PRODUCT_RULE
Multidimensional Quadrature Rule Creation

PRODUCT_RULE is a C++ 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 C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version

Related Data and Programs:

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

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

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

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

POWER_RULE, a C++ 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.

QUADRULE, a C++ library which defines quadrature rules on a variety of intervals with different weight functions.

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

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

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

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

TRUNCATED_NORMAL_RULE, a C++ 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:

• product_rule.cpp, the source code.
• product_rule.sh, commands to compile 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_o3_x.txt, the abscissas of a 1D Clenshaw Curtis rule of order 3.
• cc_d1_o3_w.txt, the weights of a 1D Clenshaw Curtis rule of order 3.
• cc_d1_o3_r.txt, defines the range of the region.

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

• gl_d1_o2_x.txt, the abscissas of a 1D Clenshaw Curtis rule of order 2.
• gl_d1_o2_w.txt, the weights of a 1D Clenshaw Curtis rule of order 2.
• gl_d1_o2_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_o3x2_x.txt, the abscissas of a 2D Clenshaw Curtis product rule of order 6.
• ccgl_d2_o3x2_w.txt, the weights of a 2D Clenshaw Curtis product rule of order 6.
• ccgl_d2_o3x2_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 true if two characters are equal, disregarding case.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• FILE_COLUMN_COUNT counts the columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• PRODUCT_RULE_SIZE returns the size of a product rule of distinct factors.
• R8_EPSILON returns the R8 roundoff unit.
• R8MAT_DATA_READ reads the data from an R8MAT file.
• R8MAT_HEADER_READ reads the header from an R8MAT file.
• R8MAT_WRITE writes an R8MAT file with no header.
• R8VEC_DIRECT_PRODUCT creates a direct product of R8VEC's.
• R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• 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.

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