QUADRATURE_RULES is a dataset directory which contains 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.
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
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
NINT_EXACTNESS, 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.
TEST_INT, a C++ library which defines test integrands for 1D quadrature rules.
Clenshaw Curtis Quadrature Rules in 1D, defined on [-1,1]:
Clenshaw Curtis Product Quadrature Rules in 2D, defined on the [-1,1] square:
Clenshaw-Curtis Product Quadrature Rules in 3D, defined on the [-1,1] cube:
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:
Fejer Type 1 Sparse Quadrature Rules in 2D, defined on the [-1,1] square:
Fejer Type 2 Sparse Quadrature Rules in 2D, defined on the [-1,1] square:
Gauss-Legendre Quadrature Rules in 1D, defined on [-1,1]:
Gauss-Legendre Product Quadrature Rules in 2D, defined on the [-1,1] square:
Gauss-Legendre Product Quadrature Rules in 3D, defined on the [-1,1] cube:
Gauss-Legendre Product Quadrature Rules in 6D, defined on the [-1,1] cube:
Gauss-Legendre Product Quadrature Rules in 6D, defined on the [-1,1] cube:
Gauss Patterson Sparse Quadrature Rules in 2D, defined on the [-1,1] square:
Newton-Cotes Closed Quadrature Rules in 1D, defined on [-1,1]:
Newton-Cotes Closed Product Quadrature Rules in 2D, defined on the [-1,1] square:
Newton-Cotes Closed Product Quadrature Rules in 3D, defined on the [-1,1] cube:
Newton-Cotes Closed Sparse Quadrature Rules in 2D, defined on the [-1,1] square:
Newton Cotes Open Sparse Quadrature Rules in 2D, defined on the [-1,1] square:
Newton Cotes Open Half Sparse Quadrature Rules in 2D, defined on the [-1,1] square:
You can go up one level to the DATASETS page.