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<head><title>LLLBases -- lattice reduction (Lenstra-Lenstra-Lovasz bases)</title>
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<div><h1>LLLBases -- lattice reduction (Lenstra-Lenstra-Lovasz bases)</h1>
<div class="single"><h2>Description</h2>
<div><em>LLLBases</em> is a package implementing several variants of LLL bases. Some of these are implemented in the Macaulay2 engine, some by Victor Shoup's NTL package, and some are implemented at top level.<p/>
A matrix over ZZ determines a lattice: this is the ZZ-module generated by the columns. It also determines a generating set. See the book: [H. Cohen, ...] for the definition and the basic algorithms for computing LLL bases.<p/>
LLL bases have nice theoretical properties, but the main benefit of LLL bases is that the entries of the resulting matrix often have dramatically small size, even smaller than theory would imply.<p/>
This package implements the following functions.</div>
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<div class="single"><h2>Author</h2>
<ul><li><div class="single"><a href="http://www.math.cornell.edu/~mike/">Michael E. Stillman</a><span> <<a href="mailto:mike@math.cornell.edu">mike@math.cornell.edu</a>></span></div>
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<div class="single"><h2>Version</h2>
This documentation describes version <b>1.1</b> of LLLBases.</div>
<div class="single"><h2>Source code</h2>
The source code from which this documentation is derived is in the file <a href="../../../../Macaulay2/LLLBases.m2">LLLBases.m2</a>.</div>
<div class="single"><h2>Exports</h2>
<ul><li><div class="single">Functions<ul><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>gramm</tt> (missing documentation<!-- tag: gramm -->)</span></li>
<li><span><tt>hermite</tt> (missing documentation<!-- tag: hermite -->)</span></li>
<li><span><a href="_is__L__L__L.html" title="is a basis an LLL basis?">isLLL</a> -- is a basis an LLL basis?</span></li>
<li><span><tt>kernelLLL</tt> (missing documentation<!-- tag: kernelLLL -->)</span></li>
<li><span><a href="___L__L__L.html" title="compute an LLL basis">LLL</a> -- compute an LLL basis</span></li>
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<li><div class="single">Symbols<ul><li><span><a href="___B__K__Z.html" title="compute BKZ reduced basis instead of an LLL reduced basis">BKZ</a> -- compute BKZ reduced basis instead of an LLL reduced basis</span></li>
<li><span><a href="___Cohen__Engine.html" title="use the original Macaulay2 LLL algorithm">CohenEngine</a> -- use the original Macaulay2 LLL algorithm</span></li>
<li><span><a href="___Cohen__Top__Level.html" title="use the Macaulay2 language LLL algorithm">CohenTopLevel</a> -- use the Macaulay2 language LLL algorithm</span></li>
<li><span><a href="___Givens.html" title="use Givens rotations instead of Gram-Schmidt during LLL">Givens</a> -- use Givens rotations instead of Gram-Schmidt during LLL</span></li>
<li><span><a href="___N__T__L.html" title="use the all-integer LLL strategy from NTL library">NTL</a> -- use the all-integer LLL strategy from NTL library</span></li>
<li><span><a href="___Real__F__P.html" title="use double precision real numbers">RealFP</a> -- use double precision real numbers</span></li>
<li><span><a href="___Real__Q__P.html" title="use quadruple precision real numbers">RealQP</a> -- use quadruple precision real numbers</span></li>
<li><span><a href="___Real__Q__P1.html" title="use a combination of double precision and quad precision real numbers">RealQP1</a> -- use a combination of double precision and quad precision real numbers</span></li>
<li><span><a href="___Real__R__R.html" title="use arbitrary precision real numbers">RealRR</a> -- use arbitrary precision real numbers</span></li>
<li><span><a href="___Real__X__D.html" title="use extended exponent real numbers">RealXD</a> -- use extended exponent real numbers</span></li>
<li><span><a href="___Threshold.html" title="the LLL threshold, in interval (1/4,1]">Threshold</a> -- the LLL threshold, in interval (1/4,1]</span></li>
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<div><h3>Menu</h3>
<h4>main LLL algorithm</h4>
<ul><li><span><a href="___L__L__L.html" title="compute an LLL basis">LLL</a> -- compute an LLL basis</span></li>
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<h4>applications and variants</h4>
<ul><li><span><tt>kernelLLL</tt> (missing documentation<!-- tag: kernelLLL -->)</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>hermite</tt> (missing documentation<!-- tag: hermite -->)</span></li>
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<h4>support routines that are occasionally useful</h4>
<ul><li><span><tt>gramm</tt> (missing documentation<!-- tag: gramm -->)</span></li>
<li><span><a href="_is__L__L__L.html" title="is a basis an LLL basis?">isLLL</a> -- is a basis an LLL basis?</span></li>
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