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<html>
<head>
<title>
GEGENBAUER_RULE - Gauss-Gegenbauer Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
GEGENBAUER_RULE <br> Gauss-Gegenbauer Quadrature Rules
</h1>
<hr>
<p>
<b>GEGENBAUER_RULE</b>
is a MATLAB program which
generates a specific Gauss-Gegenbauer quadrature rule,
based on user input.
</p>
<p>
The rule is written to three files for easy use as input
to other programs.
</p>
<p>
The <i>Gauss-Gegenbauer quadrature rule </i> is used as follows:
<pre>
Integral ( A <= x <= B ) ((x-A)(B-X))^alpha f(x) dx
</pre>
is to be approximated by
<pre>
Sum ( 1 <= i <= order ) w(i) * f(x(i))
</pre>
where <b>alpha</b> is a real parameter greater than -1.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>gegenbauer_rule</b> ( <i>order</i>, <i>alpha</i>, <i>a</i>, <i>b</i>, <i>'filename'</i> )
</blockquote>
where
<ul>
<li>
<i>order</i> is the number of points in the quadrature rule.
</li>
<li>
<i>alpha</i> is the exponent of (1-x^2), which must be greater than -1.
</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 filenames:
<i>filename</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.
</li>
</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>GEGENBAUER_RULE</b> is available in
<a href = "../../cpp_src/gegenbauer_rule/gegenbauer_rule.html">a C++ version</a> and
<a href = "../../f_src/gegenbauer_rule/gegenbauer_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/gegenbauer_rule/gegenbauer_rule.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/ccn_rule/ccn_rule.html">
CCN_RULE</a>,
a MATLAB program which
defines a nested Clenshaw Curtis quadrature rule.
</p>
<p>
<a href = "../../m_src/chebyshev1_rule/chebyshev1_rule.html">
CHEBYSHEV1_RULE</a>,
a MATLAB program which
can compute and print a Gauss-Chebyshev type 1 quadrature rule.
</p>
<p>
<a href = "../../m_src/chebyshev2_rule/chebyshev2_rule.html">
CHEBYSHEV2_RULE</a>,
a MATLAB program which
can compute and print a Gauss-Chebyshev type 2 quadrature rule.
</p>
<p>
<a href = "../../m_src/clenshaw_curtis_rule/clenshaw_curtis_rule.html">
CLENSHAW_CURTIS_RULE</a>,
a MATLAB program which
defines a Clenshaw Curtis quadrature rule.
</p>
<p>
<a href = "../../m_src/gen_hermite_rule/gen_hermite_rule.html">
GEN_HERMITE_RULE</a>,
a MATLAB program which
can compute and print a generalized Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../m_src/gen_laguerre_rule/gen_laguerre_rule.html">
GEN_LAGUERRE_RULE</a>,
a MATLAB program which
can compute and print a generalized Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../m_src/hermite_rule/hermite_rule.html">
HERMITE_RULE</a>,
a MATLAB program which
can compute and print a Gauss-Hermite quadrature rule.
</p>
<p>
<a href = "../../m_src/int_exactness_gegenbauer/int_exactness_gegenbauer.html">
INT_EXACTNESS_GEGENBAUER</a>,
a MATLAB program which
checks the polynomial exactness
of a Gauss-Gegenbauer rule.
</p>
<p>
<a href = "../../m_src/jacobi_rule/jacobi_rule.html">
JACOBI_RULE</a>,
a MATLAB program which
can compute and print a Gauss-Jacobi quadrature rule.
</p>
<p>
<a href = "../../m_src/laguerre_rule/laguerre_rule.html">
LAGUERRE_RULE</a>,
a MATLAB program which
can compute and print a Gauss-Laguerre quadrature rule.
</p>
<p>
<a href = "../../m_src/legendre_rule/legendre_rule.html">
LEGENDRE_RULE</a>,
a MATLAB program which
computes a Gauss-Legendre quadrature rule.
</p>
<p>
<a href = "../../m_src/line_felippa_rule/line_felippa_rule.html">
LINE_FELIPPA_RULE</a>,
a MATLAB 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 = "../../m_src/patterson_rule/patterson_rule.html">
PATTERSON_RULE</a>,
a MATLAB program which
computes a Gauss-Patterson quadrature rule.
</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 identical 1D factor rules.
</p>
<p>
<a href = "../../datasets/quadrature_rules_gegenbauer/quadrature_rules_gegenbauer.html">
QUADRATURE_RULES_GEGENBAUER</a>,
a dataset directory which
contains triples of files defining Gauss-Gegenbauer quadrature rules.
</p>
<p>
<a href = "../../m_src/quadrule/quadrule.html">
QUADRULE</a>,
a MATLAB library which
contains 1-dimensional quadrature rules.
</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>
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>
Sylvan Elhay, Jaroslav Kautsky,<br>
Algorithm 655:
IQPACK,
FORTRAN Subroutines for the Weights of Interpolatory Quadrature,<br>
ACM Transactions on Mathematical Software,<br>
Volume 13, Number 4, December 1987, pages 399-415.
</li>
<li>
Jaroslav Kautsky, Sylvan Elhay,<br>
Calculation of the Weights of Interpolatory Quadratures,<br>
Numerische Mathematik,<br>
Volume 40, 1982, pages 407-422.
</li>
<li>
Roger Martin, James Wilkinson,<br>
The Implicit QL Algorithm,<br>
Numerische Mathematik,<br>
Volume 12, Number 5, December 1968, pages 377-383.
</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 = "gegenbauer_rule.m">gegenbauer_rule.m</a>
the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "gegen_o4_a2.0_r.txt">gegen_o4_a2.0_r.txt</a>,
the region file created by the command
<pre><b>
gegenbauer_rule ( 4, 2.0, -1.0, +1.0, 'gegen_o4_a2.0' )
</b></pre>
</li>
<li>
<a href = "gegen_o4_a2.0_w.txt">gegen_o4_a2.0_w.txt</a>,
the weight file created by the command
<pre><b>
gegenbauer_rule ( 4, 2.0, -1.0, +1.0, 'gegen_o4_a2.0' )
</b></pre>
</li>
<li>
<a href = "gegen_o4_a2.0_x.txt">gegen_o4_a2.0_x.txt</a>,
the abscissa file created by the command
<pre><b>
gegenbauer_rule ( 4, 2.0, -1.0, +1.0, 'gegen_o4_a2.0' )
</b></pre>
</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 24 February 2010.
</i>
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