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<html>
<head>
<title>
LEGENDRE_EXACTNESS - Exactness of Gauss-Legendre Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
LEGENDRE_EXACTNESS <br> Exactness of Gauss-Legendre Quadrature Rules
</h1>
<hr>
<p>
<b>LEGENDRE_EXACTNESS</b>
is a MATLAB program which
investigates the polynomial exactness of a Gauss-Legendre
quadrature rule for the interval [-1,+1].
</p>
<p>
This program is actually appropriate for <i>any</i> quadrature
rule that estimates integrals on [-1,+1], and which does not including
a weighting function <b>w(x)</b> in the integral. This includes:
<ul>
<li>
<i>Clenshaw-Curtis</i> rules;
</li>
<li>
<i>Fejer</i> rules of Type 1 or 2;
</li>
<li>
<i>Gauss-Legendre</i> rules;
</li>
<li>
<i>Gauss-Lobatto</i> rules (Gauss rule including both endpoints);
</li>
<li>
<i>Gauss-Patterson</i> rules;
</li>
<li>
<i>Gauss-Radau</i> rules (Gauss rule including one endpoint);
</li>
<li>
<i>Newton-Cotes</i> rules, open and closed forms;
</li>
</ul>
</p>
<p>
Standard Gauss-Legendre quadrature assumes that the integrand we are
considering has a form like:
<pre>
Integral ( -1 <= x <= +1 ) f(x) dx
</pre>
</p>
<p>
A <i>standard Gauss-Legendre quadrature rule</i> is a set of <b>n</b>
positive weights <b>w</b> and abscissas <b>x</b> so that
<pre>
Integral ( -1 <= x <= +1 ) f(x) dx
</pre>
may be approximated by
<pre>
Sum ( 1 <= I <= N ) w(i) * f(x(i))
</pre>
</p>
<p>
For a standard Gauss-Legendre rule, polynomial exactness is defined in terms of
the function <b>f(x)</b>. That is, we say the rule is exact for polynomials
up to degree DEGREE_MAX if, for any polynomial <b>f(x)</b> of that degree or
less, the quadrature rule will produce the exact value of
<pre>
Integral ( -1 <= x <= +1 ) f(x) dx
</pre>
</p>
<p>
The program starts at <b>DEGREE</b> = 0, and then
proceeds to <b>DEGREE</b> = 1, 2, and so on up to a maximum degree
<b>DEGREE_MAX</b> specified by the user. At each value of <b>DEGREE</b>,
the program generates the corresponding monomial term, applies the
quadrature rule to it, and determines the quadrature error. The program
uses a scaling factor on each monomial so that the exact integral
should always be 1; therefore, each reported error can be compared
on a fixed scale.
</p>
<p>
The program is very flexible and interactive. The quadrature rule
is defined by three files, to be read at input, and the
maximum degree top be checked is specified by the user as well.
</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>
<p>
The exactness results are written to an output file with the
corresponding name:
<ul>
<li>
<i>prefix</i>_<b>exact.txt</b>
</li>
</ul>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>legendre_exactness</b> ( <i>'prefix'</i>, <i>degree_max</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, entered as a quoted string;
</li>
<li>
<i>degree_max</i> is the maximum monomial degree to check. This would normally be
a relatively small nonnegative number, such as 5, 10 or 15.
</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>LEGENDRE_EXACTNESS</b> is available in
<a href = "../../c_src/legendre_exactness/legendre_exactness.html">a C version</a> and
<a href = "../../cpp_src/legendre_exactness/legendre_exactness.html">a C++ version</a> and
<a href = "../../f77_src/legendre_exactness/legendre_exactness.html">a FORTRAN77 version</a> and
<a href = "../../f_src/legendre_exactness/legendre_exactness.html">a FORTRAN90 version</a> and
<a href = "../../m_src/legendre_exactness/legendre_exactness.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/exactness/exactness.html">
EXACTNESS</a>,
a MATLAB library which
investigates the exactness of quadrature rules that estimate the
integral of a function with a density, such as 1, exp(-x) or
exp(-x^2), over an interval such as [-1,+1], [0,+oo) or (-oo,+oo).
</p>
<p>
<a href = "../../m_src/hermite_exactness/hermite_exactness.html">
HERMITE_EXACTNESS</a>,
a MATLAB program which
tests the polynomial exactness of Gauss-Hermite quadrature rules.
</p>
<p>
<a href = "../../m_src/int_exactness/int_exactness.html">
INT_EXACTNESS</a>,
a MATLAB program which
tests the polynomial exactness of a quadrature rule for a finite interval.
