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latin_cover.html
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latin_cover.html
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<html>
<head>
<title>
LATIN_COVER - N Latin Squares which cover an NxN square.
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
LATIN_COVER <br> N Latin Squares which cover an NxN square.
</h1>
<hr>
<p>
<b>LATIN_COVER</b>
is a FORTRAN90 library which
produces a set of N Latin Squares which cover an NxN square.
</p>
<p>
Given an NxN grid, a Latin Square is a selection of N pairs of indices
(I,J) such that every possible value of I and J occurs exactly once.
A picture for the case N = 5 may suggest what is going on:
<pre>
- - - 1 -
- 1 - - -
I 1 - - - -
- - - - 1
- - 1 - -
J
</pre>
The above Latin Square can also be described by the index pairs (1,4),
(2,2), (3,1), (4,5), (5,3).
</p>
<p>
A Latin square will always contain N index pairs. Since the original
square contains NxN index pairs, it is interesting to speculate whether
it would be possible to find N Latin squares with the property that
every possible index pair (I,J) in the NxN square occurs in exactly one
of the Latin Squares. In that case, we would say that the Latin Squares
"cover" or decompose the square.
</p>
<p>
Here is an example which starts with the Latin square pictured above,
and generates 4 more Latin squares to cover the 5x5 square.
<pre>
3 2 5 1 4
2 1 4 5 3
I 1 5 3 4 2
5 4 2 3 1
4 3 1 2 5
J
</pre>
</p>
<p>
Given a value N, this program produces a Latin cover for an NxN square.
</p>
<p>
Given a description of a Latin square of order N, this program produces
N-1 more Latin squares, so that together with the input Latin square,
a Latin cover is produced.
</p>
<p>
It is also possible to generate a 3D Latin covering. Here, a single
Latin cube contains only N subcubes, so we require NxN such cubes if
we have a hope of covering all the subcubes. The routine LATIN_COVER_3D
will compute such a covering.
</p>
<p>
A small example of a 3D a Latin Cover:
<pre>
K = 1
J: 1 2 3
I:
1: 4 1 7
2: 6 3 9
3: 5 2 8
K = 2
J: 1 2 3
I:
1: 3 9 6
2: 2 8 5
3: 1 7 4
K = 3
J: 1 2 3
I:
1: 8 5 2
2: 7 4 1
3: 9 6 3
</pre>
</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>LATIN_COVER</b> is available in
<a href = "../../c_src/latin_cover/latin_cover.html">a C version</a> and
<a href = "../../cpp_src/latin_cover/latin_cover.html">a C++ version</a> and
<a href = "../../f77_src/latin_cover/latin_cover.html">a FORTRAN77 version</a> and
<a href = "../../f_src/latin_cover/latin_cover.html">a FORTRAN90 version</a> and
<a href = "../../m_src/latin_cover/latin_cover.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/latin_center/latin_center.html">
LATIN_CENTER</a>,
a FORTRAN90 library which
computes elements of a Latin Hypercube dataset, choosing center points.
</p>
<p>
<a href = "../../f_src/latin_edge/latin_edge.html">
LATIN_EDGE</a>,
a FORTRAN90 library which
computes elements of a Latin Hypercube dataset, choosing edge points.
</p>
<p>
<a href = "../../f_src/latin_random/latin_random.html">
LATIN_RANDOM</a>,
a FORTRAN90 library which
computes elements of a Latin Hypercube dataset, choosing points at random.
</p>
<p>
<a href = "../../f_src/latinize/latinize.html">
LATINIZE</a>,
a FORTRAN90 library which
adjusts N points in M dimensions to form a Latin hypercube.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Herbert Ryser,<br>
Combinatorial Mathematics,<br>
Mathematical Association of America, 1963.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "latin_cover.f90">latin_cover.f90</a>, the source code.
</li>
<li>
<a href = "latin_cover.sh">latin_cover.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "latin_cover_prb.f90">latin_cover_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "latin_cover_prb.sh">latin_cover_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "latin_cover_prb_output.txt">latin_cover_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>I4_UNIFORM</b> returns a scaled pseudorandom I4.
</li>
<li>
<b>I4_WRAP</b> forces an I4 to lie between given limits by wrapping.
</li>
<li>
<b>I4BLOCK_PRINT</b> prints an I4BLOCK.
</li>
<li>
<b>I4MAT_PRINT</b> prints an I4MAT.
</li>
<li>
<b>I4MAT_PRINT_SOME</b> prints some of an I4MAT.
</li>
<li>
<b>LATIN_COVER</b> returns a 2D Latin Square Covering.
</li>
<li>
<b>LATIN_COVER_2D</b> returns a 2D Latin Square Covering.
</li>
<li>
<b>LATIN_COVER_3D</b> returns a 3D Latin Square Covering.
</li>
<li>
<b>PERM_CHECK</b> checks that a vector represents a permutation.
</li>
<li>
<b>PERM_PRINT</b> prints a permutation.
</li>
<li>
<b>PERM_UNIFORM</b> selects a random permutation of N objects.
</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 26 June 2012.
</i>
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