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<html>
<head>
<title>
CUBE_EXACTNESS - Exactness of 3D Quadrature Rules
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
CUBE_EXACTNESS <br> Exactness of 3D Quadrature Rules
</h1>
<hr>
<p>
<b>CUBE_EXACTNESS</b>
is a C++ library which
investigates the polynomial exactness of quadrature rules
over the interior of a cube in 3D.
</p>
<p>
We assume that the integral to be approximated is of a Legendre
type, over a rectangular region:
<pre>
I(f) = integral ( z1 <= z <= z2 )
integral ( y1 <= y <= y2 )
integral ( x1 <= x <= x2 ) f(x,y,z) dx dy dz
</pre>
and that such integrals are to be approximated by:
<pre>
Q(f) = sum ( 1 <= i <= N ) w(i) * f(x(i),y(i),z(i))
</pre>
</p>
<p>
To determine the exactness of a given quadrature rule, we simply compare
the exact integral I(f) to the estimated integral Q(f) for a sequence of
monomials of increasing total degree D. This sequence begins with:
<pre>
D = 0: 1
D = 1: x y z
D = 2: x^2 xy xz y^2 yz z^2
D = 3: x^3 x^2y x^2z xy^2 xyz xz^2 y^3 y^2z yz^2 z^3
</pre>
and the exactness of a quadrature rule is defined as the largest value
of D such that I(f) and Q(f) are equal for all monomials up to and
including those of total degree D.
</p>
<p>
Note that if the 3D quadrature rule is formed as a product of
two 1D rules, then knowledge of the 1D exactness of the individual
factors gives sufficient information to determine the exactness
of the product rule, which will simply be the minimum of the exactnesses
of the three factor rules.
</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>CUBE_EXACTNESS</b> is available in
<a href = "../../c_src/cube_exactness/cube_exactness.html">a C version</a> and
<a href = "../../cpp_src/cube_exactness/cube_exactness.html">a C++ version</a> and
<a href = "../../f77_src/cube_exactness/cube_exactness.html">a FORTRAN77 version</a> and
<a href = "../../f_src/cube_exactness/cube_exactness.html">a FORTRAN90 version</a> and
<a href = "../../m_src/cube_exactness/cube_exactness.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/cube_felippa_rule/cube_felippa_rule.html">
CUBE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../cpp_src/cube_grid/cube_grid.html">
CUBE_GRID</a>,
a C++ library which
computes a grid of points
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../cpp_src/hypercube_exactness/hypercube_exactness.html">
HYPERCUBE_EXACTNESS</a>,
a C++ program which
measures the monomial exactness of an M-dimensional quadrature rule
over the interior of the unit hypercube in M dimensions.
</p>
<p>
<a href = "../../cpp_src/pyramid_exactness/pyramid_exactness.html">
PYRAMID_EXACTNESS</a>,
a C++ program which
investigates the monomial exactness of a quadrature rule
over the interior of a pyramid in 3D.
</p>
<p>
<a href = "../../cpp_src/sphere_exactness/sphere_exactness.html">
SPHERE_EXACTNESS</a>,
a C++ program which
tests the monomial exactness of a quadrature rule
on the surface of the unit sphere in 3D.
</p>
<p>
<a href = "../../cpp_src/square_exactness/square_exactness.html">
SQUARE_EXACTNESS</a>,
a C++ library which
investigates the polynomial exactness of quadrature rules for f(x,y)
over the interior of a rectangle in 2D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_exactness/tetrahedron_exactness.html">
TETRAHEDRON_EXACTNESS</a>,
a C++ program which
investigates the monomial exactness of a quadrature rule
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/triangle_exactness/triangle_exactness.html">
TRIANGLE_EXACTNESS</a>,
a C++ program which
investigates the monomial exactness quadrature rule
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/wedge_exactness/wedge_exactness.html">
WEDGE_EXACTNESS</a>,
a C++ program which
investigates the monomial exactness of a quadrature rule
over the interior of the unit wedge in 3D.
</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 = "cube_exactness.cpp">cube_exactness.cpp</a>, the source code.
</li>
<li>
<a href = "cube_exactness.hpp">cube_exactness.hpp</a>, the include file.
</li>
<li>
<a href = "cube_exactness.sh">cube_exactness.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "cube_exactness_prb.cpp">cube_exactness_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "cube_exactness_prb.sh">cube_exactness_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "cube_exactness_prb_output.txt">cube_exactness_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>LEGENDRE_3D_EXACTNESS:</b> monomial exactness for the 3D Legendre integral.
</li>
<li>
<b>LEGENDRE_3D_MONOMIAL_INTEGRAL</b> the Legendre integral of a monomial.
</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 16 August 2014.
</i>
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