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<html>
<head>
<title>
TEST_TRI_INT - Triangle Integration Test Functions
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TEST_TRI_INT <br> Triangle Integration Test Functions
</h1>
<hr>
<p>
<b>TEST_TRI_INT</b>
is a FORTRAN90 library which
defines a number of test functions to be integrated over a triangle.
</p>
<p>
It is possible to invoke a particular function by number, or to
try out all available functions,
as demonstrated in the sample calling program.
</p>
<p>
For convenience, all the integrand functions have been
scaled by a constant, so that the integral of the function
over the specific domain is exactly 1.
</p>
<p>
The test functions include <b>F(X,Y)=</b>
<ol>
<li>
<b>1</b> on the unit triangle.
</li>
<li>
<b>x</b> on the unit triangle.
</li>
<li>
<b>y</b> on the unit triangle.
</li>
<li>
<b>x<sup>2</sup></b> on the unit triangle.
</li>
<li>
<b>x*y</b> on the unit triangle.
</li>
<li>
<b>y<sup>2</sup></b> on the unit triangle.
</li>
<li>
<b>x<sup>3</sup></b> on the unit triangle.
</li>
<li>
<b>x<sup>4</sup></b> on the unit triangle.
</li>
<li>
<b>x<sup>5</sup></b> on the unit triangle.
</li>
<li>
<b>x^(-0.2)</b> remapped to (1,0), (5,0), (5,1).
</li>
<li>
<b>(x+y)^(-0.2)</b> on the unit triangle.
</li>
<li>
<b>(1-x-y)^(-0.2)</b> remapped to (-1,-3), (3,-2), (-1,2).
</li>
<li>
<b>(x*y)^(-0.2)</b> remapped to (0,0), (-7,0), (0,-3).
</li>
<li>
<b>1/sqrt(x) + 1/sqrt(y) + 1/sqrt(x+y)</b> on the unit triangle.
</li>
<li>
<b>1/sqrt(1-x-y)</b> on the unit triangle.
</li>
<li>
<b>log(x*y)</b> on the unit triangle.
</li>
<li>
<b>1/sqrt(|x-1/4|) + 1/sqrt(|y-1/2|)</b> on the unit triangle.
</li>
<li>
<b>log ( x + y )</b> on the unit triangle.
</li>
<li>
<b>sin ( x ) cos ( 5 y )</b> on the unit triangle.
</li>
<li>
<b>sin ( 11 x ) cos ( y )</b> on the unit triangle.
</li>
<li>
<b>1 / r</b> on the unit triangle,<br>
<b>r = sqrt ( x^2+y^2)</b>.
</li>
<li>
<b>log ( r ) / r</b> on the unit triangle,<br>
<b>r = sqrt ( x^2+y^2)</b>.
</li>
</ol>
</p>
<p>
The library includes a routines to define the integrand function,
the triangle over which the integral is to be carried out,
and a title for the problem.
Thus, for each integrand function, four routines are supplied. For
instance, for function #4, we have the routines:
<ul>
<li>
<b>P04_FUN</b> evaluates the integrand for problem 4.
</li>
<li>
<b>P04_TITLE</b> returns the title for problem 4.
</li>
<li>
<b>P04_VERTICES</b> returns the integration triangle for problem 4
</li>
</ul>
So once you have the calling sequences for these routines, you
can easily evaluate the function, or integrate it between the
appropriate limits, or compare your estimate of the integral
to the exact value.
</p>
<p>
Moreover, since the same interface is used for each function,
if you wish to work with problem 16 instead, you simply change
the "04" to "07" in your routine calls.
</p>
<p>
If you wish to call <i>all</i> of the functions, then you
simply use the generic interface, which again has four
routines, but which requires you to specify the problem
number as an extra input argument:
<ul>
<li>
<b>P00_FUN</b> evaluates the integrand for problem PROB.
</li>
<li>
<b>P00_SINGULARITY</b> warns if the integran for problem PROB
has vertex, edge or interior singularities that might cause
problems for some integration schemes.
</li>
<li>
<b>P00_TITLE</b> returns the title for problem PROB.
</li>
<li>
<b>P00_VERTICES</b> returns the integration triangle for problem PROB.
</li>
</ul>
</p>
<p>
Finally, some demonstration routines are built in for
simple quadrature methods. These routines include
<ul>
<li>
<b>P00_WANDZURA05_SUB</b> applies an order-5 accurate Wandzura
rule, plus subdivision of the triangle.
</li>
<li>
<b>P00_MONTE_CARLO</b> uses Monte Carlo sampling.
</li>
<li>
<b>P00_VERTEX_SUB</b> averages values at vertices, and uses
subdivision.
