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

Returns a Cholesky type decompostion of the gram matrix
of a lattice.

<p><p>

<h3>
Syntax:
</h3>
<pre>q := LatCholesky(Lambda);</pre>

<p>

<table>
<tr>
<td>matrix</td>
<td><pre>  q  </pre></td>
<td></td>
</tr>
<tr>
<td>lattice</td>
<td><pre>  Lambda  </pre></td>
<td></td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
Let \Lambda be a lattice of rank n and let
G denote its Gram matrix. Then
Q(x) = x^{t}cdot G cdot x is a positive definite
quadratic form. \smallskip

The matrix q = (q_{ij}) \in R^{n \times n} which
is returned by the <tt> LatCholesky</tt> function satisfies
 Q(x) = \sum_{i=1}^n q_{ii} \left(x_i
+ \sum_{j=i+1}^n q_{ij}x_j right)^2. 
Additionally q_{ij} = 0 for 1<i &lt;=  n, 1 &lt;=  j<i.
This matrix is used in the Fincke-Pohst enumeration algorithm.

<p><p>

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

<pre>

kash> o := Order(Poly(Zx, [1,1,1,-1]));;
kash> Lambda := Lat(o);
Basis:
[1 0.5436890126920763615708559718017479865252032976510691 0.295597742522084770\
9809965928515386138989754484467009]
[1.414213562373095048801688724209698078569671875377 -1.09155296891773362399905\
0526886152282569769981597 -0.9161259494273487353043121929335785651570335323191\
771]
[0 1.577049658848827600240931210470570288477866219778 -2.434474230834722525486\
910332524147709670192579961]

kash> LatCholesky(Lambda);
> [3 -0.3333333333333333333333333333333333333333333333336667 -0.3333333333333333\
333333333333333333333333333333327664]
[0 3.640837919617073702751368388824778481413999607308 -0.827256502557640321389\
6126463804809177631530607521363]
[0 0 4.028376706263605945611231105883947510766559979681]

</pre>

<p>


<hr>

<p>
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