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<!DOCTYPE html> <html lang="en"> <head> <title>randomSystem -- an example of a 0-dimensional square polynomial system</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="examples of polynomial systems" href="index.html">ExampleSystems</a> :: <a title="an example of a 0-dimensional square polynomial system" href="_random__System.html">randomSystem</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="_rbpl24s.html">next</a> | <a href="_random__Generalized__Eigenvalue__Problem.html">previous</a> | <a href="_rbpl24s.html">forward</a> | <a href="_random__Generalized__Eigenvalue__Problem.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>randomSystem -- an example of a 0-dimensional square polynomial system</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">F = randomSystem(n,d,K)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">n</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, positive value representing the number of polynomials</span></li> <li><span><span class="tt">d</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, positive value representing the degree of the polynomials</span></li> <li><span><span class="tt">K</span>, <span>a <a title="the class of all rings" href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, usually a field, e.g., QQ or CC_{53}</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">F</span>, <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, a system of $n$ polynomials random polynomials of degree $d$ in $n$ variables.</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>This system was solved in May 2020, using <a title="track a linear segment homotopy given start and target system" href="../../NumericalAlgebraicGeometry/html/_track_lp__List_cm__List_cm__List_rp.html">track</a> in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.</p> <p>For a system with 2 polynomials of degree 3 over rationals there were 9 solutions found in 0.06 seconds.</p> <p>For a system with 5 polynomials of degree 2 over complex numbers there were 32 solutions found in 0.297 seconds.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : randomSystem(2,3,QQ) 7 3 7 2 7 2 3 3 2 7 7 2 9 9 o1 = {--x1 + --x1 x2 + -x1*x2 + 7x2 + -x1 + -x1*x2 + -x2 + -x1 + -x2 - 10 10 3 4 4 9 2 4 ------------------------------------------------------------------------ 3 3 3 2 2 10 3 2 5 2 2 3 6 1, --x1 + -x1 x2 + 5x1*x2 + --x2 + 6x1 + -x1*x2 + -x2 + -x1 + -x2 - 10 7 9 4 9 7 7 ------------------------------------------------------------------------ 1} o1 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : randomSystem(5, 2, CC_53) -- warning: experimental computation over inexact field begun -- results not reliable (one warning given per session) 2 o2 = {(.174853 + .626892*ii)x1 + (.350611 + .379495*ii)x1*x2 + (.987173 + ------------------------------------------------------------------------ 2 .568273*ii)x2 + (.237252 + .116721*ii)x1*x3 + (.184779 + ------------------------------------------------------------------------ 2 .370833*ii)x2*x3 + (.164647 + .713493*ii)x3 + (.444183 + ------------------------------------------------------------------------ .644366*ii)x1*x4 + (.339222 + .062212*ii)x2*x4 + (.909537 + ------------------------------------------------------------------------ 2 .566034*ii)x3*x4 + (.562839 + .629991*ii)x4 + (.194945 + ------------------------------------------------------------------------ .518585*ii)x1*x5 + (.465736 + .40273*ii)x2*x5 + (.305423 + ------------------------------------------------------------------------ .732358*ii)x3*x5 + (.479826 + .815167*ii)x4*x5 + (.97723 + ------------------------------------------------------------------------ 2 .0595849*ii)x5 + (.914199 + .887381*ii)x1 + (.169813 + .965004*ii)x2 + ------------------------------------------------------------------------ (.0647412 + .877846*ii)x3 + (.0340514 + .507989*ii)x4 + (.150294 + ------------------------------------------------------------------------ 2 .656391*ii)x5 - 1, (.305946 + .53632*ii)x1 + (.205375 + ------------------------------------------------------------------------ 2 .276652*ii)x1*x2 + (.280679 + .475179*ii)x2 + (.0958269 + ------------------------------------------------------------------------ .605398*ii)x1*x3 + (.130004 + .522285*ii)x2*x3 + (.767433 + ------------------------------------------------------------------------ 2 .155346*ii)x3 + (.883549 + .942865*ii)x1*x4 + (.171029 + ------------------------------------------------------------------------ .340019*ii)x2*x4 + (.0215389 + .283851*ii)x3*x4 + (.0328338 + ------------------------------------------------------------------------ 2 .21113*ii)x4 + (.0821679 + .10394*ii)x1*x5 + (.818142 + ------------------------------------------------------------------------ .781168*ii)x2*x5 + (.971588 + .249992*ii)x3*x5 + (.611415 + ------------------------------------------------------------------------ 2 .638389*ii)x4*x5 + (.0594513 + .0625324*ii)x5 + (.0645275 + ------------------------------------------------------------------------ .283709*ii)x1 + (.977573 + .212436*ii)x2 + (.592747 + .831802*ii)x3 + ------------------------------------------------------------------------ (.501243 + .154289*ii)x4 + (.467203 + .765564*ii)x5 - 1, (.46452 + ------------------------------------------------------------------------ 2 .16645*ii)x1 + (.224737 + .00415468*ii)x1*x2 + (.905002 + ------------------------------------------------------------------------ 2 .502056*ii)x2 + (.370963 + .511938*ii)x1*x3 + (.888883 + ------------------------------------------------------------------------ 2 .0776828*ii)x2*x3 + (.35911 + .751187*ii)x3 + (.806831 + ------------------------------------------------------------------------ .532303*ii)x1*x4 + (.41855 + .138169*ii)x2*x4 + (.252537 + ------------------------------------------------------------------------ 2 .10246*ii)x3*x4 + (.475683 + .107149*ii)x4 + (.572433 + ------------------------------------------------------------------------ .269298*ii)x1*x5 + (.543508 + .212989*ii)x2*x5 + (.543294 + ------------------------------------------------------------------------ .659792*ii)x3*x5 + (.13359 + .420228*ii)x4*x5 + (.112527 + ------------------------------------------------------------------------ 2 .659052*ii)x5 + (.270725 + .075503*ii)x1 + (.581271 + .640172*ii)x2 + ------------------------------------------------------------------------ (.919548 + .560684*ii)x3 + (.847715 + .763255*ii)x4 + (.477291 + ------------------------------------------------------------------------ 2 .0048212*ii)x5 - 1, (.689783 + .231832*ii)x1 + (.0311856 + ------------------------------------------------------------------------ 2 .388518*ii)x1*x2 + (.762468 + .455803*ii)x2 + (.0258757 + ------------------------------------------------------------------------ .0288963*ii)x1*x3 + (.299391 + .77016*ii)x2*x3 + (.429347 + ------------------------------------------------------------------------ 2 .288364*ii)x3 + (.912808 + .898905*ii)x1*x4 + (.885699 + ------------------------------------------------------------------------ .761573*ii)x2*x4 + (.993594 + .895324*ii)x3*x4 + (.0770766 + ------------------------------------------------------------------------ 2 .30813*ii)x4 + (.580909 + .228196*ii)x1*x5 + (.42526 + .403967*ii)x2*x5 ------------------------------------------------------------------------ + (.34672 + .363895*ii)x3*x5 + (.633138 + .262856*ii)x4*x5 + (.559948 + ------------------------------------------------------------------------ 2 .249212*ii)x5 + (.649378 + .564001*ii)x1 + (.455949 + .00471457*ii)x2 + ------------------------------------------------------------------------ (.771746 + .74002*ii)x3 + (.20217 + .774737*ii)x4 + (.556033 + ------------------------------------------------------------------------ 2 .91523*ii)x5 - 1, (.531857 + .356922*ii)x1 + (.172651 + ------------------------------------------------------------------------ 2 .533417*ii)x1*x2 + (.738044 + .0215483*ii)x2 + (.0219272 + ------------------------------------------------------------------------ .817573*ii)x1*x3 + (.751374 + .767024*ii)x2*x3 + (.886882 + ------------------------------------------------------------------------ 2 .784812*ii)x3 + (.168645 + .358533*ii)x1*x4 + (.553621 + ------------------------------------------------------------------------ .74003*ii)x2*x4 + (.829966 + .270696*ii)x3*x4 + (.831694 + ------------------------------------------------------------------------ 2 .729951*ii)x4 + (.255456 + .466765*ii)x1*x5 + (.545222 + ------------------------------------------------------------------------ .958457*ii)x2*x5 + (.790306 + .783351*ii)x3*x5 + (.29683 + ------------------------------------------------------------------------ 2 .162247*ii)x4*x5 + (.908964 + .231486*ii)x5 + (.671836 + .707257*ii)x1 ------------------------------------------------------------------------ + (.0423322 + .839258*ii)x2 + (.821811 + .68646*ii)x3 + (.101507 + ------------------------------------------------------------------------ .253324*ii)x4 + (.223416 + .0263964*ii)x5 - 1} o2 : List</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">randomSystem</span>:</h2> <ul> <li><kbd>randomSystem(ZZ,ZZ,Ring)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="an example of a 0-dimensional square polynomial system" href="_random__System.html">randomSystem</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">ExampleSystems/randomSystem.m2:41:0</span>.</p> </div> </div> </div> </body> </html>