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<html>
<head>
<title>
NCC_TRIANGLE - Newton-Cotes Closed Quadrature Rules for the Triangle
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
NCC_TRIANGLE <br> Newton-Cotes Closed Quadrature Rules for the Triangle
</h1>
<hr>
<p>
<b>NCC_TRIANGLE</b>
is a FORTRAN90 library which
defines the weights and abscisass for a sequence of
9 Newton-Cotes closed quadrature rules for the triangle.
</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>NCC_TRIANGLE</b> is available in
<a href = "../../cpp_src/ncc_triangle/ncc_triangle.html">a C++ version</a> and
<a href = "../../f_src/ncc_triangle/ncc_triangle.html">a FORTRAN90 version</a> and
<a href = "../../m_src/ncc_triangle/ncc_triangle.html">a MATLAB version</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/dunavant/dunavant.html">
DUNAVANT</a>,
a FORTRAN90 library which
defines Dunavant rules for quadrature on a triangle.
</p>
<p>
<a href = "../../f_src/fekete/fekete.html">
FEKETE</a>,
a FORTRAN90 library which
defines Fekete rules for quadrature or interpolation
on a triangle.
</p>
<p>
<a href = "../../f_src/felippa/felippa.html">
FELIPPA</a>,
a FORTRAN90 library which
defines quadrature rules for lines, triangles, quadrilaterals,
pyramids, wedges, tetrahedrons and hexahedrons.
</p>
<p>
<a href = "../../f_src/gm_rule/gm_rule.html">
GM_RULE</a>,
a FORTRAN90 library which
defines a Grundmann-Moeller
rule for quadrature over a triangle, tetrahedron, or general
M-dimensional simplex.
</p>
<p>
<a href = "../../f_src/lyness_rule/lyness_rule.html">
LYNESS_RULE</a>,
a FORTRAN90 library which
returns Lyness-Jespersen quadrature rules for the triangle.
</p>
<p>
<a href = "../../f_src/ncc_tetrahedron/ncc_tetrahedron.html">
NCC_TETRAHEDRON</a>,
a FORTRAN90 library which
defines Newton-Cotes closed quadrature rules on a tetrahedron.
</p>
<p>
<a href = "../../f_src/nco_tetrahedron/nco_tetrahedron.html">
NCO_TETRAHEDRON</a>,
a FORTRAN90 library which
defines Newton-Cotes open quadrature rules on a tetrahedron.
</p>
<p>
<a href = "../../f_src/nco_triangle/nco_triangle.html">
NCO_TRIANGLE</a>,
a FORTRAN90 library which
defines Newton-Cotes open quadrature
rules on a triangle.
</p>
<p>
<a href = "../../f_src/nintlib/nintlib.html">
NINTLIB</a>,
a FORTRAN90 library which
defines a variety
of routines for numerical estimation of integrals in multiple dimensions.
</p>
<p>
<a href = "../../datasets/quadrature_rules_tri/quadrature_rules_tri.html">
QUADRATURE_RULES_TRI</a>,
a dataset directory which
contains triples of files which
defines various quadrature
rules on triangles.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
defines quadrature rules on a
variety of intervals with different weight functions.
</p>
<p>
<a href = "../../f_src/stroud/stroud.html">
STROUD</a>,
a FORTRAN90 library which
defines quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
</p>
<p>
<a href = "../../f_src/test_tri_int/test_tri_int.html">
TEST_TRI_INT</a>,
a FORTRAN90 library which
can be used to test algorithms
for quadrature over a triangle.
</p>
<p>
<a href = "../../f77_src/toms612/toms612.html">
TOMS612</a>,
a FORTRAN77 library which
can estimate
the integral of a function over a triangle.
</p>
<p>
<a href = "../../f77_src/toms706/toms706.html">
TOMS706</a>,
a FORTRAN77 library which
estimates the integral of a function
over a triangulated region.
