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<html>
<head>
<title>
TETRAHEDRON_NCC_RULE - Newton-Cotes Closed Quadrature for the Tetrahedron
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TETRAHEDRON_NCC_RULE <br> Newton-Cotes Closed Quadrature for the Tetrahedron
</h1>
<hr>
<p>
<b>TETRAHEDRON_NCC_RULE</b>
is a C++ library which
defines the weights and abscisass for a sequence of
7 Newton-Cotes closed quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
Newton-Cotes rules have the characteristic that the abscissas
are equally spaced. For a tetrahedron, this refers to spacing
in the unit reference tetrahedron, or in the barycentric coordinate
system. These rules may be mapped to an arbitrary tetrahedron,
and will still be valid.
</p>
<p>
The rules are said to be "closed" when they include points on
the boundary of the tetrahedron.
</p>
<p>
The use of equally spaced abscissas may be important for your
application. That may how your data was collected, for instance.
On the other hand, the use of equally spaced abscissas carries
a few costs. In particular, for a given degree of polynomial
accuracy, there will be rules that achieve this accuracy, but
use fewer abscissas than Newton-Cotes. Moreover, the Newton-Cotes
approach almost always results in negative weights for some
abscissas. This is generally an undesirable feature, particularly
when higher order quadrature rules are being used.
</p>
<p>
(Note that the first rule included in the set is not, strictly
speaking, a Newton-Cotes closed rule; it's just the rule that
uses a single point at the centroid. However, by including
this rule as the first in the set, we have a rule with each
polynomial degree of exactness from 0 to 6.)
</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>TETRAHEDRON_NCC_RULE</b> is available in
<a href = "../../c_src/tetrahedron_ncc_rule/tetrahedron_ncc_rule.html">a C version</a> and
<a href = "../../cpp_src/tetrahedron_ncc_rule/tetrahedron_ncc_rule.html">a C++ version</a> and
<a href = "../../f_src/tetrahedron_ncc_rule/tetrahedron_ncc_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/tetrahedron_ncc_rule/tetrahedron_ncc_rule.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/pyramid_felippa_rule/pyramid_felippa_rule.html">
PYRAMID_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a pyramid in 3D.
</p>
<p>
<a href = "../../cpp_src/simplex_gm_rule/simplex_gm_rule.html">
SIMPLEX_GM_RULE</a>,
a C++ library which
defines Grundmann-Moeller quadrature rules
over the interior of the simplex in M dimensions.
</p>
<p>
<a href = "../../cpp_src/square_felippa_rule/square_felippa_rule.html">
SQUARE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a square in 2D.
</p>
<p>
<a href = "../../cpp_src/stroud/stroud.html">
STROUD</a>,
a C++ library which
defines quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_arbq_rule/tetrahedron_arbq_rule.html">
TETRAHEDRON_ARBQ_RULE</a>,
a C++ library which
returns quadrature rules,
with exactness up to total degree 15,
over the interior of a tetrahedron in 3D,
by Hong Xiao and Zydrunas Gimbutas.
</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/tetrahedron_felippa_rule/tetrahedron_felippa_rule.html">
TETRAHEDRON_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_integrals/tetrahedron_integrals.html">
TETRAHEDRON_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_keast_rule/tetrahedron_keast_rule.html">
TETRAHEDRON_KEAST_RULE</a>,
a C++ library which
defines ten quadrature rules, with exactness degrees 0 through 8,
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">
TETRAHEDRON_MONTE_CARLO</a>,
a C++ library which
uses the Monte Carlo method to estimate integrals
over the interior of the unit tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">
TETRAHEDRON_NCO_RULE</a>,
a C++ library which
defines Newton-Cotes open quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/triangle_fekete_rule/triangle_fekete_rule.html">
TRIANGLE_FEKETE_RULE</a>,
a C++ library which
defines Fekete rules for interpolation or quadrature
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_felippa_rule/triangle_felippa_rule.html">
TRIANGLE_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_ncc_rule/triangle_ncc_rule.html">
TRIANGLE_NCC_RULE</a>,
a C++ library which
defines Newton-Cotes closed quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/wedge_felippa_rule/wedge_felippa_rule.html">
WEDGE_FELIPPA_RULE</a>,
a C++ library which
returns quadratures rules for approximating integrals
over the interior of the unit wedge in 3D.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Peter Silvester,<br>
Symmetric Quadrature Formulae for Simplexes,<br>
Mathematics of Computation,<br>
Volume 24, Number 109, January 1970, pages 95-100.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "tetrahedron_ncc_rule.cpp">tetrahedron_ncc_rule.cpp</a>,
the source code.
</li>
<li>
<a href = "tetrahedron_ncc_rule.hpp">tetrahedron_ncc_rule.hpp</a>,
the include file.
</li>
<li>
<a href = "tetrahedron_ncc_rule.sh">tetrahedron_ncc_rule.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "tetrahedron_ncc_rule_prb.cpp">tetrahedron_ncc_rule_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "tetrahedron_ncc_rule_prb.sh">tetrahedron_ncc_rule_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "tetrahedron_ncc_rule_prb_output.txt">tetrahedron_ncc_rule_prb_output.txt</a>,
the output from a run of the sample program.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>R8MAT_DET_4D</b> computes the determinant of a 4 by 4 R8MAT.
</li>
<li>
<b>REFERENCE_TO_PHYSICAL_T4</b> maps T4 reference points to physical points.
</li>
<li>
<b>TETRAHEDRON_NCC_DEGREE:</b> degree of an NCC rule for the tetrahedron.
</li>
<li>
<b>TETRAHEDRON_NCC_ORDER_NUM:</b> order of an NCC rule for the tetrahedron.
</li>
<li>
<b>TETRAHEDRON_NCC_RULE</b> returns the points and weights of an NCC rule.
</li>
<li>
<b>TETRAHEDRON_NCC_RULE_NUM</b> returns the number of NCC rules available.
</li>
<li>
<b>TETRAHEDRON_NCC_SUBORDER</b> returns the suborders for an NCC rule.
</li>
<li>
<b>TETRAHEDRON_NCC_SUBORDER_NUM</b> returns the number of suborders for an NCC rule.
</li>
<li>
<b>TETRAHEDRON_NCC_SUBRULE</b> returns a compressed NCC rule.
</li>
<li>
<b>TETRAHEDRON_NCC_SUBRULE_01</b> returns a compressed NCC rule 1.
</li>
<li>
<b>TETRAHEDRON_NCC_SUBRULE_02</b> returns a compressed NCC rule 2.
</li>
<li>
<b>TETRAHEDRON_NCC_SUBRULE_03</b> returns a compressed NCC rule 3.
</li>
<li>
<b>TETRAHEDRON_NCC_SUBRULE_04</b> returns a compressed NCC rule 4.
</li>
<li>
<b>TETRAHEDRON_NCC_SUBRULE_05</b> returns a compressed NCC rule 5.
</li>
<li>
<b>TETRAHEDRON_NCC_SUBRULE_06</b> returns a compressed NCC rule 6.
</li>
<li>
<b>TETRAHEDRON_NCC_SUBRULE_07</b> returns a compressed NCC rule 7.
</li>
<li>
<b>TETRAHEDRON_VOLUME</b> computes the volume of a tetrahedron.
</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 14 June 2014.
</i>
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