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<html>
<head>
<title>
TETRAHEDRON_EXACTNESS - Exactness of Quadrature on Tetrahedrons
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TETRAHEDRON_EXACTNESS <br>
Exactness of Quadrature on Tetrahedrons
</h1>
<hr>
<p>
<b>TETRAHEDRON_EXACTNESS</b>
is a FORTRAN90 program which
investigates the polynomial exactness of a
quadrature rule for a tetrahedron.
</p>
<p>
The polynomial exactness of a quadrature rule is defined as the
highest total degree <b>D</b> such that the quadrature rule is
guaranteed to integrate exactly all polynomials of total degree
<b>DEGREE_MAX</b> or less, ignoring roundoff.
The total degree of a polynomial is the maximum of the degrees
of all its monomial terms. For a tetrahedron, the degree of a monomial
term is the sum of the exponents of <b>x</b>, <b>y</b> and <b>z</b>.
Thus, for instance, the <b>DEGREE</b> of
<blockquote><b>
x<sup>2</sup>yz<sup>5</sup>
</b></blockquote>
is 2+1+5=8.
</p>
<p>
To be thorough, the program starts at <b>DEGREE</b> = 0, and then
proceeds to <b>DEGREE</b> = 1, 2, and so on up to a maximum degree
<b>DEGREE_MAX</b> specified by the user. At each value of <b>DEGREE</b>,
the program generates every possible monomial term, applies the
quadrature rule to it, and determines the quadrature error. The program
uses a scaling factor on each monomial so that the exact integral
should always be 1; therefore, each reported error can be compared
on a fixed scale.
</p>
<p>
The program is very flexible and interactive. The quadrature rule
is defined by three files, to be read at input, and the
maximum degree is specified by the user as well.
</p>
<p>
Note that the three files that define the quadrature rule
are assumed to have related names, of the form
<ul>
<li>
<i>prefix</i>_<b>x.txt</b>
</li>
<li>
<i>prefix</i>_<b>w.txt</b>
</li>
<li>
<i>prefix</i>_<b>r.txt</b>
</li>
</ul>
When running the program, the user only enters the common <i>prefix</i>
part of the file names, which is enough information for the program
to find all three files.
</p>
<p>
For information on the form of these files, see the
<b>QUADRATURE_RULES_TET</b> directory listed below.
</p>
<p>
The exactness results are written to an output file with the
corresponding name:
<ul>
<li>
<i>prefix</i>_<b>exact.txt</b>
</li>
</ul>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>tetrahedron_exactness</b> <i>prefix</i> <i>degree_max</i>
</blockquote>
where
<ul>
<li>
<i>prefix</i> is the common prefix for the files containing the abscissa, weight
and region information of the quadrature rule;
</li>
<li>
<i>degree_max</i> is the maximum total monomial degree to check. A value of 5 or 10
might be reasonable, but a value of 50 or 100 is probably never a
good input!
</li>
</ul>
</p>
<p>
If the arguments are not supplied on the command line, the
program will prompt for them.
</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_EXACTNESS</b> is available in
<a href = "../../cpp_src/tetrahedron_exactness/tetrahedron_exactness.html">a C++ version</a> and
<a href = "../../f_src/tetrahedron_exactness/tetrahedron_exactness.html">a FORTRAN90 version</a> and
<a href = "../../m_src/tetrahedron_exactness/tetrahedron_exactness.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<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/keast/keast.html">
KEAST</a>,
a FORTRAN90 library which
sets up a Keast quadrature rule for a tetrahedron.
</p>
<p>
<a href = "../../f_src/integral_test/integral_test.html">
INTEGRAL_TEST</a>,
a FORTRAN90 program which
uses test integrals to measure the effectiveness of
certain sets of quadrature rules.
</p>
<p>
<a href = "../../f_src/ncc_tetrahedron/ncc_tetrahedron.html">
NCC_TETRAHEDRON</a>,
a FORTRAN90 library which
sets up a Newton Cotes Closed quadrature rule for a tetrahedron.
</p>
<p>
<a href = "../../f_src/nco_tetrahedron/nco_tetrahedron.html">
NCO_TETRAHEDRON</a>,
a FORTRAN90 library which
sets up a Newton Cotes Open quadrature rule for a tetrahedron.
</p>
<p>
<a href = "../../f_src/nint_exactness/nint_exactness.html">
NINT_EXACTNESS</a>,
a FORTRAN90 program which
tests the exactness of integration rules over the interval,
rectangle, or multidimensional rectangle.
</p>
<p>
<a href = "../../f_src/nint_exactness_mixed/nint_exactness_mixed.html">
NINT_EXACTNESS_MIXED</a>,
a FORTRAN90 program which
measures the polynomial exactness of a multidimensional quadrature rule
based on a mixture of 1D quadrature rule factors.
