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<head><title>isLLL -- is a basis an LLL basis?</title>
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<div><a href="index.html" title="lattice reduction (Lenstra-Lenstra-Lovasz bases)">LLLBases</a> > <a href="_is__L__L__L.html" title="is a basis an LLL basis?">isLLL</a></div>
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<div><h1>isLLL -- is a basis an LLL basis?</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>isLLL 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/___Matrix.html">matrix</a></span>, over <a href="../../Macaulay2Doc/html/___Z__Z.html" title="the class of all integers">ZZ</a>, whose columns are linearly independent</span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, Whether the columns of the matrix form an LLL basis with respect to the Threshold (which has default 3/4)</span></li>
</ul>
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</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>Threshold => ...</tt> (missing documentation<!-- tag: isLLL(..., Threshold => ...) -->), </span></li>
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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/>
If the matrix is not in LLL reduced form, then the offending conditions are displayed. For example,<table class="examples"><tr><td><pre>i1 : m = matrix {{1, 0}, {1, 1}, {1, 2}, {1, 3}}
o1 = | 1 0 |
| 1 1 |
| 1 2 |
| 1 3 |
4 2
o1 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i2 : isLLL m
LLL size failure 1,0: 1.5
LLL Lovasz failure 1: -.833333
o2 = false</pre>
</td></tr>
<tr><td><pre>i3 : n = LLL m
o3 = | 1 -1 |
| 1 0 |
| 1 1 |
| 1 2 |
4 2
o3 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i4 : isLLL n
o4 = true</pre>
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If the optional argument Threshold is given, the conditions are checked using that value.<table class="examples"><tr><td><pre>i5 : m = matrix {{1, 0}, {1, 1}, {1, 2}, {1, 3}}
o5 = | 1 0 |
| 1 1 |
| 1 2 |
| 1 3 |
4 2
o5 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i6 : isLLL(m, Threshold=>1)
LLL size failure 1,0: 1.5
LLL Lovasz failure 1: -1
o6 = false</pre>
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<div class="single"><h2>Caveat</h2>
<div>A Gram-Schmidt reduction is done over QQ, so this can be computationally intensive for larger matrix sizes. It is usually easier and faster to see if LLL returns the same matrix. This routine was used to debug and test the LLL routines here, and is provided as an alternate check of correctness.The matrix must be defined over the ring ZZ. It should be possible to allow real and rational matrices too, but this is not yet implemented.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___L__L__L.html" title="compute an LLL basis">LLL</a> -- compute an LLL basis</span></li>
<li><span><a href="_gcd__L__L__L.html" title="compute the gcd of integers, and small multipliers">gcdLLL</a> -- compute the gcd of integers, and small multipliers</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>
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<div class="waystouse"><h2>Ways to use <tt>isLLL</tt> :</h2>
<ul><li>isLLL(Matrix)</li>
</ul>
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