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<html>
<head>
<title>
SIMPLEX_GM_RULE - Grundmann-Moeller Quadrature Rules for the Simplex in M dimensions
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SIMPLEX_GM_RULE <br> Grundmann-Moeller Quadrature Rules for the Simplex in M dimensions
</h1>
<hr>
<p>
<b>SIMPLEX_GM_RULE</b>
is a MATLAB library which
defines Grundmann-Moeller quadrature rules
over the interior of a triangle in 2D, a tetrahedron in 3D, or
over the interior of the simplex in M dimensions.
</p>
<p>
The user can choose the spatial dimension M, thus defining the region
to be a triangle (M = 2), tetrahedron (M = 3) or a general M-dimensional
simplex.
</p>
<p>
The user chooses the index <b>S</b> of the rule. Rules are available
with index <b>S</b> = 0 on up. A rule of index <b>S</b> will exactly
integrate any polynomial of total degree <b>2*S+1</b> or less.
</p>
<p>
The rules are defined on the unit M-dimensional simplex. A simple
linear transformation can be used to map the vertices and weights
to an arbitrary simplex, while preserving the accuracy of the rule.
</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>SIMPLEX_GM_RULE</b> is available in
<a href = "../../c_src/simplex_gm_rule/simplex_gm_rule.html">a C version</a> and
<a href = "../../cpp_src/simplex_gm_rule/simplex_gm_rule.html">a C++ version</a> and
<a href = "../../f77_src/simplex_gm_rule/simplex_gm_rule.html">a FORTRAN77 version</a> and
<a href = "../../f_src/simplex_gm_rule/simplex_gm_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/simplex_gm_rule/simplex_gm_rule.html">a MATLAB version</a>.
<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_grid/simplex_grid.html">
SIMPLEX_GRID</a>,
a MATLAB library which
generates a regular grid of points
over the interior of an arbitrary 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
defines quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and M-dimensions.
</p>
<p>
<a href = "../../m_src/tetrahedron_exactness/tetrahedron_exactness.html">
TETRAHEDRON_EXACTNESS</a>,
a MATLAB program which
investigates the monomial exactness of a quadrature rule
over the interior of a tetrahedron in 3D.
</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_integrals/tetrahedron_integrals.html">
TETRAHEDRON_INTEGRALS</a>,
a MATLAB 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 = "../../m_src/tetrahedron_keast_rule/tetrahedron_keast_rule.html">
TETRAHEDRON_KEAST_RULE</a>,
a MATLAB library which
defines ten quadrature rules, with exactness degrees 0 through 8,
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">
TETRAHEDRON_MONTE_CARLO</a>,
a MATLAB library which
uses the Monte Carlo method to estimate integrals
over the interior of the unit 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 (NCC) quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">
TETRAHEDRON_NCO_RULE</a>,
a MATLAB library which
defines Newton-Cotes Open (NCO) 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_ncc_rule/triangle_ncc_rule.html">
TRIANGLE_NCC_RULE</a>,
a MATLAB library which
defines Newton-Cotes Closed (NCC) 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 (NCO) quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_wandzura_rule/triangle_wandzura_rule.html">
TRIANGLE_WANDZURA_RULE</a>,
a MATLAB library which
defines Wandzura rules for quadrature
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>
Paul Bratley, Bennett Fox, Linus Schrage,<br>
A Guide to Simulation,<br>
Second Edition,<br>
Springer, 1987,<br>
ISBN: 0387964673,<br>
LC: QA76.9.C65.B73.
</li>
<li>
Bennett Fox,<br>
Algorithm 647:
Implementation and Relative Efficiency of Quasirandom
Sequence Generators,<br>
ACM Transactions on Mathematical Software,<br>
Volume 12, Number 4, December 1986, pages 362-376.
</li>
<li>
Axel Grundmann, Michael Moeller,<br>
Invariant Integration Formulas for the N-Simplex
by Combinatorial Methods,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 15, Number 2, April 1978, pages 282-290.
</li>
<li>
Pierre LEcuyer,<br>
Random Number Generation,<br>
in Handbook of Simulation,<br>
edited by Jerry Banks,<br>
Wiley, 1998,<br>
ISBN: 0471134031,<br>
LC: T57.62.H37.