</li>
</ul>
and can be used with any of the sample integrands.
</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>TEST_TRI_INT</b> is available in
<a href = "../../cpp_src/test_tri_int/test_tri_int.html">a C++ version</a> and
<a href = "../../f_src/test_tri_int/test_tri_int.html">a FORTRAN90 version</a> and
<a href = "../../m_src/test_tri_int/test_tri_int.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/cubpack/cubpack.html">
CUBPACK</a>,
a FORTRAN90 library which
estimates the integral of a function over a collection of N-dimensional
hyperrectangles and simplices.
</p>
<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 interpolation or quadrature 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 Grundmann-Moeller rules 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_triangle/ncc_triangle.html">
NCC_TRIANGLE</a>,
a FORTRAN90 library which
defines Newton-Cotes Closed quadrature rules on a triangle.
</p>
<p>
<a href = "../../cpp_src/nco_triangle/nco_triangle.html">
NCO_TRIANGLE</a>,
a C++ library which
defines Newton-Cotes Open quadrature rules on a triangle.
</p>
<p>
<a href = "../../f_src/stroud/stroud.html">
STROUD</a>,
a FORTRAN90 library which
defines quadrature rules for a variety of multidimensional regions.
</p>
<p>
<a href = "../../f77_src/toms612/toms612.html">
TOMS612</a>,
a FORTRAN77 library which
estimates the integral of a function over a triangle.
</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/triangulation/triangulation.html">
TRIANGULATION</a>,
a FORTRAN90 library which
performs various operations on order 3 ("linear") or order 6 ("quadratic") triangulations.
</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>
Elise deDoncker, Ian Robinson,<br>
An Algorithm for Automatic Integration Over a Triangle
Using Nonlinear Extrapolation,<br>
ACM Transactions on Mathematical Software,<br>
Volume 10, Number 1, March 1984, pages 1-16.
</li>
<li>
Elise deDoncker, Ian Robinson,<br>
Algorithm 612:
Integration over a Triangle Using Nonlinear Extrapolation,<br>
ACM Transactions on Mathematical Software,<br>
Volume 10, Number 1, March 1984, pages 17-22.
</li>
<li>
Stephen Wandzura, Hong Xiao,<br>
Symmetric Quadrature Rules on a Triangle,<br>
Computers and Mathematics with Applications,<br>
Volume 45, pages 1829-1840, 2003.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "test_tri_int.f90">test_tri_int.f90</a>, the source code.
</li>
<li>
<a href = "test_tri_int.sh">test_tri_int.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "test_tri_int_prb.f90">test_tri_int_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "test_tri_int_prb.sh">test_tri_int_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "test_tri_int_prb_output.txt">test_tri_int_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>GET_PROB_NUM</b> returns the number of test integration problems.
</li>
<li>
<b>P00_FUN</b> evaluates the integrand for any problem.
</li>
<li>
<b>P00_MONTE_CARLO</b> applies the Monte Carlo rule to integrate a function.
</li>
<li>
<b>P00_SINGULARITY</b> warns of common singularities for any problem.
</li>
<li>
<b>P00_TITLE</b> returns the title for any problem.
</li>
<li>
<b>P00_VERTEX_SUB</b> approximates an integral in a triangle by subdivision.
</li>
<li>
<b>P00_VERTICES</b> returns the vertices for any problem.
</li>
<li>
<b>P00_WANDZURA05_SUB</b> uses subdivision and a Wandzura rule.
</li>
<li>
<b>P01_FUN</b> evaluates the integrand for problem 1.
</li>
<li>
<b>P01_TITLE</b> returns the title of problem 1.
</li>
<li>
<b>P01_VERTICES</b> returns the vertices for problem 1.
</li>
<li>
<b>P02_FUN</b> evaluates the integrand for problem 2.
</li>
<li>
<b>P02_TITLE</b> returns the title of problem 2.
</li>
<li>
<b>P02_VERTICES</b> returns the vertices for problem 2.
</li>
<li>
<b>P03_FUN</b> evaluates the integrand for problem 3.
</li>
<li>
<b>P03_TITLE</b> returns the title of problem 3.
</li>
<li>
<b>P03_VERTICES</b> returns the vertices for problem 3.
</li>
<li>
<b>P04_FUN</b> evaluates the integrand for problem 4.
</li>
<li>
<b>P04_TITLE</b> returns the title of problem 4.
</li>
<li>
<b>P04_VERTICES</b> returns the vertices for problem 4.
</li>
<li>
<b>P05_FUN</b> evaluates the integrand for problem 5.
</li>
<li>
<b>P05_TITLE</b> returns the title of problem 5.
</li>
<li>
<b>P05_VERTICES</b> returns the vertices for problem 5.
