# PRODUCT_RULE Multidimensional Quadrature Rule Creation

PRODUCT_RULE, 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
```
When the program read the first 1D rule, it would be searching for three files:
• cc_d1_o003_x.txt
• cc_d1_o003_w.txt
• cc_d1_o003_r.txt
and similarly for the second file.

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 )
where
• 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.

### Languages:

PRODUCT_RULE is available in a C version and 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.

TRUNCATED_NORMAL_RULE, a MATLAB program which computes a quadrature rule for a normal probability density function (PDF), also called a Gaussian distribution, that has been truncated to [A,+oo), (-oo,B] or [A,B].

### Reference:

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

### Source Code:

Last revised on 06 March 2019.