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<html>
<head>
<title>
TETRAHEDRON_NCO_RULE - Newton-Cotes Open Quadrature for the Tetrahedron
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TETRAHEDRON_NCO_RULE <br> Newton-Cotes Open Quadrature for the Tetrahedron
</h1>
<hr>
<p>
<b>TETRAHEDRON_NCO_RULE</b>
is a MATLAB library which
defines the weights and abscisass for a sequence of
7 Newton-Cotes open 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 "open" when they do not 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>
<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_NCO_RULE</b> is available in
<a href = "../../c_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">a C version</a> and
<a href = "../../cpp_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">a C++ version</a> and
<a href = "../../f_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">a MATLAB version</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/cube_felippa_rule/cube_felippa_rule.html">
CUBE_FELIPPA_RULE</a>,
a MATLAB library which
returns the points and weights of a Felippa quadrature rule
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../m_src/line_nco_rule/line_nco_rule.html">
LINE_NCO_RULE</a>,
a MATLAB library which
computes a Newton Cotes Open (NCO) quadrature rule,
using equally spaced points,
over the interior of a line segment in 1D.
</p>
<p>
<a href = "../../m_src/pyramid_felippa_rule/pyramid_felippa_rule.html">
PYRAMID_FELIPPA_RULE</a>,
a MATLAB library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a pyramid in 3D.
</p>
<p>
<a href = "../../m_src/simplex_gm_rule/simplex_gm_rule.html">
SIMPLEX_GM_RULE</a>,
a MATLAB library which
defines Grundmann-Moeller quadrature rules
over the interior of a simplex in M dimensions.
</p>
<p>
<a href = "../../m_src/square_felippa_rule/square_felippa_rule.html">
SQUARE_FELIPPA_RULE</a>,
a MATLAB library which
returns the points and weights of a Felippa quadrature rule
over the interior of a square in 2D.
</p>
<p>
<a href = "../../m_src/stroud/stroud.html">
STROUD</a>,
a MATLAB library which
contains quadrature rules for a variety of unusual areas, surfaces
and volumes in 2D, 3D and N-dimensions.
</p>
<p>
<a href = "../../m_src/tetrahedron_arbq_rule/tetrahedron_arbq_rule.html">
TETRAHEDRON_ARBQ_RULE</a>,
a MATLAB 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 = "../../m_src/tetrahedron_exactness/tetrahedron_exactness.html">
TETRAHEDRON_EXACTNESS</a>,
a MATLAB program which
investigates the monomial exactness of a quadrature rule
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_felippa_rule/tetrahedron_felippa_rule.html">
TETRAHEDRON_FELIPPA_RULE</a>,
a MATLAB library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_integrals/tetrahedron_integrals.html">
TETRAHEDRON_INTEGRALS</a>,
a MATLAB 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 = "../../m_src/tetrahedron_keast_rule/tetrahedron_keast_rule.html">
TETRAHEDRON_KEAST_RULE</a>,
a MATLAB library which
defines ten quadrature rules, with exactness degrees 0 through 8,
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">
TETRAHEDRON_MONTE_CARLO</a>,
a MATLAB program which
uses the Monte Carlo method to estimate integrals
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_ncc_rule/tetrahedron_ncc_rule.html">
TETRAHEDRON_NCC_RULE</a>,
a MATLAB library which
defines Newton-Cotes closed quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/triangle_fekete_rule/triangle_fekete_rule.html">
TRIANGLE_FEKETE_RULE</a>,
a MATLAB library which
defines Fekete rules for quadrature or interpolation
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_felippa_rule/triangle_felippa_rule.html">
TRIANGLE_FELIPPA_RULE</a>,
a MATLAB library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_ncc_rule/triangle_ncc_rule.html">
TRIANGLE_NCC_RULE</a>,
a MATLAB library which
defines Newton-Cotes closed quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/wedge_felippa_rule/wedge_felippa_rule.html">
WEDGE_FELIPPA_RULE</a>,
a MATLAB 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_nco_degree.m">tetrahedron_nco_degree.m</a>
returns the degree of a given NCO rule for the tetrahedron.
</li>
<li>
<a href = "tetrahedron_nco_order_num.m">tetrahedron_nco_order_num.m</a>
returns the order of a given NCO rule for the tetrahedron.
</li>
<li>
<a href = "tetrahedron_nco_rule.m">tetrahedron_nco_rule.m</a>
returns the points and weights of an NCO rule.
</li>
<li>
<a href = "tetrahedron_nco_rule_num.m">tetrahedron_nco_rule_num.m</a>
returns the number of NCO rules available.
</li>
<li>
<a href = "tetrahedron_nco_suborder.m">tetrahedron_nco_suborder.m</a>
returns the suborders for an NCO rule.
</li>
<li>
<a href = "tetrahedron_nco_suborder_num.m">tetrahedron_nco_suborder_num.m</a>
returns the number of suborders for an NCO rule.
</li>
<li>
<a href = "tetrahedron_nco_subrule.m">tetrahedron_nco_subrule.m</a>
returns a compressed NCO rule.
</li>
<li>
<a href = "r8mat_det_4d.m">r8mat_det_4d.m</a>
returns the determinant of a 4x4 matrix.
</li>
<li>
<a href = "reference_to_physical_t4.m">reference_to_physical_t4.m</a>
maps T4 reference points to physical points.
</li>
<li>
<a href = "tetrahedron_volume.m">tetrahedron_volume.m</a>
computes the volume of a tetrahedron.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>
prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "tetrahedron_nco_test.m">tetrahedron_nco_test.m</a>,
runs all the tests.
</li>
<li>
<a href = "tetrahedron_nco_test01.m">tetrahedron_nco_test01.m</a>,
tests TETRAHEDRON_NCO_RULE_NUM, TETRAHEDRON_NCO_DEGREE, and
TETRAHEDRON_NCO_ORDER_NUM.
</li>
<li>
<a href = "tetrahedron_nco_test02.m">tetrahedron_nco_test02.m</a>,
tests TETRAHEDRON_NCO_RULE by summing the weights.
</li>
<li>
<a href = "tetrahedron_nco_test03.m">tetrahedron_nco_test03.m</a>,
tests TETRAHEDRON_NCO_RULE by summing the barycentric coordinates.
</li>
<li>
<a href = "tetrahedron_nco_test04.m">tetrahedron_nco_test04.m</a>,
tests TETRAHEDRON_NCO_ORDER by integrating monomials in the
unit tetrahedron.
</li>
<li>
<a href = "tetrahedron_nco_test05.m">tetrahedron_nco_test05.m</a>,
tests REFERENCE_TO_PHYSICAL_T4 by transforming an NCO rule
from the unit tetrahedron to another tetrahedron.
</li>
<li>
<a href = "tetrahedron_nco_test06.m">tetrahedron_nco_test06.m</a>,
tests TETRAHEDRON_NCO_RULE by printing out one rule.
</li>
<li>
<a href = "tetrahedron_nco_test_output.txt">tetrahedron_nco_test_output.txt</a>,
the output from a run of the sample program.
</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 18 June 2014.
</i>
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