</li>
<li>
<b>P06_FUN</b> evaluates the integrand for problem 6.
</li>
<li>
<b>P06_TITLE</b> returns the title of problem 6.
</li>
<li>
<b>P06_VERTICES</b> returns the vertices for problem 6.
</li>
<li>
<b>P07_FUN</b> evaluates the integrand for problem 7.
</li>
<li>
<b>P07_TITLE</b> returns the title of problem 7.
</li>
<li>
<b>P07_VERTICES</b> returns the vertices for problem 7.
</li>
<li>
<b>P08_FUN</b> evaluates the integrand for problem 8.
</li>
<li>
<b>P08_TITLE</b> returns the title of problem 8.
</li>
<li>
<b>P08_VERTICES</b> returns the vertices for problem 8.
</li>
<li>
<b>P09_FUN</b> evaluates the integrand for problem 9.
</li>
<li>
<b>P09_TITLE</b> returns the title of problem 9.
</li>
<li>
<b>P09_VERTICES</b> returns the vertices for problem 9.
</li>
<li>
<b>P10_FUN</b> evaluates the integrand for problem 10.
</li>
<li>
<b>P10_TITLE</b> returns the title of problem 10.
</li>
<li>
<b>P10_VERTICES</b> returns the vertices for problem 10.
</li>
<li>
<b>P11_FUN</b> evaluates the integrand for problem 11.
</li>
<li>
<b>P11_TITLE</b> returns the title of problem 11.
</li>
<li>
<b>P11_VERTICES</b> returns the vertices for problem 11.
</li>
<li>
<b>P12_FUN</b> evaluates the integrand for problem 12.
</li>
<li>
<b>P12_TITLE</b> returns the title of problem 12.
</li>
<li>
<b>P12_VERTICES</b> returns the vertices for problem 12.
</li>
<li>
<b>P13_FUN</b> evaluates the integrand for problem 13.
</li>
<li>
<b>P13_TITLE</b> returns the title of problem 13.
</li>
<li>
<b>P13_VERTICES</b> returns the vertices for problem 13.
</li>
<li>
<b>P14_FUN</b> evaluates the integrand for problem 14.
</li>
<li>
<b>P14_TITLE</b> returns the title of problem 14.
</li>
<li>
<b>P14_VERTICES</b> returns the vertices for problem 14.
</li>
<li>
<b>P15_FUN</b> evaluates the integrand for problem 15.
</li>
<li>
<b>P15_TITLE</b> returns the title of problem 15.
</li>
<li>
<b>P15_VERTICES</b> returns the vertices for problem 15.
</li>
<li>
<b>P16_FUN</b> evaluates the integrand for problem 16.
</li>
<li>
<b>P16_TITLE</b> returns the title of problem 16.
</li>
<li>
<b>P16_VERTICES</b> returns the vertices for problem 16.
</li>
<li>
<b>P17_FUN</b> evaluates the integrand for problem 17.
</li>
<li>
<b>P17_TITLE</b> returns the title of problem 17.
</li>
<li>
<b>P17_VERTICES</b> returns the vertices for problem 17.
</li>
<li>
<b>P18_FUN</b> evaluates the integrand for problem 18.
</li>
<li>
<b>P18_TITLE</b> returns the title of problem 18.
</li>
<li>
<b>P18_VERTICES</b> returns the vertices for problem 18.
</li>
<li>
<b>P19_FUN</b> evaluates the integrand for problem 19.
</li>
<li>
<b>P19_TITLE</b> returns the title of problem 19.
</li>
<li>
<b>P19_VERTICES</b> returns the vertices for problem 19.
</li>
<li>
<b>P20_FUN</b> evaluates the integrand for problem 20.
</li>
<li>
<b>P20_TITLE</b> returns the title of problem 20.
</li>
<li>
<b>P20_VERTICES</b> returns the vertices for problem 20.
</li>
<li>
<b>P21_FUN</b> evaluates the integrand for problem 21.
</li>
<li>
<b>P21_TITLE</b> returns the title of problem 21.
</li>
<li>
<b>P21_VERTICES</b> returns the vertices for problem 21.
</li>
<li>
<b>P22_FUN</b> evaluates the integrand for problem 22.
</li>
<li>
<b>P22_TITLE</b> returns the title of problem 22.
</li>
<li>
<b>P22_VERTICES</b> returns the vertices for problem 22.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>SUBTRIANGLE_NEXT</b> computes the next subtriangle of a triangle.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TRIANGLE_AREA</b> computes the area of a triangle in 2D.
</li>
<li>
<b>TRIANGLE_SAMPLE</b> returns random points in a triangle.
</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 10 April 2007.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>