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<html>
<head>
<title>
PRODUCT_RULE - Multidimensional Quadrature Rule Creation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
PRODUCT_RULE <br> Multidimensional Quadrature Rule Creation
</h1>
<hr>
<p>
<b>PRODUCT_RULE</b>
is a MATLAB program which
creates a multidimensional quadrature rule by using
a product of <i>distinct</i> one-dimensional quadrature rules.
</p>
<p>
The program reads a single input file, which contains a list
that defines the 1D rules to be used as factors.
</p>
<p>
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 <i>prefix</i> have 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>
</p>
<p>
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:
<pre>
cc_d1_o003
gl_d1_o002
</pre>
When the program read the first 1D rule, it would be searching for
three files:
<ul>
<li>
<b>cc_d1_o003_x.txt</b>
</li>
<li>
<b>cc_d1_o003_w.txt</b>
</li>
<li>
<b>cc_d1_o003_r.txt</b>
</li>
</ul>
and similarly for the second file.
</p>
<p>
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, <i>product</i>, in which case the files
would be:
<ul>
<li>
<b>product_x.txt</b>
</li>
<li>
<b>product_w.txt</b>
</li>
<li>
<b>product_r.txt</b>
</li>
</ul>
</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>product_rule</b> ( <i>list_file</i>, <i>product_prefix</i> )
</blockquote>
where
<ul>
<li>
<i>list_file</i> 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.
</li>
<li>
<i>product_prefix</i> is the prefix to be used when writing out the X, W and R
files of the computed multidimensional product rule.
</li>
</ul>
</p>
<p>
If the arguments are not supplied on the command line, the
program will prompt for them.
</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>PRODUCT_RULE</b> is available in
<a href = "../../c_src/product_rule/product_rule.html">a C version</a> and
<a href = "../../cpp_src/product_rule/product_rule.html">a C++ version</a> and
<a href = "../../f77_src/product_rule/product_rule.html">a FORTRAN77 version</a> and
<a href = "../../f_src/product_rule/product_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/product_rule/product_rule.html">a MATLAB version</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<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 = "../../m_src/nint_exactness/nint_exactness.html">
NINT_EXACTNESS</a>,
a MATLAB program which
can test a quadrature rule for polynomial exactness.
</p>
<p>
<a href = "../../m_src/nint_exactness_mixed/nint_exactness_mixed.html">
NINT_EXACTNESS_MIXED</a>,
a MATLAB program which
measures the polynomial exactness of a multidimensional quadrature rule
based on a mixture of 1D quadrature rule factors.
</p>
<p>
<a href = "../../m_src/nintlib/nintlib.html">
NINTLIB</a>,
a MATLAB library which
numerically estimates integrals in multiple dimensions.
</p>
<p>
<a href = "../../m_src/power_rule/power_rule.html">
POWER_RULE</a>,
a MATLAB program which
constructs a power rule, that is, a product quadrature rule
from <i>identical</i> 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 = "../../m_src/quadrature_test/quadrature_test.html">
QUADRATURE_TEST</a>,
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.
</p>
<p>
<a href = "../../m_src/quadrule/quadrule.html">
QUADRULE</a>,
a MATLAB library which
defines quadrature rules on a
variety of intervals with different weight functions.
</p>
<p>
<a href = "../../m_src/sparse_grid_mixed/sparse_grid_mixed.html">
SPARSE_GRID_MIXED</a>,
a MATLAB library which
creates a sparse grid dataset based on a mixed set of 1D factor rules.
</p>
<p>
<a href = "../../m_src/stroud/stroud.html">
STROUD</a>,
a MATLAB library which
contains quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
</p>
<p>
<a href = "../../m_src/test_nint/test_nint.html">
TEST_NINT</a>,
a MATLAB library which
defines integrand functions for testing
multidimensional quadrature routines.
</p>
<p>
<a href = "../../m_src/testpack/testpack.html">
TESTPACK</a>,
a MATLAB library which
defines a set of integrands used to test multidimensional quadrature.
</p>
<p>
<a href = "../../m_src/truncated_normal_rule/truncated_normal_rule.html">
TRUNCATED_NORMAL_RULE</a>,
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].
</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 = "product_rule.m">product_rule.m</a>,
the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
As an example, we make a 2D product rule.
</p>
<p>
The first factor is a 1D Clenshaw Curtis rule of order 3:
<ul>
<li>
<a href = "cc_d1_o3_x.txt">
cc_d1_o3_x.txt</a>,
the abscissas of a 1D Clenshaw Curtis rule of order 3.
</li>
<li>
<a href = "cc_d1_o3_w.txt">
cc_d1_o3_w.txt</a>,
the weights of a 1D Clenshaw Curtis rule of order 3.
</li>
<li>
<a href = "cc_d1_o3_r.txt">
cc_d1_o3_r.txt</a>,
defines the range of the region.
</li>
</ul>
</p>
<p>
The second factor is a 1D Gauss Legendre rule of order 2:
<ul>
<li>
<a href = "gl_d1_o2_x.txt">
gl_d1_o2_x.txt</a>,
the abscissas of a 1D Clenshaw Curtis rule of order 2.
</li>
<li>
<a href = "gl_d1_o2_w.txt">
gl_d1_o2_w.txt</a>,
the weights of a 1D Clenshaw Curtis rule of order 2.
</li>
<li>
<a href = "gl_d1_o2_r.txt">
gl_d1_o2_r.txt</a>,
defines the range of the region.
</li>
</ul>
</p>
<p>
We give the command
<blockquote>
<b>product_rule</b> factors.txt
</blockquote>
<ul>
<li>
<a href = "factors.txt">
factors.txt</a>,
a file containing the list of factors.
</li>
</ul>
</p>
<p>
The resulting product rule files:
<ul>
<li>
<a href = "ccgl_d2_o3x2_x.txt">
ccgl_d2_o3x2_x.txt</a>,
the abscissas of a 2D Clenshaw Curtis product rule of order 6.
</li>
<li>
<a href = "ccgl_d2_o3x2_w.txt">
ccgl_d2_o3x2_w.txt</a>,
the weights of a 2D Clenshaw Curtis product rule of order 6.
</li>
<li>
<a href = "ccgl_d2_o3x2_r.txt">
ccgl_d2_o3x2_r.txt</a>,
defines the range of the product region.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 29 June 2009.
</i>
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