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<!DOCTYPE html> <html lang="en"> <head> <title>normaliz -- calls Normaliz</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="an interface to use Normaliz in Macaulay 2" href="index.html">Normaliz</a> :: <a title="calls Normaliz" href="_normaliz.html">normaliz</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="_normaliz_lp__List_rp.html">next</a> | <a href="_nmz__Version.html">previous</a> | <a href="_normaliz_lp__List_rp.html">forward</a> | <a href="_nmz__Version.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>normaliz -- calls Normaliz</h1> <div> <h2>Description</h2> <p></p> This function applies <span class="tt">Normaliz</span> to the input data, which can be a matrix specifying a cone and an integer indicating the type for <span class="tt">Normaliz</span> or a list consisting of pairs of such a matrix and an integer. The function returns an object of type <a title="class of rational cones" href="___Rational__Cone.html">RationalCone</a>. The type determines how the rows of the matrix are interpreted, see also <a title="creates an input file for Normaliz" href="_write__Nmz__Data.html">writeNmzData</a>: <ul> <li>integral_closure: Computes the Hilbert basis of the rational cone generated by the rows with respect to the ambient lattice \ZZ^n;</li> <li>normalization: The same as integral_closure, but with respect to the sublattice of \ZZ^n generated by the rows;</li> <li>polytope: Computes the integral points in the polytope spanned by the rows and its Ehrhart semigroup (the semigroup determined by the polytope);</li> <li>rees_algebra: Computes the integral closure of the Rees algebra of the ideal generated by the monomials with exponent vectors the rows;</li> <li>inequalities: Computes the Hilbert basis of the rational cone in \RR^m given by the system of homogeneous inequalities <span class="tt">mat </span>x\ \geq\ 0;</li> <li>equations: Computes the Hilbert basis of the rational cone given by the nonnegative solutions of the homogeneous system <span class="tt">mat </span>x\ =\ 0.</li> <li>congruences: Computes the Hilbert basis of the rational cone given by the nonnegative solutions of the system of congruences defined by the rows as follows: Each row (x_1,\dots,x_n,c) represents a congruence x_1 z_1+\dots+x_n z_n \equiv \ 0 \mod \ c.</li> <li>inhom_inequalities: Computes the Hilbert basis of the rational cone in \RR^m given by the system of inhomogeneous inequalities. Each row (x_1,\dots,x_n,b) represents an inequality x_1 z_1+\dots+x_n z_n + b \geq \ 0.</li> <li>inhom_equations: Computes the Hilbert basis of the rational cone given by the nonnegative solutions of the inhomogeneous system <span class="tt">mat </span>x\ =\ b.</li> <li>inhom_congruences: Computes the Hilbert basis of the rational cone given by the nonnegative solutions of the system of congruences defined by the rows as follows: Each row (x_1,\dots,x_n,b,c) represents a congruence x_1 z_1+\dots+x_n z_n + b \equiv \ 0 \mod \ c.</li> <li>normal_toric_ideal: Computes the monoid as a quotient of \ZZ_+^n modulo a system of congruences (in the semigroup sense) defined by the rows of the input matrix.</li> </ul> <p></p> It is possible to combine certain input types, see <a title="calls Normaliz with several input matrices" href="_normaliz_lp__List_rp.html">normaliz(List)</a>. If you want to input only one matrix you can also use <a title="calls Normaliz" href="_normaliz_lp__Matrix_cm__String_rp.html">normaliz(Matrix,String)</a>. <p></p> By default, the cone returned contains only the content of the output file .gen, under the key "gen", i.e. the generators that have been computed, line by line, and the content of the output file .inv, under the key "inv". <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : setNmzOption("allf",true);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : eq=matrix {{1, 1, 1, -1, -1, -1, 0, 0, 0}, {1, 1, 1, 0, 0, 0, -1, -1, -1}, {0, 1, 1, -1, 0, 0, -1, 0, 0}, {1, 0, 1, 0, -1, 0, 0, -1, 0}, {1, 1, 0, 0, 0, -1, 0, 0, -1}, {0, 1, 1, 0, -1, 0, 0, 0, -1}, {1, 1, 0, 0, -1, 0, -1, 0, 0}}; 7 9 o2 