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<!DOCTYPE html> <html lang="en"> <head> <title>bertiniZeroDimSolve -- a main method to solve a zero-dimensional system of equations</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 to solve a zero-dimensional system of equations" href="_bertini__Zero__Dim__Solve.html">bertiniZeroDimSolve</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="___Copy__B_sq__File.html">next</a> | <a href="_bertini__User__Homotopy.html">previous</a> | <a href="___Copy__B_sq__File.html">forward</a> | <a href="_bertini__User__Homotopy.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>bertiniZeroDimSolve -- a main method to solve a zero-dimensional system of equations</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">S = bertiniZeroDimSolve F</code></dd> <dd><code class="language-macaulay2">S = bertiniZeroDimSolve I</code></dd> <dd><code class="language-macaulay2">S = bertiniZeroDimSolve(I, UseRegeneration=>1)</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 (system need not be square)</span></li> <li><span><span class="tt">I</span>, <span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, an ideal defining a variety</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><a title="an option to group variables and use multihomogeneous homotopies" href="___Variable_spgroups.html">AffVariableGroup</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span>, <span>an option to group variables and use multihomogeneous homotopies</span></span></li> <li><span><a title="an option to designate the constants for a Bertini Input file" href="___B_sq__Constants.html">B'Constants</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span>, <span>an option to designate the constants for a Bertini Input file</span></span></li> <li><span><span class="tt">B'Functions</span> (missing documentation)<!--tag: [bertiniZeroDimSolve, B'Functions]--> <span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span>, <span></span></span></li> <li><span><span class="tt">BertiniInputConfiguration</span> (missing documentation)<!--tag: [bertiniZeroDimSolve, BertiniInputConfiguration]--> <span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span>, <span></span></span></li> <li><span><a title="an option to group variables and use multihomogeneous homotopies" href="___Variable_spgroups.html">HomVariableGroup</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span>, <span>an option to group variables and use multihomogeneous homotopies</span></span></li> <li><span><span class="tt">IsProjective</span> (missing documentation)<!--tag: [bertiniZeroDimSolve, IsProjective]--> <span class="tt"> => </span><span class="tt">...</span>, <span>default value -1</span>, <span></span></span></li> <li><span><span class="tt">M2Precision</span> (missing documentation)<!--tag: [bertiniZeroDimSolve, M2Precision]--> <span class="tt"> => </span><span class="tt">...</span>, <span>default value 53</span>, <span></span></span></li> <li><span><span class="tt">NameMainDataFile</span> (missing documentation)<!--tag: [bertiniZeroDimSolve, NameMainDataFile]--> <span class="tt"> => </span><span class="tt">...</span>, <span>default value "main_data"</span>, <span></span></span></li> <li><span><span class="tt">NameSolutionsFile</span> (missing documentation)<!--tag: [bertiniZeroDimSolve, NameSolutionsFile]--> <span class="tt"> => </span><span class="tt">...</span>, <span>default value "raw_solutions"</span>, <span></span></span></li> <li><span><span class="tt">OutputStyle</span> (missing documentation)<!--tag: [bertiniZeroDimSolve, OutputStyle]--> <span class="tt"> => </span><span class="tt">...</span>, <span>default value "OutPoints"</span>, <span></span></span></li> <li><span><a title="an option which designates symbols/strings/variables that will be set to be a random real number or random complex number" href="___Bertini_spinput_spfile_spdeclarations_co_sprandom_spnumbers.html">RandomComplex</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span>, <span>an option which designates symbols/strings/variables that will be set to be a random real number or random complex number</span></span></li> <li><span><a title="an option which designates symbols/strings/variables that will be set to be a random real number or random complex number" href="___Bertini_spinput_spfile_spdeclarations_co_sprandom_spnumbers.html">RandomReal</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span>, <span>an option which designates symbols/strings/variables that will be set to be a random real number or random complex number</span></span></li> <li><span><a title="Option to change directory for file storage." href="___Top__Directory.html">TopDirectory</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value "/tmp/M2-269518-0/0"</span>, <span>Option to change directory for file storage.</span></span></li> <li><span><span class="tt">UseRegeneration</span> (missing documentation)<!--tag: [bertiniZeroDimSolve, UseRegeneration]--> <span class="tt"> => </span><span class="tt">...</span>, <span>default value -1</span>, <span></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">S</span>, <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list of points that are contained in the variety of F</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>This method finds isolated solutions to the system F via numerical polynomial homotopy continuation by (1) building a Bertini input file from the system F, (2) calling Bertini on this input file, (3) returning solutions from a machine readable file that is an output from Bertini.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = CC[x,y];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : F = {x^2-1,y^2-2};</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : S = bertiniZeroDimSolve F o3 = {{1, 1.41421}, {1, -1.41421}, {-1, 1.41421}, {-1, -1.41421}} o3 : List</code></pre> </td> </tr> </table> <div> <p>Each solution is of type <a title="a type used to store a point in complex space" href="../../NAGtypes/html/___Abstract__Point.html">Point</a>. Additional information about the solution can be accessed by using <a title="examine contents of an object" href="../../Macaulay2Doc/html/_peek.html">peek</a>.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i4 : peek S_0 o4 = Point{cache => CacheTable{...14...}} Coordinates => {1, 1.41421}</code></pre> </td> </tr> </table> <div> <p>Bertini uses a multihomogeneous homotopy as a default, but regeneration can be deployed with the option UseRegeneration=>1 .</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : R = CC[x];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : F = {x^2*(x-1)};</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : S = bertiniZeroDimSolve F o7 = {{1}, {-1.55589e-15-2.46051e-15*ii}} o7 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : B = bertiniZeroDimSolve(F,UseRegeneration=>1) o8 = {{1}} o8 : List</code></pre> </td> </tr> </table> <div> <p>Variables must begin with a letter (lowercase or capital) and can only contain letters, numbers, underscores, and square brackets. Regeneration in bertiniZeroDimSolve only finds nonsingular isolated points.</p> </div> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">bertiniZeroDimSolve</span>:</h2> <ul> <li><kbd>bertiniZeroDimSolve(Ideal)</kbd></li> <li><kbd>bertiniZeroDimSolve(List)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="a main method to solve a zero-dimensional system of equations" href="_bertini__Zero__Dim__Solve.html">bertiniZeroDimSolve</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.m2:2674:0</span>.</p> </div> </div> </div> </body> </html>