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<!DOCTYPE html> <html lang="en"> <head> <title>gcdLLL -- compute the gcd of integers, and small multipliers</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="compute the gcd of integers, and small multipliers" href="_gcd__L__L__L.html">gcdLLL</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> <a href="_hermite.html">next</a> | <a href="_kernel__L__L__L.html">previous</a> | <a href="_hermite.html">forward</a> | <a href="_kernel__L__L__L.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>gcdLLL -- compute the gcd of integers, and small multipliers</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">(g,z) = gcdLLL m</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">m</span>, <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, of integers</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><span class="tt">Strategy</span> (missing documentation)<!--tag: [gcdLLL, Strategy]--> <span class="tt"> => </span><span class="tt">...</span>, <span>default value null</span>, <span></span></span></li> <li><span><span class="tt">Threshold</span> (missing documentation)<!--tag: [gcdLLL, 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 class="tt">g</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the gcd of the integers in the list s</span></li> <li><span><span class="tt">z</span>, <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, of integers</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> 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></p> The method used is described in the paper: <p></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></p> For an example, <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : s = apply(5,i->372*(random 1000000)) o1 = {306370272, 229247604, 135272220, 220821804, 229345440} o1 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : (g,z) = gcdLLL s o2 = (372, | 1 -2 11 48 -20 |) | -5 -24 -2 -19 7 | | -12 15 -15 -7 7 | | 7 5 -31 0 11 | | 4 13 26 -41 5 | o2 : Sequence</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : matrix{s} * z o3 = | 0 0 0 0 372 | 1 5 o3 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="lattice reduction (Lenstra-Lenstra-Lovasz bases)" href="index.html">LLLBases</a> -- lattice reduction (Lenstra-Lenstra-Lovasz bases)</span></li> <li><span><a title="compute an LLL basis" href="___L__L__L.html">LLL</a> -- compute an LLL basis</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">gcdLLL</span>:</h2> <ul> <li><kbd>gcdLLL(List)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="compute the gcd of integers, and small multipliers" href="_gcd__L__L__L.html">gcdLLL</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.m2:1112:0</span>.</p> </div> </div> </div> </body> </html>