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<html>
<head>
<title>
CCN_RULE - Nested Clenshaw Curtis Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
CCN_RULE <br> Nested Clenshaw Curtis Quadrature Rules
</h1>
<hr>
<p>
<b>CCN_RULE</b>
is a C++ program which
generates a quadrature rule based on a nested set of points
inspired by the Clenshaw Curtis quadrature rule.
</p>
<p>
The data defining the rule is written to three files for easy use as input
to other programs.
</p>
<p>
The nested Clenshaw Curtis quadrature rule is used as follows:
<pre>
Integral ( A <= x <= B ) f(x) dx
</pre>
is to be approximated by
<pre>
Sum ( 1 <= i <= order ) w(i) * f(x(i))
</pre>
</p>
<p>
If the order of the CCN rule is 1, 3, 5, 9, 17, 33, or in general
2^L+1, then the rule is identical to the Clenshaw Curtis rule.
</p>
<p>
Otherwise, the rule is based on a subset of the points in the
Clenshaw Curtis rule of next highest order in the sequence 2^L+1.
</p>
<p>
The CCN rule has no special accuracy properties, except that the
rules of odd order are symmetric, and hence get one extra degree of
precision. Moreover, the rules of even order have a single unpaired
point which is assigned weight zero, so that it is equivalent
to the immediately preceding rule of odd order.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>ccn_rule</b> <i>n</i> <i>a</i> <i>b</i> <i>filename</i>
</blockquote>
where
<ul>
<li>
<i>n</i> is the order, or number of points in the quadrature rule;
</li>
<li>
<i>a</i> is the left endpoint;
</li>
<li>
<i>b</i> is the right endpoint;
</li>
<li>
<i>filename</i> specifies the output filenames:
name<i>file</i><b>_w.txt</b>,
<i>filename</i><b>_x.txt</b>, and <i>filename</i><b>_r.txt</b>,
containing the weights, abscissas, and interval limits.
</ul>
</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>CCN_RULE</b> is available in
<a href = "../../c_src/ccn_rule/ccn_rule.html">a C version</a> and
<a href = "../../cpp_src/ccn_rule/ccn_rule.html">a C++ version</a> and
<a href = "../../f77_src/ccn_rule/ccn_rule.html">a FORTRAN77 version</a> and
<a href = "../../f_src/ccn_rule/ccn_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/ccn_rule/ccn_rule.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/chebyshev1_rule/chebyshev1_rule.html">
CHEBYSHEV1_RULE</a>,
a C++ program which
can compute and print a Gauss-Chebyshev type 1 quadrature rule.
</p>
<p>
<a href = "../../cpp_src/chebyshev2_rule/chebyshev2_rule.html">
CHEBYSHEV2_RULE</a>,
a C++ program which
can compute and print a Gauss-Chebyshev type 2 quadrature rule.
</p>
<p>
<a href = "../../cpp_src/clenshaw_curtis_rule/clenshaw_curtis_rule.html">
CLENSHAW_CURTIS_RULE</a>,
a C++ program which
can compute and print a Gauss-Chebyshev type 1 quadrature rule.
</p>
<p>
<a href = "../../cpp_src/gegenbauer_rule/gegenbauer_rule.html">
GEGENBAUER_RULE</a>,
a C++ program which
can compute and print a Gauss-Gegenbauer quadrature rule.
</p>
<p>
<a href = "../../cpp_src/gen_hermite_rule/gen_hermite_rule.html">
GEN_HERMITE_RULE</a>,
a C++ program which
can compute and print a generalized Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../cpp_src/gen_laguerre_rule/gen_laguerre_rule.html">
GEN_LAGUERRE_RULE</a>,
a C++ program which
can compute and print a generalized Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../cpp_src/hermite_rule/hermite_rule.html">
HERMITE_RULE</a>,
a C++ program which
can compute and print a Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../cpp_src/int_exactness_legendre/int_exactness_legendre.html">
INT_EXACTNESS_LEGENDRE</a>,
a C++ program which
checks the polynomial exactness of a Gauss-Legendre quadrature rule.
