PRODUCT_RULE
Multidimensional Quadrature Rule Creation

PRODUCT_RULE is a C++ program, using double precision arithmetic, which creates a multidimensional quadrature rule by using a product of one-dimensional quadrature rules.

The current version of the program requires that all the "factor rules" be identical, that is, the product rule is created as the product of n identical one-dimensional rules. It is possible to create product rules in which the factors are not equal, and one day this program may be extended to handle that case.

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:

product_rule prefix dim_num

prefix

the common prefix for the files containing the abscissa, weight and region information of the quadrature rule;

dim_num

the dimension of the product 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:

product_x.txt

a file containing the dim_num-dimensional abscissas of the product rule;

product_w.txt

a file containing the weights of the product rule;

product_r.txt

a file containing the range of the product rule;

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:

CLENSHAW_CURTIS is a C++ library which sets up a Clenshaw Curtis quadrature grid in multiple dimensions.

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

NINT_EXACTNESS is a C++ program which can test a quadrature rule for polynomial exactness.

NINTLIB is a C++ library of routines for numerically estimating integrals in multiple dimensions.

PRODUCT_FACTOR is an executable C++ program which creates a multi-dimensional quadrature rule as a tensor product of 1-D rules, which may be distinct.

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

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

QUADRULE is a C++ library of routines for defining quadrature rules on a variety of intervals with different weight functions.

STROUD is a FORTRAN90 library containing quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TEST_NINT is a C++ library of integrand functions for testing multidimensional quadrature routines.

TESTPACK is a C++ library of functions which define 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.C, the source code.
• product_rule.csh, commands to compile 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

product_rule cc_d1_o003 2

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

List of Routines:

• MAIN is the main program for PRODUCT_RULE.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• DTABLE_CLOSE_WRITE closes a file used to write a DTABLE.
• DTABLE_DATA_READ reads data from a DTABLE file.
• DTABLE_DATA_WRITE writes DTABLE data to a file.
• DTABLE_HEADER_READ reads the header from a DTABLE file.
• DTABLE_HEADER_WRITE writes the header to a DTABLE file.
• DTABLE_OPEN_WRITE opens a file to write a DTABLE.
• DTABLE_WRITE writes a DTABLE to a file.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• PRODUCT_RULE sets up a product rule.
• PRODUCT_RULE_SIZE returns the size of a product rule.
• S_BLANK_DELETE removes blanks from a string, left justifying the remainder.
• S_TO_I4 reads an I4 from a string.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING writes the current YMDHMS date into a string.
• TUPLE_NEXT computes the next element of a tuple space.

You can go up one level to the C++ source codes.