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<p>&nbsp;<p>
<hr noshade size="5">
<h2><a name="MatMLLL"></a>
MatMLLL
</h2>

Performs an MLLL reduction on the columns of an integer or
real matrix.

<p><p>

<h3>
Syntax:
</h3>
<pre>L := MatLLL(M);</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td></td>
</tr>
<tr>
<td>matrix</td>
<td><pre>  M  </pre></td>
<td></td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
Performs a MLLL reduction on the columns of an integer or real
matrix <tt> M</tt>. Let <tt> M</tt> \in {R}^{m \times
(n+1)} and let a|1, &hellip; ,a|{n+1} \in {R}^m be the
columns of <tt> M</tt>. The <tt> MatMLLL</tt> function expects that
a|1, &hellip; ,a|n are linearly independent and that a|{n+1}
has a non-zero entry. It computes columns
b|1, &hellip; ,b|{n+1} \in {R}^m of <tt> B</tt> such that 
{Z} cdot a|1 + cdots + {Z} cdot a|{n+1} =
{Z} cdot b|1 + cdots + {Z} cdot b|{n+1} 
and additionally a transformation matrix <tt> T</tt> \in
{R}^{(n+1) \times (n+1)} with <tt> B</tt> = <tt> M</tt>
cdot <tt> T</tt>. Furthermore a flag is returned. If it is
true, b|1, &hellip; ,b|{n+1} are linearly independent.
Otherwise all entries of b|{n+1} are zero. These results
are stored in a list containing the reduced matrix <tt> B</tt>,
the unimodular transformation matrix <tt> T</tt> and the flag.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
MLLL-Reduction of a matrix: 
<pre>

kash> M := Mat(Z, [ [ 234, 469, -54411 ],
> [ -6546, -13126, 1531763 ], [ -6312, -12657, 1477352 ] ]);
[    234     469  -54411]
[  -6546  -13126 1531763]
[  -6312  -12657 1477352]
kash> L := MatMLLL(M);
[ [ 1  1  0]
    [-1  0  0]
    [ 0  1  0], [ -794943 -1327888  4198061]
    [  427600   714271 -2258136]
    [     267      446    -1410], true ]
kash> L[1];
[ 1  1  0]
[-1  0  0]
[ 0  1  0]
kash> M*L[2];
> [ 1  1  0]
[-1  0  0]
[ 0  1  0]

</pre>

<p>


<hr>

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