</p>
<p>
<a href = "../../cpp_src/jacobi_rule/jacobi_rule.html">
JACOBI_RULE</a>,
a C++ program which
can compute and print a Gauss-Jacobi quadrature rule.
</p>
<p>
<a href = "../../cpp_src/laguerre_rule/laguerre_rule.html">
LAGUERRE_RULE</a>,
a C++ program which
can compute and print a Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../cpp_src/legendre_rule/legendre_rule.html">
LEGENDRE_RULE</a>,
a C++ program which
can compute and print a Gauss-Legendre quadrature rule.
</p>
<p>
<a href = "../../cpp_src/line_felippa_rule/line_felippa_rule.html">
LINE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a line segment in 1D.
</p>
<p>
<a href = "../../cpp_src/line_ncc_rule/line_ncc_rule.html">
LINE_NCC_RULE</a>,
a C++ library which
computes a Newton Cotes Closed (NCC) quadrature rule for the line,
that is, for an interval of the form [A,B], using equally spaced points
which include the endpoints.
</p>
<p>
<a href = "../../cpp_src/line_nco_rule/line_nco_rule.html">
LINE_NCO_RULE</a>,
a C++ library which
computes a Newton Cotes Open (NCO) quadrature rule,
using equally spaced points,
over the interior of a line segment in 1D.
</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 = "../../datasets/quadrature_rules_ccn/quadrature_rules_ccn.html">
QUADRATURE_RULES_CCN</a>,
a dataset directory which
contains quadrature rules for integration on [-1,+1],
using a nested Clenshaw Curtis rule.
</p>
<p>
<a href = "../../cpp_src/tanh_sinh_rule/tanh_sinh_rule.html">
TANH_SINH_RULE</a>,
a C++ program which
computes and writes out a tanh-sinh quadrature rule of given order.
</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>
Milton Abramowitz, Irene Stegun,<br>
Handbook of Mathematical Functions,<br>
National Bureau of Standards, 1964,<br>
ISBN: 0-486-61272-4,<br>
LC: QA47.A34.
</li>
<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>
<li>
Arthur Stroud, Don Secrest,<br>
Gaussian Quadrature Formulas,<br>
Prentice Hall, 1966,<br>
LC: QA299.4G3S7.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "ccn_rule.cpp">ccn_rule.cpp</a>, the source code.
</li>
<li>
<a href = "ccn_rule.sh">ccn_rule.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>CCN_09</b> is a nested Clenshaw Curtis rule of order 9,
which will exactly match the standard Clenshaw Curtis rule.
<ul>
<li>
<a href = "ccn_o9_r.txt">ccn_o9_r.txt</a>,
the region file created by the command
<pre><b>
ccn_rule 9 -1 +1 ccn_o9
</b></pre>
</li>
<li>
<a href = "ccn_o9_w.txt">ccn_o9_w.txt</a>,
the weight file created by the command
<pre><b>
ccn_rule 9 -1 +1 ccn_o9
</b></pre>
</li>
<li>
<a href = "ccn_o9_x.txt">ccn_o9_x.txt</a>,
the abscissa file created by the command
<pre><b>
ccn_rule 9 -1 +1 ccn_o9
</b></pre>
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for CCN_RULE.
</li>
<li>
<b>CCN_COMPUTE_POINTS:</b> compute Clenshaw Curtis Nested points.
</li>
<li>
<b>I4_MIN</b> returns the minimum of two I4's.
</li>
<li>
<b>NC_COMPUTE</b> computes a Newton-Cotes quadrature rule.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file with no header.
</li>
<li>
<b>RESCALE</b> rescales a Legendre quadrature rule from [-1,+1] to [A,B].
</li>
<li>
<b>RULE_WRITE</b> writes a quadrature rule to three files.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</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 06 March 2011.
</i>
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