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<!DOCTYPE html> <html lang="en"> <head> <title>sortSolutions(List) -- sort the list of solutions</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="Common types used in Numerical Algebraic Geometry" href="index.html">NAGtypes</a> :: <a title="sort the list of solutions" href="_sort__Solutions_lp__List_rp.html">sortSolutions(List)</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="_substitute_lp__Poly__System_cm__Ring_rp.html">next</a> | <a href="_sort__Solutions_lp..._cm__Tolerance_eq_gt..._rp.html">previous</a> | <a href="_substitute_lp__Poly__System_cm__Ring_rp.html">forward</a> | <a href="_sort__Solutions_lp..._cm__Tolerance_eq_gt..._rp.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>sortSolutions(List) -- sort the list of solutions</h1> <ul> <li><span>Function: <a title="sort the list of solutions" href="_sort__Solutions_lp__List_rp.html">sortSolutions</a></span></li> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">t = sortSolutions s</code></dd> </dl> </li> <li>Inputs: <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>, contains solutions (represented either by lists of coordinates or <a title="a type used to store a point in complex space" href="___Abstract__Point.html">points</a>)</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><a title="the tolerance of a numerical computation" href="_sort__Solutions_lp..._cm__Tolerance_eq_gt..._rp.html">Tolerance</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value .000001</span>, <span>the tolerance of a numerical computation</span></span></li> <li><span><span class="tt">Weights</span><span class="tt"> => </span><span class="tt">...</span>, <span>default value null</span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">t</span>, <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, contains solutions sorted as described below</span></li> </ul> </li> </ul> <div> <h2>Description</h2> The sorting is done lexicographically regarding each complex n-vector as real 2n-vector. The output format of <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> and <a title="solve a system of polynomial equations" href="../../NumericalAlgebraicGeometry/html/_solve__System_lp__List_rp.html">solveSystem</a> is respected.<br>For the corresponding coordinates a and b (of two real 2n-vectors) a < b if b-a is larger than <a title="the tolerance of a numerical computation" href="../../NumericalLinearAlgebra/html/___Tolerance.html">Tolerance</a>. <p></p> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : needsPackage "NumericalAlgebraicGeometry" o1 = NumericalAlgebraicGeometry o1 : Package</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : R = CC[x,y];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : s = solveSystem {x^2+y^2-1, x*y} o3 = {{3.02251e-20+2.05294e-20*ii, -1}, {-3.02251e-20-2.05294e-20*ii, 1}, {1, ------------------------------------------------------------------------ 8.30314e-21+4.44918e-20*ii}, {-1, -8.30314e-21-4.44918e-20*ii}} o3 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : sortSolutions s o4 = {{-1, -8.30314e-21-4.44918e-20*ii}, {3.02251e-20+2.05294e-20*ii, -1}, ------------------------------------------------------------------------ {-3.02251e-20-2.05294e-20*ii, 1}, {1, 8.30314e-21+4.44918e-20*ii}} o4 : List</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> The sorting described above does not possess good properties, since there may be near ties in specific coordinate values between several points. A better way is to specify a random weight (of length 2n where n=#points) as an optional parameter <a title="sort the list of solutions" href="_sort__Solutions_lp__List_rp.html">sortSolutions(...,Weights=>...)</a>, which provides a linear functional that evaluates to distinct (and sufficiently) real numbers on the given points. </div> <div> <h2>See also</h2> <ul> <li><span><a title="solve a system of polynomial equations" href="../../NumericalAlgebraicGeometry/html/_solve__System_lp__List_rp.html">solveSystem</a> -- solve a system of polynomial equations</span></li> <li><span><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> -- track a linear segment homotopy given start and target system</span></li> <li><span><a title="determine if solutions are equal" href="_are__Equal.html">areEqual</a> -- determine if solutions are equal</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use this method:</h2> <ul> <li><span><a title="sort the list of solutions" href="_sort__Solutions_lp__List_rp.html">sortSolutions(List)</a> -- sort the list of solutions</span></li> </ul> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">NAGtypes/doc-NAGtypes.m2:145:0</span>.</p> </div> </div> </div> </body> </html>