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<html>
<head>
<title>
EXACTNESS - Exactness of Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
EXACTNESS <br> Exactness of Quadrature Rules
</h1>
<hr>
<p>
<b>EXACTNESS</b>
is a C++ 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 1D quadrature rule estimates I(f), the integral of a function f(x)
over an interval [a,b] with density rho(x):
<pre>
I(f) = integral ( a < x < b ) f(x) rho(x) dx
</pre>
by a n-point quadrature rule of weights w and points x:
<pre>
Q(f) = sum ( 1 <= i <= n ) w(i) f(x(i))
</pre>
</p>
<p>
Most quadrature rules come in a family of various sizes. A quadrature
rule of size n is said to have exactness p if it is true that the
quadrature estimate is exactly equal to the exact integral for every
monomial (and hence, polynomial) whose degree is p or less.
</p>
<p>
This program allows the user to specify a quadrature rule, a size n,
and a degree p_max. It then computes and compares the exact integral
and quadrature estimate for monomials of degree 0 through p_max, so
that the user can analyze the results.
</p>
<p>
Common quadrature rules include:
<ul>
<li>
Gauss-Hermite quadrature, with density exp(-x^2), over (-oo,+oo);
exactness = 2 * n - 1;
</li>
<li>
Gauss-Laguerre quadrature, with density exp(-1), over [0,+oo);
exactness = 2 * n - 1;
</li>
<li>
Gauss-Legendre quadrature, with density 1, over [-1,+1];
exactness = 2 * n - 1;
</li>
</ul>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files 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>EXACTNESS</b> is available in
<a href = "../../c_src/exactness/exactness.html">a C version</a> and
<a href = "../../cpp_src/exactness/exactness.html">a C++ version</a> and
<a href = "../../f77_src/exactness/exactness.html">a FORTRAN77 version</a> and
<a href = "../../f_src/exactness/exactness.html">a FORTRAN90 version</a> and
<a href = "../../m_src/exactness/exactness.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/exactness_2d/exactness_2d.html">
EXACTNESS_2D</a>,
a C++ library which
investigates the exactness of 2D quadrature rules that estimate the
integral of a function f(x,y) over a 2D domain.
</p>
<p>
<a href = "../../cpp_src/hermite_exactness/hermite_exactness.html">
HERMITE_EXACTNESS</a>,
a C++ program which
tests the monomial exactness of Gauss-Hermite quadrature rules
for estimating the integral of a function with density exp(-x^2)
over the interval (-oo,+oo).
</p>
<p>
<a href = "../../cpp_src/laguerre_exactness/laguerre_exactness.html">
LAGUERRE_EXACTNESS</a>,
a C++ program which
tests the monomial exactness of Gauss-Laguerre quadrature rules
for estimating the integral of a function with density exp(-x)
over the interval [0,+oo).
</p>
<p>
<a href = "../../cpp_src/legendre_exactness/legendre_exactness.html">
LEGENDRE_EXACTNESS</a>,
a C++ program which
tests the monomial exactness of Gauss-Legendre quadrature rules
for estimating the integral of a function with density 1
over the interval [-1,+1].
</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 = "exactness.cpp">exactness.cpp</a>, the source code.
</li>
<li>
<a href = "exactness.hpp">exactness.hpp</a>, the include file.
</li>
<li>
<a href = "exactness.sh">exactness.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "exactness_prb.cpp">exactness_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "exactness_prb.sh">exactness_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "exactness_prb_output.txt">exactness_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>HERMITE_EXACTNESS</b> investigates exactness of Hermite quadrature.
</li>
<li>
<b>HERMITE_INTEGRAL</b> evaluates a monomial Hermite integral.
</li>
<li>
<b>HERMITE_MONOMIAL_QUADRATURE</b> applies a quadrature rule to a monomial.
</li>
<li>
<b>LAGUERRE_EXACTNESS</b> investigates exactness of Laguerre quadrature.
</li>
<li>
<b>LAGUERRE_INTEGRAL</b> evaluates a monomial integral associated with L(n,x).
</li>
<li>
<b>LAGUERRE_MONOMIAL_QUADRATURE</b> applies Laguerre quadrature to a monomial.
</li>
<li>
<b>LEGENDRE_EXACTNESS</b> investigates exactness of Legendre quadrature.
</li>
<li>
<b>LEGENDRE_INTEGRAL</b> evaluates a monomial Legendre integral.
</li>
<li>
<b>LEGENDRE_MONOMIAL_QUADRATURE</b> applies a quadrature rule to a monomial.
</li>
<li>
<b>R8_FACTORIAL</b> computes the factorial of N.
</li>
<li>
<b>R8_FACTORIAL2</b> computes the double factorial function.
</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 18 May 2014.
</i>
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