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<head><title>gcdLLL -- compute the gcd of integers, and small multipliers</title>
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<div><a href="index.html" title="lattice reduction (Lenstra-Lenstra-Lovasz bases)">LLLBases</a> > <a href="_gcd__L__L__L.html" title="compute the gcd of integers, and small multipliers">gcdLLL</a></div>
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<div><h1>gcdLLL -- compute the gcd of integers, and small multipliers</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>(g,z) = gcdLLL m</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>m</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, of integers</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><tt>g</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the gcd of the integers in the list s</span></li>
<li><span><tt>z</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, of integers</span></li>
</ul>
</div>
</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>Strategy => ...</tt> (missing documentation<!-- tag: gcdLLL(..., Strategy => ...) -->), </span></li>
<li><span><tt>Threshold => ...</tt> (missing documentation<!-- tag: gcdLLL(..., Threshold => ...) -->), </span></li>
</ul>
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</li>
</ul>
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<div class="single"><h2>Description</h2>
<div>This function is provided by the package <a href="index.html" title="lattice reduction (Lenstra-Lenstra-Lovasz bases)">LLLBases</a>.<p/>
The first n-1 columns of the matrix z form a basis of the kernel of the n integers of the list s, and the dot product of the last column of z and s is the gcd g.<p/>
The method used is described in the paper:<p/>
Havas, Majewski, Matthews, <em>Extended GCD and Hermite Normal Form Algorithms via Lattice Basis Reduction</em>, Experimental Mathematics 7:2 p. 125 (1998).<p/>
For an example,<table class="examples"><tr><td><pre>i1 : s = apply(5,i->372*(random 1000000))
o1 = {330441276, 107988252, 326524488, 42328020, 219662280}
o1 : List</pre>
</td></tr>
<tr><td><pre>i2 : (g,z) = gcdLLL s
o2 = (372, | 2 7 -1 -44 7 |)
| -14 -18 -9 -15 -6 |
| 7 -7 13 18 -11 |
| 18 -17 -28 -1 4 |
| -10 12 -8 47 8 |
o2 : Sequence</pre>
</td></tr>
<tr><td><pre>i3 : matrix{s} * z
o3 = | 0 0 0 0 372 |
1 5
o3 : Matrix ZZ <--- ZZ</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="index.html" title="lattice reduction (Lenstra-Lenstra-Lovasz bases)">LLLBases</a> -- lattice reduction (Lenstra-Lenstra-Lovasz bases)</span></li>
<li><span><a href="___L__L__L.html" title="compute an LLL basis">LLL</a> -- compute an LLL basis</span></li>
<li><span><tt>kernelLLL</tt> (missing documentation<!-- tag: kernelLLL -->)</span></li>
<li><span><tt>hermite</tt> (missing documentation<!-- tag: hermite -->)</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>gcdLLL</tt> :</h2>
<ul><li>gcdLLL(List)</li>
</ul>
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