</li>
<li>
Peter Lewis, Allen Goodman, James Miller,<br>
A Pseudo-Random Number Generator for the System/360,<br>
IBM Systems Journal,<br>
Volume 8, 1969, pages 136-143.
</li>
<li>
Albert Nijenhuis, Herbert Wilf,<br>
Combinatorial Algorithms for Computers and Calculators,<br>
Second Edition,<br>
Academic Press, 1978,<br>
ISBN: 0-12-519260-6,<br>
LC: QA164.N54.
</li>
<li>
ML Wolfson, HV Wright,<br>
Algorithm 160:
Combinatorial of M Things Taken N at a Time,<br>
Communications of the ACM,<br>
Volume 6, Number 4, April 1963, page 161.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "comp_next.m">comp_next.m</a>,
computes the compositions of the integer N into K parts.
</li>
<li>
<a href = "gm_rule_set.m">gm_rule_set.m</a>,
sets a Grundmann-Moeller rule.
</li>
<li>
<a href = "gm_rule_set_old.m">gm_rule_set_old.m</a>,
sets a Grundmann-Moeller rule. (OBSOLETE VERSION)
</li>
<li>
<a href = "gm_rule_size.m">gm_rule_size.m</a>,
determines the size of a Grundmann-Moeller rule.
</li>
<li>
<a href = "i4_choose.m">i4_choose.m</a>,
computes the binomial coefficient C(N,K).
</li>
<li>
<a href = "i4_huge.m">i4_huge.m</a>,
returns a "huge" I4.
</li>
<li>
<a href = "monomial_value.m">monomial_value.m</a>,
evaluates a monomial.
</li>
<li>
<a href = "r8_factorial.m">r8_factorial.m</a>,
computes the factorial.
</li>
<li>
<a href = "r8vec_uniform_01.m">r8vec_uniform_01.m</a>,
returns a unit pseudorandom R8VEC.
</li>
<li>
<a href = "simplex_unit_monomial_int.m">simplex_unit_monomial_int.m</a>,
integrates a monomial over a simplex.
</li>
<li>
<a href = "simplex_unit_monomial_quadrature.m">simplex_unit_monomial_quadrature.m</a>,
quadrature of monomials in a unit simplex.
</li>
<li>
<a href = "simplex_unit_sample.m">simplex_unit_sample.m</a>,
returns uniformly random points from a general simplex.
</li>
<li>
<a href = "simplex_unit_to_general.m">simplex_unit_to_general.m</a>,
maps the unit simplex to a general simplex.
</li>
<li>
<a href = "simplex_unit_volume.m">simplex_unit_volume.m</a>,
computes the volume of the unit simplex.
</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 = "simplex_gm_rule_test.m">simplex_gm_rule_test.m</a>,
calls all the problems.
</li>
<li>
<a href = "simplex_gm_rule_test01.m">simplex_gm_rule_test01.m</a>,
tests SIMPLEX_UNIT_TO_GENERAL.
</li>
<li>
<a href = "simplex_gm_rule_test02.m">simplex_gm_rule_test02.m</a>,
tests SIMPLEX_UNIT_TO_GENERAL.
</li>
<li>
<a href = "simplex_gm_rule_test03.m">simplex_gm_rule_test03.m</a>,
tests GM_RULE_SIZE.
</li>
<li>
<a href = "simplex_gm_rule_test04.m">simplex_gm_rule_test04.m</a>,
tests GM_RULE_SET.
</li>
<li>
<a href = "simplex_gm_rule_test05.m">simplex_gm_rule_test05.m</a>,
tests GM_RULE_SET.
</li>
<li>
<a href = "simplex_gm_rule_test06.m">simplex_gm_rule_test06.m</a>,
tests GM_RULE_SET.
</li>
<li>
<a href = "simplex_gm_rule_test07.m">simplex_gm_rule_test07.m</a>,
tests GM_RULE_SET.
</li>
<li>
<a href = "simplex_gm_rule_test_output.txt">simplex_gm_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 07 February 2011.
</i>
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