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

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

<p><p>

<h3>
Syntax:
</h3>
<pre>L := MatLLL(M [,r]);
L := MatLLL(M,"short" | "long" [,r]);</pre>

<p>

<table>
<tr>
<td>list of <tt> A</tt>, <tt> T</tt></td>
<td><pre>  L  </pre></td>
<td></td>
</tr>
<tr>
<td>matrix</td>
<td><pre>  M  </pre></td>
<td></td>
</tr>
<tr>
<td>LLL constant (default 0.75)</td>
<td><pre>  r  </pre></td>
<td></td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
Performs an LLL reduction on the columns of an integer or real
matrix <tt> M</tt>. The result is a list containing the reduced
matrix <tt> A</tt> and the unimodular transformation matrix {\tt
T} such that <tt> M</tt> = <tt> A</tt>*<tt> T</tt>. The <tt> MatLLL</tt>
function requires that the columns of <tt> M</tt> are linearly
independent. \smallskip

If <tt> M</tt> is an integer matrix the <tt> MatLLL</tt> function
normally omits any floating-point arithmetic. When the {\tt
"SHORT"} option is specified the <tt> MatLLL</tt> function uses
floating-point arithmetic basing on machine
doubles. \smallskip

If <tt> M</tt> is a real matrix the <tt> MatLLL</tt> function
normally makes use of the long floating-point arithmetic of
kash. When the <tt> "SHORT"</tt> option is specified the {\tt
MatLLL} function uses floating-point arithmetic basing on
machine doubles

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>

<pre>

kash> M := Mat(Z, [[234, 469], [-6546, -13126],
> [-6312, -12657]]);
[   234    469]
[ -6546 -13126]
[ -6312 -12657]
kash> L := MatLLL(M);
[ [  1  42]
    [-34 -18]
    [-33  24], [  -2  385]
    [   1 -192] ]
kash> L[1];
[  1  42]
[-34 -18]
[-33  24]
kash> M*L[2];
> [  1  42]
[-34 -18]
[-33  24]

</pre>

<p>


<hr>

<p>
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