POWER_RULE
Multidimensional Quadrature Rule Creation

POWER_RULE is a C++ 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 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:

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 C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CLENSHAW_CURTIS_RULE, a C++ program which defines a Clenshaw Curtis quadrature rule.

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

LATTICE_RULE, a C++ library which approximates M-dimensional integrals using lattice rules.

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

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

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

PRODUCT_RULE, a C++ 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.

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

STROUD, a C++ library which defines 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:

• power_rule.cpp, the source code.
• power_rule.sh, commands to compile the source code.

Examples and Tests:

As an example, we start with a 1D Clenshaw Curtis rule of order 3:

• cc3_x.txt, the abscissas of a 1D Clenshaw Curtis rule of order 3.
• cc3_w.txt, the weights of a 1D Clenshaw Curtis rule of order 3.
• cc3_r.txt, defines the range of the region.

We give the command

power_rule cc3 2

• power_rule_output.txt, the output file.
• power_x.txt, the abscissas of the power rule.
• power_w.txt, the weights of the power rule.
• power_r.txt, defines the range of the power region.

List of Routines:

• MAIN is the main program for POWER_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 number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• I4_POWER returns the value of I^J.
• POWER_RULE_SET sets up a power rule.
• POWER_RULE_SIZE returns the size of a power rule.
• 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.
• 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.
• TUPLE_NEXT computes the next element of a tuple space.

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