</p>
<p>
<a href = "../../m_src/int_exactness_chebyshev1/int_exactness_chebyshev1.html">
INT_EXACTNESS_CHEBYSHEV1</a>,
a MATLAB program which
tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.
</p>
<p>
<a href = "../../m_src/int_exactness_chebyshev2/int_exactness_chebyshev2.html">
INT_EXACTNESS_CHEBYSHEV2</a>,
a MATLAB program which
tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.
</p>
<p>
<a href = "../../m_src/int_exactness_gegenbauer/int_exactness_gegenbauer.html">
INT_EXACTNESS_GEGENBAUER</a>,
a MATLAB program which
tests the polynomial exactness of Gauss-Gegenbauer quadrature rules.
</p>
<p>
<a href = "../../m_src/int_exactness_gen_hermite/int_exactness_gen_hermite.html">
INT_EXACTNESS_GEN_HERMITE</a>,
a MATLAB program which
tests the polynomial exactness of generalized Gauss-Hermite quadrature rules.
</p>
<p>
<a href = "../../m_src/int_exactness_gen_laguerre/int_exactness_gen_laguerre.html">
INT_EXACTNESS_GEN_LAGUERRE</a>,
a MATLAB program which
tests the polynomial exactness of generalized Gauss-Laguerre quadrature rules.
</p>
<p>
<a href = "../../m_src/int_exactness_jacobi/int_exactness_jacobi.html">
INT_EXACTNESS_JACOBI</a>,
a MATLAB program which
tests the polynomial exactness of Gauss-Jacobi quadrature rules.
</p>
<p>
<a href = "../../m_src/laguerre_exactness/laguerre_exactness.html">
LAGUERRE_EXACTNESS</a>,
a MATLAB program which
tests the polynomial exactness of Gauss-Laguerre quadrature rules
for integration over [0,+oo) with density function exp(-x).
</p>
<p>
<a href = "../../m_src/legendre_rule/legendre_rule.html">
LEGENDRE_RULE</a>,
a MATLAB program which
generates a Gauss-Legendre quadrature
rule on request.
</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 = "legendre_exactness.m">legendre_exactness.m</a>,
is the source code.
</li>
</ul>
</p>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>LEG_O1</b> is a standard Gauss-Legendre order 1 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o1_x.txt">
leg_o1_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o1_w.txt">
leg_o1_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o1_r.txt">
leg_o1_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o1_exact.txt">leg_o1_exact.txt</a>,
the results of the command
<blockquote><b>
legendre_exactness ( 'leg_o1', 5 )
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>LEG_O2</b> is a standard Gauss-Legendre order 2 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o2_x.txt">
leg_o2_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o2_w.txt">
leg_o2_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o2_r.txt">
leg_o2_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o2_exact.txt">leg_o2_exact.txt</a>,
the results of the command
<blockquote><b>
legendre_exactness ( 'leg_o2', 5 )
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>LEG_O4</b> is a standard Gauss-Legendre order 4 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o4_x.txt">
leg_o4_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o4_w.txt">
leg_o4_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o4_r.txt">
leg_o4_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o4_exact.txt">leg_o4_exact.txt</a>,
the results of the command
<blockquote><b>
legendre_exactness ( 'leg_o4', 10 )
</b></blockquote>
</li>
</ul>
</p>
<p>
<b>LEG_O8</b> is a standard Gauss-Legendre order 8 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o8_x.txt">
leg_o8_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o8_w.txt">
leg_o8_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o8_r.txt">
leg_o8_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o8_exact.txt">leg_o8_exact.txt</a>,
the results of the exactness test.
</li>
the results of the command
<blockquote><b>
legendre_exactness ( 'leg_o8', 18 )
</b></blockquote>
</ul>
</p>
<p>
<b>LEG_O16</b> is a standard Gauss-Legendre order 16 rule.
<ul>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o16_x.txt">
leg_o16_x.txt</a>,
the abscissas of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o16_w.txt">
leg_o16_w.txt</a>,
the weights of the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o16_r.txt">
leg_o16_r.txt</a>,
defines the region for the rule.
</li>
<li>
<a href = "../../datasets/quadrature_rules_legendre/leg_o16_exact.txt">leg_o16_exact.txt</a>,
the results of the command
<blockquote><b>
legendre_exactness ( 'leg_o16', 35 )
</b></blockquote>
</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 05 August 2009.
</i>
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