</p>
<p>
<a href = "../../f_src/nintlib/nintlib.html">
NINTLIB</a>,
a FORTRAN90 library which
numerically estimates integrals in multiple dimensions.
</p>
<p>
<a href = "../../f_src/pyramid_exactness/pyramid_exactness.html">
PYRAMID_EXACTNESS</a>,
a FORTRAN90 program which
investigates the polynomial exactness of a quadrature rule for the pyramid.
</p>
<p>
<a href = "../../datasets/quadrature_rules_tet/quadrature_rules_tet.html">
QUADRATURE_RULES_TET</a>,
a dataset directory which
contains sets of files that define quadrature
rules over a tetrahedron.
</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/sphere_exactness/sphere_exactness.html">
SPHERE_EXACTNESS</a>,
a FORTRAN90 program which
tests the polynomial exactness of a quadrature rule for the unit sphere;
</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 multiple dimensions.
</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 a tetrahedron.
</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>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Philip Davis, Philip Rabinowitz,<br>
Methods of Numerical Integration,<br>
Second Edition,<br>
Dover, 2007,<br>
ISBN: 0486453391,<br>
LC: QA299.3.D28.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "tetrahedron_exactness.f90">tetrahedron_exactness.f90</a>, the source code.
</li>
<li>
<a href = "tetrahedron_exactness.sh">tetrahedron_exactness.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
The "W", "X" and "R" files associated with the point sets tested
here may be found in the <b>QUADRATURE_RULES_TET</b> directory.
</p>
<p>
<ul>
<li>
<a href = "keast0_exact.txt">keast0_exact.txt</a>.
</li>
<li>
<a href = "keast1_exact.txt">keast1_exact.txt</a>.
</li>
<li>
<a href = "keast2_exact.txt">keast2_exact.txt</a>.
</li>
<li>
<a href = "keast3_exact.txt">keast3_exact.txt</a>.
</li>
<li>
<a href = "keast4_exact.txt">keast4_exact.txt</a>.
</li>
<li>
<a href = "keast5_exact.txt">keast5_exact.txt</a>.
</li>
<li>
<a href = "keast6_exact.txt">keast6_exact.txt</a>.
</li>
<li>
<a href = "keast7_exact.txt">keast7_exact.txt</a>.
</li>
<li>
<a href = "keast8_exact.txt">keast8_exact.txt</a>.
</li>
<li>
<a href = "keast9_exact.txt">keast9_exact.txt</a>.
</li>
<li>
<a href = "ncc0_exact.txt">ncc0_exact.txt</a>.
</li>
<li>
<a href = "ncc1_exact.txt">ncc1_exact.txt</a>.
</li>
<li>
<a href = "ncc2_exact.txt">ncc2_exact.txt</a>.
</li>
<li>
<a href = "ncc3_exact.txt">ncc3_exact.txt</a>.
</li>
<li>
<a href = "ncc4_exact.txt">ncc4_exact.txt</a>.
</li>
<li>
<a href = "ncc5_exact.txt">ncc5_exact.txt</a>.
</li>
<li>
<a href = "ncc6_exact.txt">ncc6_exact.txt</a>.
</li>
<li>
<a href = "nco0_exact.txt">nco0_exact.txt</a>.
</li>
<li>
<a href = "nco1_exact.txt">nco1_exact.txt</a>.
</li>
<li>
<a href = "nco2_exact.txt">nco2_exact.txt</a>.
</li>
<li>
<a href = "nco3_exact.txt">nco3_exact.txt</a>.
</li>
<li>
<a href = "nco4_exact.txt">nco4_exact.txt</a>.
</li>
<li>
<a href = "nco5_exact.txt">nco5_exact.txt</a>.
</li>
<li>
<a href = "nco6_exact.txt">nco6_exact.txt</a>.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for TETRAHEDRON_EXACTNESS.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>COMP_NEXT</b> computes the compositions of the integer N into K parts.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>MONOMIAL_INT_TET_UNIT</b> integrates a monomial over the unit tetrahedron.
</li>
<li>
<b>MONOMIAL_QUADRATURE_TET</b> applies quadrature to a monomial in a tetrahedron.
</li>
<li>
<b>MONOMIAL_VALUE</b> evaluates a monomial.
</li>
<li>
<b>R8MAT_DATA_READ</b> reads data from an R8MAT file.
</li>
<li>
<b>R8MAT_DET_4D</b> computes the determinant of a 4 by 4 matrix.
</li>
<li>
<b>R8MAT_HEADER_READ</b> reads the header from an R8MAT file.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</li>
<li>
<b>TETRAHEDRON_ORDER4_PHYSICAL_TO_REFERENCE</b> maps physical points to reference points.
</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 = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 14 October 2009.
</i>
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