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_solve__Rational__System.html
<!DOCTYPE html> <html lang="en"> <head> <title>solveRationalSystem -- approximates solutions to a rational 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="a package for Polynomial Homotopy Continuation" href="index.html">PHCpack</a> :: <a title="approximates solutions to a rational system of equations" href="_solve__Rational__System.html">solveRationalSystem</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="_solve__Rational__System_lp..._cm__Verbose_eq_gt..._rp.html">next</a> | <a href="_see__Progress.html">previous</a> | <a href="_solve__Rational__System_lp..._cm__Verbose_eq_gt..._rp.html">forward</a> | <a href="_see__Progress.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>solveRationalSystem -- approximates solutions to a rational system of equations</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">solveRationalSystem (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 system of rational equations with a finite number of solutions</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><a title="option to specify whether additional output is wanted" href="_solve__Rational__System_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 specify whether additional output is wanted</span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, containing the solutions of f, each of type <a title="a type used to store a point in complex space" href="../../NAGtypes/html/___Abstract__Point.html">Point</a></span></li> </ul> </li> <li> <div> Consequences: <ul> <li>converts the rational system into a Laurent system, invokes the commands <span class="tt">phc -b</span> and <span class="tt">phc -z</span>,</li> <li>adds slack variables if needed (i.e. if system is overdetermined),</li> <li>writes the system to temporary file,</li> <li>launches the blackbox solver,</li> <li>stores the output of phc in a temporary file,</li> <li>parses and outputs the solutions.</li> </ul> </div> </li> </ul> <div> <h2>Description</h2> <div> <p>This function returns numerical approximations of all complex solutions of a rational system. The function converts the system to a Laurent polynomial system and then calls <span class="tt">PHCpack</span>'s blackbox solver.</p> <p></p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = QQ[x,y,z];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : system = {y-x^2, z-x^3, (x+y+z-1)/x};</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : sols = solveRationalSystem(system) o3 = {{.543689, .295598, .160713}, {-.771845-1.11514*ii, -.647799+1.72143*ii, ------------------------------------------------------------------------ 2.41964-.606291*ii}, {-.771845+1.11514*ii, -.647799-1.72143*ii, ------------------------------------------------------------------------ 2.41964+.606291*ii}} o3 : List</code></pre> </td> </tr> </table> <div> <p>The solutions are of type <a title="a type used to store a point in complex space" href="../../NAGtypes/html/___Abstract__Point.html">Point</a>. Each point <span class="tt">p</span> comes with cached diagnostics. For example, <span class="tt">p.cache.LastT</span> is the end value of the continuation parameter; if it equals 1, then the solver reached the end of the path properly.</p> <p></p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i4 : peek first sols o4 = Point{cache => CacheTable{...3...} } Coordinates => {.543689, .295598, .160713}</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="a numerical blackbox solver" href="_solve__System.html">solveSystem</a> -- a numerical blackbox solver</span></li> <li><span><a title="converts a list of rational polynomials into Laurent polynomials" href="_to__Laurent__Polynomial.html">toLaurentPolynomial</a> -- converts a list of rational polynomials into Laurent polynomials</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">solveRationalSystem</span>:</h2> <ul> <li><kbd>solveRationalSystem(List)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="approximates solutions to a rational system of equations" href="_solve__Rational__System.html">solveRationalSystem</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">PHCpack/PHCpackDoc.m2:979:0</span>.</p> </div> </div> </div> </body> </html>
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