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<html>
<head>
<title>
POWER_RULE - Multidimensional Quadrature Rule Creation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
POWER_RULE <br> Multidimensional Quadrature Rule Creation
</h1>
<hr>
<p>
<b>POWER_RULE</b>
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.
</p>
<p>
Note that the three files that define the quadrature rule
are assumed to have related names, of the form
<ul>
<li>
<i>prefix</i>_<b>x.txt</b>
</li>
<li>
<i>prefix</i>_<b>w.txt</b>
</li>
<li>
<i>prefix</i>_<b>r.txt</b>
</li>
</ul>
When running the program, the user only enters the common <i>prefix</i>
part of the file names, which is enough information for the program
to find all three files.
</p>
<p>
For information on the form of these files, see the
<b>QUADRATURE_RULES</b> directory listed below.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>power_rule</b> <i>prefix</i> <i>dim_num</i>
</blockquote>
where
<ul>
<li>
<i>prefix</i> is the common prefix for the files containing the abscissa, weight
and region information of the quadrature rule;
</li>
<li>
<i>dim_num</i> is the dimension of the product space.
</li>
</ul>
</p>
<p>
If the arguments are not supplied on the command line, the
program will prompt for them.
</p>
<p>
The program will create these output files, which define
the multidimensional quadrature rule:
<dl>
<dt>
power_x.txt
</dt>
<dd>
a file containing the abscissas of the
power rule;
</dd>
<dt>
power_w.txt
</dt>
<dd>
a file containing the weights of the power rule;
</dd>
<dt>
power_r.txt
</dt>
<dd>
a file containing the range of the power rule;
</dd>
</dl>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>POWER_RULE</b> is available in
<a href = "../../c_src/power_rule/power_rule.html">a C version</a> and
<a href = "../../cpp_src/power_rule/power_rule.html">a C++ version</a> and
<a href = "../../f77_src/power_rule/power_rule.html">a FORTRAN77 version</a> and
<a href = "../../f_src/power_rule/power_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/power_rule/power_rule.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/clenshaw_curtis_rule/clenshaw_curtis_rule.html">
CLENSHAW_CURTIS_RULE</a>,
a C++ program which
defines a Clenshaw Curtis quadrature rule.
</p>
<p>
<a href = "../../f_src/integral_test/integral_test.html">
INTEGRAL_TEST</a>,
a FORTRAN90 program which
uses some of these test integrals to evaluate sets of quadrature points.
</p>
<p>
<a href = "../../cpp_src/lattice_rule/lattice_rule.html">
LATTICE_RULE</a>,
a C++ library which
approximates M-dimensional integrals using lattice rules.
</p>
<p>
<a href = "../../cpp_src/nint_exactness/nint_exactness.html">
NINT_EXACTNESS</a>,
a C++ program which
can test a quadrature rule for polynomial exactness.
</p>
<p>
<a href = "../../cpp_src/nintlib/nintlib.html">
NINTLIB</a>,
a C++ library which
numerically estimates integrals
in multiple dimensions.
</p>
<p>
<a href = "../../cpp_src/patterson_rule/patterson_rule.html">
PATTERSON_RULE</a>,
a C++ program which
computes a Gauss-Patterson quadrature rule.
</p>
<p>
<a href = "../../cpp_src/product_rule/product_rule.html">
PRODUCT_RULE</a>,
a C++ program which
constructs a product quadrature rule from distinct 1D factor rules.
</p>
<p>
<a href = "../../datasets/quadrature_rules/quadrature_rules.html">
QUADRATURE_RULES</a>,
a dataset directory which
contains sets of files that define quadrature
rules over various 1D intervals or multidimensional hypercubes.
</p>
<p>
<a href = "../../cpp_src/quadrule/quadrule.html">
QUADRULE</a>,
a C++ library which
defines quadrature rules on a
variety of intervals with different weight functions.
</p>
<p>
<a href = "../../cpp_src/stroud/stroud.html">
STROUD</a>,
a C++ library which
defines quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
</p>
<p>
<a href = "../../cpp_src/test_nint/test_nint.html">
TEST_NINT</a>,
a C++ library which
defines integrand functions for testing
multidimensional quadrature routines.
</p>
<p>
<a href = "../../cpp_src/testpack/testpack.html">
TESTPACK</a>,
a C++ library which
defines a set of integrands used to test multidimensional quadrature.
</p>
<p>
<a href = "../../cpp_src/truncated_normal_rule/truncated_normal_rule.html">
TRUNCATED_NORMAL_RULE</a>,
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].
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Philip Davis, Philip Rabinowitz,<br>
Methods of Numerical Integration,<br>
Second Edition,<br>
Dover, 2007,<br>
ISBN: 0486453391,<br>
LC: QA299.3.D28.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "power_rule.cpp">power_rule.cpp</a>, the source code.
</li>
<li>
<a href = "power_rule.sh">power_rule.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
As an example, we start with a 1D Clenshaw Curtis rule of order 3:
<ul>
<li>
<a href = "cc3_x.txt">
cc3_x.txt</a>,
the abscissas of a 1D Clenshaw Curtis rule of order 3.
</li>
<li>
<a href = "cc3_w.txt">
cc3_w.txt</a>,
the weights of a 1D Clenshaw Curtis rule of order 3.
</li>
<li>
<a href = "cc3_r.txt">
cc3_r.txt</a>,
defines the range of the region.
</li>
</ul>
</p>
<p>
We give the command
<blockquote>
<b>power_rule</b> cc3 2
</blockquote>
to create a Clenshaw Curtis product rule of order 3x3 for a 2D region.
The files which are created include:
<ul>
<li>
<a href = "power_rule_output.txt">
power_rule_output.txt</a>,
the output file.
</li>
<li>
<a href = "power_x.txt">
power_x.txt</a>,
the abscissas of the power rule.
</li>
<li>
<a href = "power_w.txt">
power_w.txt</a>,
the weights of the power rule.
</li>
<li>
<a href = "power_r.txt">
power_r.txt</a>,
defines the range of the power region.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for POWER_RULE.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is true if two characters are equal, disregarding case.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>I4_POWER</b> returns the value of I^J.
</li>
<li>
<b>POWER_RULE_SET</b> sets up a power rule.
</li>
<li>
<b>POWER_RULE_SIZE</b> returns the size of a power rule.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8MAT_DATA_READ</b> reads the data from an R8MAT file.
</li>
<li>
<b>R8MAT_HEADER_READ</b> reads the header from an R8MAT file.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file with no header.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TUPLE_NEXT</b> computes the next element of a tuple space.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 03 February 2014.
</i>
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