POWER_RULE
Multidimensional Quadrature Rule Creation

POWER_RULE is a MATLAB program which creates a power rule, that is, a multidimensional quadrature rule formed as a repeated product of a single one-dimensional quadrature rule.

Note that the three files that define the quadrature rule are assumed to have related names, of the form

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

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

Usage:

power_rule prefix dim_num

• prefix is the common prefix for the files containing the abscissa, weight and region information of the quadrature rule;
• dim_num is the dimension of the space.

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

The program will create these output files, which define the multidimensional quadrature rule:

power_x.txt

a file containing the abscissas of the power rule;

power_w.txt

a file containing the weights of the power rule;

power_r.txt

a file containing the range of the power rule;

Licensing:

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

Languages:

POWER_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.

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

PATTERSON_RULE, a MATLAB program which computes a Gauss-Patterson quadrature rule.

PRODUCT_RULE, a MATLAB program which constructs a product quadrature rule from distinct 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.

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.

Reference:

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

Source Code:

• power_rule.m, the source code.

Examples and Tests:

As an example, we start with 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.

We give the command

power_rule ( 'cc_d1_o003', 2 )

• cc_d2_o009_x.txt, the abscissas of a 2D Clenshaw Curtis power rule of order 9.
• cc_d2_o009_w.txt, the weights of a 2D Clenshaw Curtis power rule of order 9.
• cc_d2_o009_r.txt, defines the range of the power region.

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