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<html>
<head>
<title>
ASA299 - Lattice Points in N-dimensional Simplex
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
ASA299 <br> Lattice Points in N-dimensional Simplex
</h1>
<hr>
<p>
<b>ASA299</b>
is a C++ library which
generates, one at a time,
the lattice points (integer coordinates) contained in a simplex.
</p>
<p>
<b>ASA299</b> is Applied Statistics Algorithm 299. Source code for many
Applied Statistics Algorithms is available through
<a href = "http://lib.stat.cmu.edu/apstat">STATLIB</a>.
</p>
<p>
The simplex is defined by N-dimensional points <b>X</b> such that:
<blockquote><b>
0 <= X(1:N)
</b></blockquote>
and
<blockquote><b>
sum ( X(1:N) ) <= T
</b></blockquote>
where T is an integer.
</p>
<p>
Lattice points are points <b>X</b> which satisfy the simplex conditions
and for which all the components are integers.
</p>
<p>
This routine generates all the lattice points in a given simplex, one at
a time, in a reverse lexicographic order.
</p>
<p>
The output for <b>N</b> = 3, <b>T</b> = 4 would be:
<pre>
1 4 0 0
2 3 1 0
3 3 0 1
4 2 2 0
5 2 1 1
6 2 0 2
7 1 3 0
8 1 2 1
9 1 1 2
10 1 0 3
11 0 4 0
12 0 3 1
13 0 2 2
14 0 1 3
15 0 0 4
</pre>
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote><b>
call simplex_lattice_point_next ( n, t, more, x )
</b></blockquote>
</p>
<p>
To use the routine, initialize by setting the spatial dimension <b>N</b>
and the simplex size parameter <b>T</b> to appropriate values, and
set <b>MORE</b> to FALSE. The initial value of <b>X</b> is not important.
</p>
<p>
Call the routine. On return, <b>X</b> will contain the first lattice
point in the simplex. If <b>MORE</b> is TRUE, then the routine may
be called again to get the next point. In fact, as long as the output
value of <b>MORE</b> is TRUE, there is at least one more lattice point
that can be found by making another call. When <b>MORE</b> is returned
as FALSE, then there are no more lattice points; the value of <b>X</b>
returned at that time is the "last" such point.
</p>
<p>
During the computation of a sequence of lattice points, the user should
not change the values of <b>N</b>, <b>T</b>, <b>MORE</b> or <b>X</b>.
</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>ASA299</b> is available in
<a href = "../../c_src/asa299/asa299.html">a C version</a> and
<a href = "../../cpp_src/asa299/asa299.html">a C++ version</a> and
<a href = "../../f77_src/asa299/asa299.html">a FORTRAN77 version</a> and
<a href = "../../f_src/asa299/asa299.html">a FORTRAN90 version</a> and
<a href = "../../m_src/asa299/asa299.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/combo/combo.html">
COMBO</a>,
a FORTRAN90 library which
enumerates combinations, partitions, subsets, index sets,
and other combinatorial objects.
</p>
<p>
<a href = "../../f_src/subset/subset.html">
SUBSET</a>,
a FORTRAN90 library which
enumerates combinations, partitions, subsets, index sets,
and other combinatorial objects.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Scott Chasalow, Richard Brand,<br>
Algorithm AS 299:
Generation of Simplex Lattice Points,<br>
Applied Statistics,<br>
Volume 44, Number 4, 1995, pages 534-545.
</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>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "asa299.cpp">asa299.cpp</a>, the source code.
</li>
<li>
<a href = "asa299.hpp">asa299.hpp</a>, the source code.
</li>
<li>
<a href = "asa299.sh">asa299.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "asa299_prb.cpp">asa299_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "asa299_prb.sh">asa299_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "asa299_prb_output.txt">asa299_prb_output.txt</a>,
the output ffile.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>SIMPLEX_LATTICE_POINT_NEXT</b> generates lattice points in a simplex.
</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 = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 10 January 2007.
</i>
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