# QUADRATURE_RULES Rules for Approximate Integration

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.

### 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
``````

### 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:

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.

### Sample Files:

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.

Last revised on 13 September 2007.