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<!DOCTYPE html> <html lang="en"> <head> <title>isLLL -- is a basis an LLL basis?</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="lattice reduction (Lenstra-Lenstra-Lovasz bases)" href="index.html">LLLBases</a> » <a title="is a basis an LLL basis?" href="_is__L__L__L.html">isLLL</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> next | <a href="_gramm.html">previous</a> | <a href="___B__K__Z.html">forward</a> | <a href="_gramm.html">backward</a> | <a href="index.html">up</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>isLLL -- is a basis an LLL basis?</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">isLLL m</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">m</span>, <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, over <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">ZZ</a>, whose columns are linearly independent</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><span class="tt">Threshold</span> (missing documentation)<!--tag: [isLLL, Threshold]--> <span class="tt"> => </span><span class="tt">...</span>, <span>default value 3/4</span>, <span></span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of boolean values" 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> </li> </ul> <div> <h2>Description</h2> This function is provided by the package <a title="lattice reduction (Lenstra-Lenstra-Lovasz bases)" href="index.html">LLLBases</a>. <p></p> If the matrix is not in LLL reduced form, then the offending conditions are displayed. For example, <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">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</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : isLLL m LLL size failure 1,0: 1.5 LLL Lovasz failure 1: -.833333 o2 = false</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : n = LLL m o3 = | 1 -1 | | 1 0 | | 1 1 | | 1 2 | 4 2 o3 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : isLLL n o4 = true</code></pre> </td> </tr> </table> If the optional argument Threshold is given, the conditions are checked using that value. <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">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</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : isLLL(m, Threshold=>1) LLL size failure 1,0: 1.5 LLL Lovasz failure 1: -1 o6 = false</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> 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> <div> <h2>See also</h2> <ul> <li><span><a title="compute an LLL basis" href="___L__L__L.html">LLL</a> -- compute an LLL basis</span></li> <li><span><a title="compute the gcd of integers, and small multipliers" href="_gcd__L__L__L.html">gcdLLL</a> -- compute the gcd of integers, and small multipliers</span></li> <li><span><span class="tt">kernelLLL</span> (missing documentation)<!--tag: kernelLLL--> </span></li> <li><span><span class="tt">hermite</span> (missing documentation)<!--tag: hermite--> </span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">isLLL</span>:</h2> <ul> <li><kbd>isLLL(Matrix)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="is a basis an LLL basis?" href="_is__L__L__L.html">isLLL</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">LLLBases.m2:1081:0</span>.</p> </div> </div> </div> </body> </html>