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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 C++ program which
investigates the polynomial exactness of a quadrature rule
over the interior of a tetrahedron in 3D.
</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>
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 = "../../c_src/tetrahedron_exactness/tetrahedron_exactness.html">a C version</a> and
<a href = "../../cpp_src/tetrahedron_exactness/tetrahedron_exactness.html">a C++ version</a> and
<a href = "../../f77_src/tetrahedron_exactness/tetrahedron_exactness.html">a FORTRAN77 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 = "../../cpp_src/cube_exactness/cube_exactness.html">
CUBE_EXACTNESS</a>,
a C++ library which
investigates the polynomial exactness of quadrature rules
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../cpp_src/hypercube_exactness/hypercube_exactness.html">
HYPERCUBE_EXACTNESS</a>,
a C++ program which
measures the monomial exactness of an M-dimensional quadrature rule
over the interior of the unit hypercube in M dimensions.
</p>
<p>
<a href = "../../cpp_src/pyramid_exactness/pyramid_exactness.html">
PYRAMID_EXACTNESS</a>,
a C++ program which
investigates the polynomial exactness of a quadrature rule
over the interior of the unit pyramid in 3D.
</p>
<p>
<a href = "../../cpp_src/sphere_exactness/sphere_exactness.html">
SPHERE_EXACTNESS</a>,
a C++ program which
tests the monomial exactness of a quadrature rule
on the surface of the unit sphere in 3D.
</p>
<p>
<a href = "../../cpp_src/square_exactness/square_exactness.html">
SQUARE_EXACTNESS</a>,
a C++ library which
investigates the polynomial exactness of quadrature rules for f(x,y)
over the interior of a rectangle in 2D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_arbq_rule/tetrahedron_arbq_rule.html">
TETRAHEDRON_ARBQ_RULE</a>,
a C++ 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 = "../../cpp_src/tetrahedron_felippa_rule/tetrahedron_felippa_rule.html">
TETRAHEDRON_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_integrals/tetrahedron_integrals.html">
TETRAHEDRON_INTEGRALS</a>,
a C++ 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 = "../../cpp_src/tetrahedron_keast_rule/tetrahedron_keast_rule.html">
TETRAHEDRON_KEAST_RULE</a>,
a C++ library which
defines ten quadrature rules, with exactness degrees 0 through 8,
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">
TETRAHEDRON_MONTE_CARLO</a>,
a C++ program which
uses the Monte Carlo method to estimate integrals
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_ncc_rule/tetrahedron_ncc_rule.html">
TETRAHEDRON_NCC_RULE</a>,
a C++ library which
defines Newton-Cotes Closed (NCC) quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">
TETRAHEDRON_NCO_RULE</a>,
a C++ library which
defines Newton-Cotes Open (NCO) quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/triangle_exactness/triangle_exactness.html">
TRIANGLE_EXACTNESS</a>,
a C++ program which
investigates the polynomial exactness of a quadrature rule
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/wedge_exactness/wedge_exactness.html">
WEDGE_EXACTNESS</a>,
a C++ program which
investigates the monomial exactness of a quadrature rule
over the interior of the unit wedge in 3D.
</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.cpp">tetrahedron_exactness.cpp</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>DTABLE_DATA_READ</b> reads data from a DTABLE file.
</li>
<li>
<b>DTABLE_HEADER_READ</b> reads the header from a DTABLE file.
</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_DET_4D</b> computes the determinant of a 4 by 4 matrix.
</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>
<li>
<b>TIMESTRING</b> writes the current YMDHMS date into a string.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 05 July 2007.
</i>
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