PRODUCT_RULE
Multidimensional Quadrature Rule Creation

PRODUCT_RULE is a MATLAB program which creates a multidimensional quadrature rule by using a product of distinct one-dimensional quadrature rules.

The program reads a single input file, which contains a list that defines the 1D rules to be used as factors.

Each 1D rule is stored in three files, an "X", "W", and "R" file, which are assumed to share a common filename prefix, so that the files defined by a given prefix have the form:

• prefix_x.txt
• prefix_w.txt
• prefix_r.txt

For instance, let us suppose we want to compute a 2D product rule formed from a 3 point Clenshaw Curtis rule and a 2 point Gauss-Legendre rule. If the prefixes for these files were "cc_d1_o003" and "gl_d1_o002" respectively, then the input file to the program would read as follows:

        cc_d1_o003
        gl_d1_o002
      

• cc_d1_o003_x.txt
• cc_d1_o003_w.txt
• cc_d1_o003_r.txt

Once the program has computed the multidimensional product rule, it again writes out three files describing the product rule, with a common filename prefix specified by the user, which might be, for instance, product, in which case the files would be:

• product_x.txt
• product_w.txt
• product_r.txt

For information on the form of these files, see the QUADRATURE_RULES directory listed below.

Usage:

product_rule ( list_file, product_prefix )

• list_file is the name of the file containing the list of filename prefixes that can be used to locate the X, W and R files of each 1D quadrature rule.
• product_prefix is the prefix to be used when writing out the X, W and R files of the computed multidimensional product rule.

If the arguments are not supplied on the command line, the program will prompt for them.

Licensing:

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

Languages:

PRODUCT_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version

Related Data and Programs:

INTEGRAL_TEST, a FORTRAN90 program which uses some of these test integrals to evaluate sets of quadrature points.

NINT_EXACTNESS, a MATLAB program which can test a quadrature rule for polynomial exactness.

NINT_EXACTNESS_MIXED, a MATLAB program which measures the polynomial exactness of a multidimensional quadrature rule based on a mixture of 1D quadrature rule factors.

NINTLIB, a MATLAB library which numerically estimates integrals in multiple dimensions.

POWER_RULE, a MATLAB program which constructs a power rule, that is, a product quadrature rule from identical 1D factor rules.

QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_TEST, a 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, a MATLAB library which defines quadrature rules on a variety of intervals with different weight functions.

SPARSE_GRID_MIXED, a MATLAB library which creates a sparse grid dataset based on a mixed set of 1D factor rules.

STROUD, a MATLAB library which contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TEST_NINT, a MATLAB library which defines integrand functions for testing multidimensional quadrature routines.

TESTPACK, a MATLAB library which defines a set of integrands used to test multidimensional quadrature.

Reference:

1. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.

Source Code:

• product_rule.m, the source code.

Examples and Tests:

As an example, we make a 2D product rule.

The first factor is a 1D Clenshaw Curtis rule of order 3:

• cc_d1_o003_x.txt, the abscissas of a 1D Clenshaw Curtis rule of order 3.
• cc_d1_o003_w.txt, the weights of a 1D Clenshaw Curtis rule of order 3.
• cc_d1_o003_r.txt, defines the range of the region.

The second factor is a 1D Gauss Legendre rule of order 2:

• gl_d1_o002_x.txt, the abscissas of a 1D Clenshaw Curtis rule of order 2.
• gl_d1_o002_w.txt, the weights of a 1D Clenshaw Curtis rule of order 2.
• gl_d1_o002_r.txt, defines the range of the region.

We give the command

product_rule factors.txt

• factors.txt, a file containing the list of factors.

The resulting product rule files:

• ccgl_d2_o006_x.txt, the abscissas of a 2D Clenshaw Curtis product rule of order 6.
• ccgl_d2_o006_w.txt, the weights of a 2D Clenshaw Curtis product rule of order 6.
• ccgl_d2_o006_r.txt, defines the range of the product region.

You can go up one level to the MATLAB source codes.