One Hat Cyber Team
Your IP :
216.73.216.14
Server IP :
194.44.31.54
Server :
Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64
Server Software :
Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k
PHP Version :
5.6.40
Buat File
|
Buat Folder
Eksekusi
Dir :
~
/
usr
/
share
/
doc
/
Macaulay2
/
Bertini
/
html
/
View File Name :
_bertini__Pos__Dim__Solve.html
<!DOCTYPE html> <html lang="en"> <head> <title>bertiniPosDimSolve -- a main method that is used to produce witness sets</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="software for numerical algebraic geometry" href="index.html">Bertini</a> :: <a title="a main method that is used to produce witness sets" href="_bertini__Pos__Dim__Solve.html">bertiniPosDimSolve</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="_bertini__Refine__Sols.html">next</a> | <a href="_bertini__Parameter__Homotopy.html">previous</a> | <a href="_bertini__Refine__Sols.html">forward</a> | <a href="_bertini__Parameter__Homotopy.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>bertiniPosDimSolve -- a main method that is used to produce witness sets</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">V = bertiniPosDimSolve I</code></dd> <dd><code class="language-macaulay2">V = bertiniPosDimSolve F</code></dd> </dl> </li> <li>Inputs: <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 list of ring elements defining a variety</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><span class="tt">BertiniInputConfiguration</span> (missing documentation)<!--tag: [bertiniPosDimSolve, BertiniInputConfiguration]--> <span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span>, <span></span></span></li> <li><span><a title="optional argument to specify whether to use homogeneous coordinates" href="___Is__Projective.html">IsProjective</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value -1</span>, <span>optional argument to specify whether to use homogeneous coordinates</span></span></li> <li><span><a title="Option to silence additional output" href="_bertini__Track__Homotopy_lp..._cm__Verbose_eq_gt..._rp.html">Verbose</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value false</span>, <span>Option to silence additional output</span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">V</span>, <span>a <a title="a numerical variety" href="../../NAGtypes/html/___Numerical__Variety.html">numerical variety</a></span>, a numerical irreducible decomposition of the variety defined by F</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>The method <span class="tt">bertiniPosDimSolve</span> calls <span class="tt">Bertini</span> to find a numerical irreducible decomposition of the zero-set of F. The decomposition is returned as the <a title="a numerical variety" href="../../NAGtypes/html/___Numerical__Variety.html">NumericalVariety</a> NV. Witness sets of NV contain approximations to solutions of the system F=0. Bertini (1) writes the system to temporary files, (2) invokes <span class="tt">Bertini</span>'s solver with <span class="tt">TrackType => 1</span>, (3) Bertini uses a cascade homotopy to find witness supersets in each dimension, (4) removes extra points using a membership test or local dimension test, (5) deflates singular witness points, and finally (6) decomposes using a combination of monodromy and a linear trace test</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : F = {(y^2+x^2+z^2-1)*x,(y^2+x^2+z^2-1)*y} 3 2 2 2 3 2 o2 = {x + x*y + x*z - x, x y + y + y*z - y} o2 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : S = bertiniPosDimSolve F o3 = S o3 : NumericalVariety</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : S#1_0#Points -- 1_0 chooses the first witness set in dimension 1 o4 = {{2.64468e-59+1.83949e-59*ii, -1.0877e-60+3.37583e-59*ii, .261246+.146018*ii}} o4 : VerticalList</code></pre> </td> </tr> </table> <div> <p>Each <a title="a witness set" href="../../NAGtypes/html/___Witness__Set.html">WitnessSet</a> is accessed by dimension and then list position.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : S#1 --first specify dimension o5 = {(dim=1,deg=1)} o5 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : peek oo_0 --then list position o6 = WitnessSet{cache => CacheTable{...3...} } Equations => {-3} | x3+xy2+xz2-x | {-3} | x2y+y3+yz2-y | Points => {{2.64468e-59+1.83949e-59*ii, -1.0877e-60+3.37583e-59*ii, .261246+.146018*ii}} Slice => | .073883+1.51329ii 1.3836+.186588ii -2.02193+.757676ii .638855+.0972991ii |</code></pre> </td> </tr> </table> <div> <p>In the example, we find two components, one component has dimension 1 and degree 1 and the other has dimension 2 and degree 2. We get the same results using symbolic methods.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i7 : PD=primaryDecomposition( ideal F) 2 2 2 o7 = {ideal(x + y + z - 1), ideal (y, x)} o7 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : dim PD_0 o8 = 2</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : degree PD_0 o9 = 2</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : dim PD_1 o10 = 1</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : degree PD_1 o11 = 1</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">bertiniPosDimSolve</span>:</h2> <ul> <li><kbd>bertiniPosDimSolve(Ideal)</kbd></li> <li><kbd>bertiniPosDimSolve(List)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="a main method that is used to produce witness sets" href="_bertini__Pos__Dim__Solve.html">bertiniPosDimSolve</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">Bertini.m2:2756:0</span>.</p> </div> </div> </div> </body> </html>