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<html>
<head>
<title>
TRIANGLE_NCC_RULE - Newton-Cotes Closed Quadrature Rules for the Triangle
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGLE_NCC_RULE <br> Newton-Cotes Closed Quadrature Rules for the Triangle
</h1>
<hr>
<p>
<b>TRIANGLE_NCC_RULE</b>
is a MATLAB library which
defines the weights and abscisass for a sequence of
9 Newton-Cotes closed quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
Newton-Cotes rules have the characteristic that the abscissas
are equally spaced. For a triangle, this refers to spacing
in the unit reference triangle, or in the barycentric coordinate
system. These rules may be mapped to an arbitrary triangle,
and will still be valid.
</p>
<p>
The rules are said to be "closed" when they include points on
the boundary of the triangle.
</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 8.)
</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>TRIANGLE_NCC_RULE</b> is available in
<a href = "../../c_src/triangle_ncc_rule/triangle_ncc_rule.html">a C version</a> and
<a href = "../../cpp_src/triangle_ncc_rule/triangle_ncc_rule.html">a C++ version</a> and
<a href = "../../f_src/triangle_ncc_rule/triangle_ncc_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangle_ncc_rule/triangle_ncc_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/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 M-dimensions.
</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_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_dunavant_rule/triangle_dunavant_rule.html">
TRIANGLE_DUNAVANT_RULE</a>,
a MATLAB library which
sets up a Dunavant quadrature rule
over the interior of a triangle in 2D.
</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_lyness_rule/triangle_lyness_rule.html">
TRIANGLE_LYNESS_RULE</a>,
a MATLAB library which
returns Lyness-Jespersen quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_nco_rule/triangle_nco_rule.html">
TRIANGLE_NCO_RULE</a>,
a MATLAB library which
defines Newton-Cotes open quadrature
rules on a triangle.
</p>
<p>
<a href = "../../m_src/triangle_symq_rule/triangle_symq_rule.html">
TRIANGLE_SYMQ_RULE</a>,
a MATLAB library which
returns efficient symmetric quadrature rules,
with exactness up to total degree 50,
over the interior of an arbitrary triangle in 2D,
by Hong Xiao and Zydrunas Gimbutas.
</p>
<p>
<a href = "../../m_src/triangle_wandzura_rule/triangle_wandzura_rule.html">
TRIANGLE_WANDZURA_RULE</a>,
a MATLAB library which
sets up a quadrature rule of exactness 5, 10, 15, 20, 25 or 30
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>
Gisela Engeln-Muellges, Frank Uhlig,<br>
Numerical Algorithms with C,<br>
Springer, 1996,<br>
ISBN: 3-540-60530-4,<br>
LC: QA297.E56213.
</li>
<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 = "i4_modp.m">i4_modp.m</a>
returns the nonnegative remainder of integer division.
</li>
<li>
<a href = "i4_wrap.m">i4_wrap.m</a>
forces an integer to lie between given limits by wrapping.
</li>
<li>
<a href = "reference_to_physical_t3.m">reference_to_physical_t3.m</a>
maps T3 reference points to physical points.
</li>
<li>
<a href = "triangle_area.m">triangle_area.m</a>
computes the area of a triangle.
</li>
<li>
<a href = "triangle_ncc_degree.m">triangle_ncc_degree.m</a>
returns the degree of a given NCC rule for the triangle.
</li>
<li>
<a href = "triangle_ncc_order_num.m">triangle_ncc_order_num.m</a>
returns the order of a given NCC rule for the triangle.
</li>
<li>
<a href = "triangle_ncc_rule.m">triangle_ncc_rule.m</a>
returns the points and weights of an NCC rule.
</li>
<li>
<a href = "triangle_ncc_rule_num.m">triangle_ncc_rule_num.m</a>
returns the number of NCC rules available.
</li>
<li>
<a href = "triangle_ncc_suborder.m">triangle_ncc_suborder.m</a>
returns the suborders for an NCC rule.
</li>
<li>
<a href = "triangle_ncc_suborder_num.m">triangle_ncc_suborder_num.m</a>
returns the number of suborders for an NCC rule.
</li>
<li>
<a href = "triangle_ncc_subrule.m">triangle_ncc_subrule.m</a>
returns a compressed NCC rule.
</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 = "triangle_ncc_rule_test.m">triangle_ncc_rule_test.m</a>,
runs all the tests.
</li>
<li>
<a href = "triangle_ncc_rule_test01.m">triangle_ncc_rule_test01.m</a>,
tests TRIANGLE_NCC_RULE_NUM, TRIANGLE_NCC_DEGREE, and
TRIANGLE_NCC_ORDER_NUM.
</li>
<li>
<a href = "triangle_ncc_rule_test02.m">triangle_ncc_rule_test02.m</a>,
tests TRIANGLE_NCC_RULE by summing the weights.
</li>
<li>
<a href = "triangle_ncc_rule_test03.m">triangle_ncc_rule_test03.m</a>,
tests TRIANGLE_NCC_RULE by summing the barycentric coordinates.
</li>
<li>
<a href = "triangle_ncc_rule_test04.m">triangle_ncc_rule_test04.m</a>,
tests TRIANGLE_NCC_ORDER by integrating monomials in the
unit triangle.
</li>
<li>
<a href = "triangle_ncc_rule_test05.m">triangle_ncc_rule_test05.m</a>,
tests REFERENCE_TO_PHYSICAL_T3 by transforming an NCC rule
from the unit triangle to another triangle.
</li>
<li>
<a href = "triangle_ncc_rule_test_output.txt">triangle_ncc_rule_test_output.txt</a>,
the output file.
</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 17 June 2014.
</i>
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