</p>
<p>
<a href = "../../f_src/triangle_exactness/triangle_exactness.html">
TRIANGLE_EXACTNESS</a>,
a FORTRAN90 program which
investigates the polynomial
exactness of a quadrature rule for the triangle.
</p>
<p>
<a href = "../../f_src/triangle_monte_carlo/triangle_monte_carlo.html">
TRIANGLE_MONTE_CARLO</a>,
a FORTRAN90 program which
uses the Monte Carlo method to estimate integrals over a triangle.
</p>
<p>
<a href = "../../f_src/wandzura/wandzura.html">
WANDZURA</a>,
a FORTRAN90 library which
defines Wandzura rules for quadrature
on a triangle.
</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 = "ncc_triangle.f90">ncc_triangle.f90</a>, the source code.
</li>
<li>
<a href = "ncc_triangle.sh">ncc_triangle.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "ncc_triangle_prb.f90">ncc_triangle_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "ncc_triangle_prb.sh">ncc_triangle_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "ncc_triangle_prb_output.txt">ncc_triangle_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
One of the tests in the sample calling program creates
<a href = "../../data/eps/eps.html">EPS</a> files of
the abscissas in the unit triangle. These have been converted
to <a href = "../../data/png/png.html">PNG</a> files for
display here.
<ul>
<li>
<a href = "ncc_triangle_rule_01.png">ncc_triangle_rule_01.png</a>,
a plot of rule 1.
</li>
<li>
<a href = "ncc_triangle_rule_02.png">ncc_triangle_rule_02.png</a>,
a plot of rule 2.
</li>
<li>
<a href = "ncc_triangle_rule_03.png">ncc_triangle_rule_03.png</a>,
a plot of rule 3.
</li>
<li>
<a href = "ncc_triangle_rule_04.png">ncc_triangle_rule_04.png</a>,
a plot of rule 4.
</li>
<li>
<a href = "ncc_triangle_rule_05.png">ncc_triangle_rule_05.png</a>,
a plot of rule 5.
</li>
<li>
<a href = "ncc_triangle_rule_06.png">ncc_triangle_rule_06.png</a>,
a plot of rule 6.
</li>
<li>
<a href = "ncc_triangle_rule_07.png">ncc_triangle_rule_07.png</a>,
a plot of rule 7.
</li>
<li>
<a href = "ncc_triangle_rule_08.png">ncc_triangle_rule_08.png</a>,
a plot of rule 8.
</li>
<li>
<a href = "ncc_triangle_rule_09.png">ncc_triangle_rule_09.png</a>,
a plot of rule 9.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>FILE_NAME_INC</b> increments a partially numeric filename.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of integer division.
</li>
<li>
<b>I4_WRAP</b> forces an integer to lie between given limits by wrapping.
</li>
<li>
<b>NCC_TRIANGLE_DEGREE</b> returns the degree of an NCC rule for the triangle.
</li>
<li>
<b>NCC_TRIANGLE_ORDER_NUM</b> returns the order of an NCC rule for the triangle.
</li>
<li>
<b>NCC_TRIANGLE_RULE</b> returns the points and weights of an NCC rule.
</li>
<li>
<b>NCC_TRIANGLE_RULE_NUM</b> returns the number of Newton-Cotes closed rules available.
</li>
<li>
<b>NCC_TRIANGLE_SUBORDER</b> returns the suborders for an NCC rule.
</li>
<li>
<b>NCC_TRIANGLE_SUBORDER_NUM</b> returns the number of suborders for an NCC rule.
</li>
<li>
<b>NCC_TRIANGLE_SUBRULE</b> returns a compressed NCC rule.
</li>
<li>
<b>REFERENCE_TO_PHYSICAL_T3</b> maps T3 reference points to physical points.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TIMESTRING</b> writes the current YMDHMS date into a string.
</li>
<li>
<b>TRIANGLE_AREA</b> computes the area of a triangle.
</li>
<li>
<b>TRIANGLE_POINTS_PLOT</b> plots a triangle and some points.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 29 January 2007.
</i>
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