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<html>
<head>
<title>
PATTERSON_RULE - Gauss-Patterson Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
PATTERSON_RULE <br> Gauss-Patterson Quadrature Rules
</h1>
<hr>
<p>
<b>PATTERSON_RULE</b>
is a MATLAB program which
generates a specific Gauss-Patterson 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 Gauss-Patterson quadrature is a nested family which begins with
the Gauss-Legendre rules of orders 1 and 3, and then succesively inserts
one new abscissa in each subinterval. Thus, after the second rule, the
Gauss-Patterson rules do not have the super-high precision of the
Gauss-Legendre rules. They trade this precision in exchange for the
advantages of nestedness. This means that Gauss-Patterson rules are
only available for orders of 1, 3, 7, 15, 31, 63, 127, 255 or 511.
</p>
<p>
The <i>standard Gauss-Patterson quadrature rule </i> 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>
The polynomial precision of a Gauss-Patterson rule can be checked
numerically by the <b>INT_EXACTNESS_LEGENDRE</b> program. We should expect
<table border=1>
<tr>
<th>Index</th><th>Order</th><th>Free+Fixed</th><th>Expected Precision</th><th>Actual Precision</th>
</tr>
<tr>
<td>0</td><td>1</td><td>1 + 0</td><td>2*1+0-1=1</td><td>1</td>
</tr>
<tr>
<td>1</td><td>3</td><td>3 + 0</td><td>2*3+0-1=5</td><td>5</td>
</tr>
<tr>
<td>2</td><td>7</td><td>4 + 3</td><td>2*4+3-1=10</td><td>10 + 1 = 11</td>
</tr>
<tr>
<td>3</td><td>15</td><td>8 + 7</td><td>2*8+7-1=22</td><td>22 + 1 = 23</td>
</tr>
<tr>
<td>4</td><td>31</td><td>16 + 15</td><td>2*16+15-1=46</td><td>46 + 1 = 47</td>
</tr>
<tr>
<td>5</td><td>63</td><td>32 + 31</td><td>2*32+31-1=94</td><td>94 + 1 = 95</td>
</tr>
<tr>
<td>6</td><td>127</td><td>64 + 63</td><td>2*64+63-1=190</td><td>190 + 1 = 191</td>
</tr>
<tr>
<td>7</td><td>255</td><td>128 + 127</td><td>2*128+127-1=382</td><td>382 + 1 = 383</td>
</tr>
<tr>
<td>8</td><td>511</td><td>256 + 255</td><td>2*256+255-1=766</td><td>766 + 1 = 767</td>
</tr>
</table>
where the extra 1 degree of precision comes about because the rules are symmetric,
and can integrate any odd monomial exactly. Thus, after the first rule, the
precision is 3*2^index - 1.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>patterson_rule</b> ( <i>order</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. Acceptable values are
1, 3, 7, 15, 31, 63, 127, 255 or 511.
</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:
<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>PATTERSON_RULE</b> is available in
<a href = "../../c_src/patterson_rule/patterson_rule.html">a C version</a> and
<a href = "../../cpp_src/patterson_rule/patterson_rule.html">a C++ version</a> and
<a href = "../../f77_src/patterson_rule/patterson_rule.html">a FORTRAN77 version</a> and
<a href = "../../f_src/patterson_rule/patterson_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/patterson_rule/patterson_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/gegenbauer_rule/gegenbauer_rule.html">
GEGENBAUER_RULE</a>,
a MATLAB program which
can compute and print a Gauss-Gegenbauer 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_legendre/int_exactness_legendre.html">
INT_EXACTNESS_LEGENDRE</a>,
a MATLAB program which
checks the polynomial exactness of a Gauss-Legendre quadrature rule.
</p>
<p>
<a href = "../../f_src/intlib/intlib.html">
INTLIB</a>,
a FORTRAN90 library which
contains routines for numerical estimation of integrals in 1D.
</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
can compute and print 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 = "../../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 = "../../datasets/quadrature_rules_legendre/quadrature_rules_legendre.html">
QUADRATURE_RULES_LEGENDRE</a>,
a dataset directory which
contains triples of files defining standard Gauss-Legendre
quadrature rules.
</p>
<p>
<a href = "../../m_src/quadrule/quadrule.html">
QUADRULE</a>,
a MATLAB library which
defines 1-dimensional quadrature rules.
</p>
<p>
<a href = "../../f77_src/toms699/toms699.html">
TOMS699</a>,
a FORTRAN77 library which
implements a new representation of Patterson's quadrature formula;<br>
this is ACM TOMS algorithm 699.
</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>
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 = "patterson_rule.m">patterson_rule.m</a>
the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "gp_o15_r.txt">gp_o15_r.txt</a>,
the region file created by the command
<pre><b>
patterson_rule ( 15, 'gp_o15' )
</b></pre>
</li>
<li>
<a href = "gp_o15_w.txt">gp_o15_w.txt</a>,
the weight file created by the command
<pre><b>
patterson_rule ( 15, 'gp_o15' )
</b></pre>
</li>
<li>
<a href = "gp_o15_x.txt">gp_o15_x.txt</a>,
the abscissa file created by the command
<pre><b>
patterson_rule ( 15, 'gp_o15' )
</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 12 February 2010.
</i>
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