules in 2D, defined on the [-1,1] square:

• f1_d2_level2_x.txt, the abscissas for the level 2, order 17 sparse rule.
• f1_d2_level2_w.txt, the weights for the level 2, order 17 rule.
• f1_d2_level2_r.txt, the range of the integration region.

Fejer Type 2 Sparse Quadrature Rules in 2D, defined on the [-1,1] square:

• f2_d2_level2_x.txt, the abscissas for the level 2, order 17 sparse rule.
• f2_d2_level2_w.txt, the weights for the level 2, order 17 rule.
• f2_d2_level2_r.txt, the range of the integration region.

Gauss-Legendre Quadrature Rules in 1D, defined on [-1,1]:

• gl_d1_o1_x.txt, the abscissas for the order 1 rule.
• gl_d1_o1_w.txt, the weights for the order 1 rule.
• gl_d1_o1_r.txt, the range of the integration region.
• gl_d1_o2_x.txt, the abscissas for the order 2 rule.
• gl_d1_o2_w.txt, the weights for the order 2 rule.
• gl_d1_o2_r.txt, the range of the integration region.
• gl_d1_o3_x.txt, the abscissas for the order 3 rule.
• gl_d1_o3_w.txt, the weights for the order 3 rule.
• gl_d1_o3_r.txt, the range of the integration region.
• gl_d1_o4_x.txt, the abscissas for the order 4 rule.
• gl_d1_o4_w.txt, the weights for the order 4 rule.
• gl_d1_o4_r.txt, the range of the integration region.
• gl_d1_o5_x.txt, the abscissas for the order 5 rule.
• gl_d1_o5_w.txt, the weights for the order 5 rule.
• gl_d1_o5_r.txt, the range of the integration region.

Gauss-Legendre Product Quadrature Rules in 2D, defined on the [-1,1] square:

• gl_d2_o3x3_x.txt, the abscissas for the order 9 = (3x3) product rule.
• gl_d2_o3x3_w.txt, the weights for the order 9 = (3x3) product rule.
• gl_d2_o3x3_r.txt, the range of the integration region.

Gauss-Legendre Product Quadrature Rules in 3D, defined on the [-1,1] cube:

• gl_d3_o3x3x3_x.txt, the abscissas for the order 27 = (3x3x3) product rule.
• gl_d3_o3x3x3_w.txt, the weights for the order 27 = (3x3x3) product rule.
• gl_d3_o3x3x3_r.txt, the range of the integration region.

Gauss-Legendre Product Quadrature Rules in 6D, defined on the [-1,1] cube:

• gl_d6_oone_x.txt, the abscissas for the order 1 = (1^6) product rule.
• gl_d6_oone_w.txt, the weights.
• gl_d6_oone_r.txt, the range of the integration region.
• gl_d6_otwo_x.txt, the abscissas for the order 64 = (2^6) product rule.
• gl_d6_otwo_w.txt, the weights.
• gl_d6_otwo_r.txt, the range of the integration region.
• gl_d6_othree_x.txt, the abscissas for the order 729 = (3^6) product rule.
• gl_d6_othree_w.txt, the weights.
• gl_d6_othree_r.txt, the range of the integration region.
• gl_d6_ofour_x.txt, the abscissas for the order 4096 = (4^6) product rule.
• gl_d6_ofour_w.txt, the weights.
• gl_d6_ofour_r.txt, the range of the integration region.
• gl_d6_ofive_x.txt, the abscissas for the order 15625 = (5^6) product rule.
• gl_d6_ofive_w.txt, the weights.
• gl_d6_ofive_r.txt, the range of the integration region.

Gauss-Legendre Product Quadrature Rules in 6D, defined on the [-1,1] cube:

• gl_d10_oone_x.txt, the abscissas for the order 1 = (1^10) product rule.
• gl_d10_oone_w.txt, the weights.
• gl_d10_oone_r.txt, the range of the integration region.
• gl_d10_otwo_x.txt, the abscissas for the order 1024 = (2^10) product rule.
• gl_d10_otwo_w.txt, the weights.
• gl_d10_otwo_r.txt, the range of the integration region.
• gl_d10_othree_x.txt, the abscissas for the order 59,049 = (3^10) product rule.
• gl_d10_othree_w.txt, the weights.
• gl_d10_othree_r.txt, the range of the integration region.

Gauss Patterson Sparse Quadrature Rules in 2D, defined on the [-1,1] square:

• gp_d2_level2_x.txt, the abscissas for the level 2, order 17 sparse rule.
• gp_d2_level2_w.txt, the weights for the level 2, order 17 rule.
• gp_d2_level2_r.txt, the range of the integration region.

Newton-Cotes Closed Quadrature Rules in 1D, defined on [-1,1]:

• ncc_d1_o5_x.txt, the abscissas for the order 5 rule.
• ncc_d1_o5_w.txt, the weights for the order 5 rule.
• ncc_d1_o5_r.txt, the range of the integration region.

Newton-Cotes Closed Product Quadrature Rules in 2D, defined on the [-1,1] square:

• ncc_d2_o5x5_x.txt, the abscissas for the 5x5 product rule.
• ncc_d2_o5x5_w.txt, the weights for the 5x5 product rule.
• ncc_d2_o5x5_r.txt, the range of the integration region.

Newton-Cotes Closed Product Quadrature Rules in 3D, defined on the [-1,1] cube:

• ncc_d3_o5x5x5_x.txt, the abscissas for the 5x5x5 product rule.
• ncc_d3_o5x5x5_w.txt, the weights for the 5x5x5 product rule.
• ncc_d3_o5x5x5_r.txt, the range of the integration region.

Newton-Cotes Closed Sparse Quadrature Rules in 2D, defined on the [-1,1] square:

• ncc_d2_level3_x.txt, the abscissas for the level 3, order 29 sparse rule.
• ncc_d2_level3_w.txt, the weights for the level 3, order 29 rule.
• ncc_d2_level3_r.txt, the range of the integration region.

Newton Cotes Open Sparse Quadrature Rules in 2D, defined on the [-1,1] square:

• nco_d2_level2_x.txt, the abscissas for the level 2, order 17 sparse rule.
• nco_d2_level2_w.txt, the weights for the level 2, order 17 rule.
• nco_d2_level2_r.txt, the range of the integration region.

Newton Cotes Open Half Sparse Quadrature Rules in 2D, defined on the [-1,1] square:

• ncoh_d2_level2_x.txt, the abscissas for the level 2, order 17 sparse rule.
• ncoh_d2_level2_w.txt, the weights for the level 2, order 17 rule.
• ncoh_d2_level2_r.txt, the range of the integration region.

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