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : rc=normaliz(eq,"equations") o3 = RationalCone{"gen" => | 0 2 1 2 1 0 1 0 2 | } | 1 0 2 2 1 0 0 2 1 | | 1 1 1 1 1 1 1 1 1 | | 1 2 0 0 1 2 2 0 1 | | 2 0 1 0 1 2 1 2 0 | "inv" => HashTable{"" => (1, 1, 1) } "class group" => (1, 2, 2) "degree 1 elements" => 5 "dim max subspace" => 0 "embedding dim" => 9 "external index" => 1 "graded" => true "grading denom" => 3 "grading" => (1, 1, 1, 0, 0, 0, 0, 0, 0) "hilbert basis elements" => 5 "hilbert quasipolynomial denom" => 1 "hilbert series denom" => (1, 1, 1) "hilbert series num" => (1, 2, 1) "inhomogeneous" => false "multiplicity denom" => 1 "multiplicity" => 4 "number extreme rays" => 4 "number support hyperplanes" => 4 "rank" => 3 "size triangulation" => 2 "sum dets" => 4 o3 : RationalCone</code></pre> </td> </tr> </table> <p></p> To obtain all the information written by <span class="tt">Normaliz</span> set the option <a href="_all__Computations.html">allComputations</a> to true (to decide which information shall be written by <span class="tt">Normaliz</span> use the options for <span class="tt">Normaliz</span>, see <a title="sets a command line option for Normaliz" href="_set__Nmz__Option.html">setNmzOption</a>). Then the method returns an object of type RationalCone whose keys are the suffixes of all the output files written, with value the content of the corresponding output file, which is an matrix whose rows contain the data computed, except for the suffix <span class="tt">inv</span>, for which the type is a <a title="the class of all hash tables" href="../../Macaulay2Doc/html/___Hash__Table.html">HashTable</a> (see also <a title="returns the numerical invariants computed" href="_get__Num__Invs.html">getNumInvs</a>). <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i4 : arc=normaliz(allComputations=>true,eq,"equations");</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : arc#"gen" o5 = | 0 2 1 2 1 0 1 0 2 | | 1 0 2 2 1 0 0 2 1 | | 1 1 1 1 1 1 1 1 1 | | 1 2 0 0 1 2 2 0 1 | | 2 0 1 0 1 2 1 2 0 | 5 9 o5 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : arc#"ext" o6 = | 0 2 1 2 1 0 1 0 2 | | 1 0 2 2 1 0 0 2 1 | | 1 2 0 0 1 2 2 0 1 | | 2 0 1 0 1 2 1 2 0 | 4 9 o6 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : arc#"inv" o7 = HashTable{"" => (1, 1, 1) } "class group" => (1, 2, 2) "degree 1 elements" => 5 "dim max subspace" => 0 "embedding dim" => 9 "external index" => 1 "graded" => true "grading denom" => 3 "grading" => (1, 1, 1, 0, 0, 0, 0, 0, 0) "hilbert basis elements" => 5 "hilbert quasipolynomial denom" => 1 "hilbert series denom" => (1, 1, 1) "hilbert series num" => (1, 2, 1) "inhomogeneous" => false "multiplicity denom" => 1 "multiplicity" => 4 "number extreme rays" => 4 "number support hyperplanes" => 4 "rank" => 3 "size triangulation" => 2 "sum dets" => 4 o7 : HashTable</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="class of rational cones" href="___Rational__Cone.html">RationalCone</a> -- class of rational cones</span></li> <li><span><a title="reads an output file of Normaliz containing one matrix" href="_read__Nmz__Data.html">readNmzData</a> -- reads an output file of Normaliz containing one matrix</span></li> <li><span><a href="___Keeping_spresults_spof_spthe_spcomputation_spby_sp__Normaliz.html">Keeping results of the computation by Normaliz</a></span></li> <li><span><a href="_output_spfiles_spwritten_spby_sp__Normaliz.html">output files written by Normaliz</a></span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">normaliz</span>:</h2> <ul> <li><span><a title="calls Normaliz with several input matrices" href="_normaliz_lp__List_rp.html">normaliz(List)</a> -- calls Normaliz with several input matrices</span></li> <li><span><a title="calls Normaliz" href="_normaliz_lp__Matrix_cm__String_rp.html">normaliz(Matrix,String)</a> -- calls Normaliz</span></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="calls Normaliz" href="_normaliz.html">normaliz</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">Normaliz.m2:1247:0</span>.</p> </div> </div> </div> </body> </html>
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