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<!DOCTYPE html> <html lang="en"> <head> <title>Chordal -- exploiting chordal structure in polynomial ideals</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="exploiting chordal structure in polynomial ideals" href="index.html">Chordal</a> :: <a title="exploiting chordal structure in polynomial ideals" href="index.html">Chordal</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="_adjacent__Minors__Ideal.html">next</a> | previous | <a href="_adjacent__Minors__Ideal.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>Chordal -- exploiting chordal structure in polynomial ideals</h1> <div> <h2>Description</h2> <p>This package provides several specialized routines for structured polynomial ideals. The sparsity structure of a polynomial set can be described with a graph. By exploiting some suitable "chordal completion" of this graph, it is possible to develop more efficient algorithms for several problems in computational algebraic geometry. </p> See <a title="of the Chordal package" href="_installation_spand_spconfiguration.html">installation and configuration</a> for instructions on how to install this package. <p>The examples below illustrate how to use this package to get the following properties of a structured ideal: compute elimination ideals, count the number of zeros, determine the dimension, decompose the variety. </p> <h5>graphical structure of an ideal</h5> We can abstract the sparsity structure of a polynomial system in a graph. By exploiting the chordal completions of this graph more efficient algorithms can be developed. <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = QQ[a,b,c,d];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : I = ideal {a^2-1, a^2+a*b+1, a^3+c^2, b*d + d, c^3+c*d}; o2 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : G = constraintGraph I o3 = Graph{a => {b, c}} b => {a, d} c => {a, d} d => {b, c} o3 : Graph</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : Gc = chordalGraph G o4 = ChordalGraph{a => {b, c} } b => {c, d} c => {d} d => {} o4 : ChordalGraph</code></pre> </td> </tr> </table> <h5>chordal elimination</h5> Consider the <a title="chromatic ideal of a graph" href="_chromatic__Ideal.html">3-chromatic ideal</a> of the cycle graph. Its elimination ideals have have a simple generating set. <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : I = chromaticIdeal(QQ, cycleGraph 10, 3); o5 : Ideal of QQ[a..j]</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : N = chordalNet I;</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : chordalElim N;</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : N 2 2 2 2 o8 = ChordalNet{ a => {{a*b - a*j + b - j , a + a*j + j }} } 2 2 b => {b + b*c + c } 2 2 c => {c + c*d + d } 2 2 d => {d + d*e + e } 2 2 e => {e + e*f + f } 2 2 f => {f + f*g + g } 2 2 g => {g + g*h + h } 2 2 h => {h + h*i - i*j - j } 2 2 i => {i + i*j + j } 3 j => {j - 1} o8 : ChordalNet</code></pre> </td> </tr> </table> On the other hand, its Groebner basis is complicated. <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i9 : sum for f in gbList I list #terms f o9 = 253</code></pre> </td> </tr> </table> <h5>chordal networks</h5> Consider the <a title="ideal of adjacent minors" href="_adjacent__Minors__Ideal.html">ideal of adjacent minors</a> of a 2xn matrix. This ideal decomposes into F<sub>n</sub> components, where F<sub>n</sub> is the Fibonacci number. These (exponentially many) components can be represented compactly with a chordal network. <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i10 : I = adjacentMinorsIdeal(QQ,2,10); o10 : Ideal of QQ[a..t]</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : N = chordalNet I;</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : chordalTria N;</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i13 : N o13 = ChordalNet{ a => { , a*d - b*c} } b => { , b, } c => {c, , , c*f - d*e} d => {d, d, , } e => { , , e*h - f*g, e, , e*h - f*g} f => { , f, , f, , } g => {g, , , g*j - h*i, , g*j - h*i} h => {h, h, , , , } i => { , , i*l - j*k, i, , i*l - j*k} j => { , j, , j, , } k => {k, , , k*n - l*m, , k*n - l*m} l => {l, l, , , , } m => { , , m*p - n*o, m, , m*p - n*o} n => { , n, , n, , } o => {o, , , o*r - p*q, , o*r - p*q} p => {p, p, , , , } q => { , , q*t - r*s, q, , q*t - r*s} r => { , r, , r, , } s => {s, , , } t => {t, , } o13 : ChordalNet</code></pre> </td> </tr> </table> Several properties of an ideal can be computed efficiently given a chordal representation; for instance, its dimension. <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i14 : dim N o14 = 11</code></pre> </td> </tr> </table> We can also extract the top dimensional part, and count the number of components. <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i15 : topComponents N</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i16 : codimCount N 9 o16 = 55t o16 : ZZ[t]</code></pre> </td> </tr> </table> <br>For further example see <ul> <li><span><a title="a new representation of polynomial ideals" href="_chordal_spnetworks_spexamples.html">chordal networks examples</a> -- a new representation of polynomial ideals</span></li> </ul> <h2>Overview of methods</h2> Graphical structure of a polynomial ideal: <ul> <li><span><a title="constraint graph of a polynomial set" href="_constraint__Graph.html">constraintGraph</a> -- constraint graph of a polynomial set</span></li> <li><span><a title="chordal completion of a graph" href="_chordal__Graph.html">chordalGraph</a> -- chordal completion of a graph</span></li> <li><span><a title="constructs a chordal network from a polynomial set" href="_chordal__Net.html">chordalNet</a> -- constructs a chordal network from a polynomial set</span></li> </ul> Elimination routines: <ul> <li><span><a title="performs elimination on the chordal network" href="_chordal__Elim.html">chordalElim</a> -- performs elimination on the chordal network</span></li> <li><span><a title="makes a chordal network triangular" href="_chordal__Tria.html">chordalTria</a> -- makes a chordal network triangular</span></li> </ul> Methods for triangular chordal networks: <ul> <li><span><a title="counts the number of roots of a chordal network" href="_root__Count.html">rootCount</a> -- counts the number of roots of a chordal network</span></li> <li><span><a title="dimension of a chordal network" href="_dim_lp__Chordal__Net_rp.html">dim(ChordalNet)</a> -- dimension of a chordal network</span></li> <li><span><a title="dimension of a chordal network" href="_dim_lp__Chordal__Net_rp.html">codim(ChordalNet)</a> -- dimension of a chordal network</span></li> <li><span><a title="codimension counts of the chains of a chordal network" href="_codim__Count.html">codimCount</a> -- codimension counts of the chains of a chordal network</span></li> <li><span><a title="top dimension of a chordal network" href="_top__Components_lp__Chordal__Net_rp.html">topComponents(ChordalNet)</a> -- top dimension of a chordal network</span></li> <li><span><a title="ideal membership test" href="___Ring__Element_sp_pc_sp__Chordal__Net.html">RingElement % ChordalNet</a> -- ideal membership test</span></li> <li><span><a title="iterates over the chains of a chordal network" href="_next__Chain.html">nextChain</a> -- iterates over the chains of a chordal network</span></li> <li><span><a title="components of a chordal network" href="_components_lp__Chordal__Net_cm__Z__Z_rp.html">components(ChordalNet)</a> -- components of a chordal network</span></li> </ul> <h2>References</h2> <ul> <li>D. Cifuentes, P.A. Parrilo (2016), "Exploiting chordal structure in polynomial ideals: a Groebner basis approach", in "SIAM J. Discrete Math", 30(3):1534-–1570</li> <li>D. Cifuentes, P.A. Parrilo (2017), "Chordal networks of polynomial ideals", in "SIAM J. Appl. Algebra Geometry", 1(1):73-–170</li> </ul> </div> <div> <div> <div> <h2>Authors</h2> <ul> <li><a href="http://www.mit.edu/~diegcif">Diego Cifuentes</a><span> <<a href="mailto:diegcif%40mit.edu">diegcif@mit.edu</a>></span></li> <li><a href="http://www.mit.edu/~parrilo/">Pablo A. Parrilo</a><span> <<a href="mailto:parrilo%40mit.edu">parrilo@mit.edu</a>></span></li> </ul> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>0.2</b> of Chordal, released <b>May 15, 2021</b>.</p> </div> <div> <h2>Citation</h2> <p>If you have used this package in your research, please cite it as follows:</p> <table class="examples"> <tr> <td> <pre><code class="language-bib">@misc{ChordalSource, title = {{Chordal: routines that exploit chordal structure. Version~0.2}}, author = {Diego Cifuentes and Pablo A. Parrilo}, howpublished = {A \emph{Macaulay2} package available at \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Types <ul> <li><span><a title="a chordal graph" href="___Chordal__Graph.html">ChordalGraph</a> -- a chordal graph</span></li> <li><span><a title="a chordal network" href="___Chordal__Net.html">ChordalNet</a> -- a chordal network</span></li> <li><span><a title="a chain of a chordal network" href="___Chordal__Net__Chain.html">ChordalNetChain</a> -- a chain of a chordal network</span></li> <li><span><a title="a node of a chordal network" href="___Chordal__Net__Node.html">ChordalNetNode</a> -- a node of a chordal network</span></li> <li><span><a title="a rank of a chordal network" href="___Chordal__Net__Rank.html">ChordalNetRank</a> -- a rank of a chordal network</span></li> <li><span><a title="the elimination tree of a chordal graph" href="___Elim__Tree.html">ElimTree</a> -- the elimination tree of a chordal graph</span></li> </ul> </li> <li>Functions and commands <ul> <li><span><a title="ideal of adjacent minors" href="_adjacent__Minors__Ideal.html">adjacentMinorsIdeal</a> -- ideal of adjacent minors</span></li> <li><span><a title="performs elimination on the chordal network" href="_chordal__Elim.html">chordalElim</a> -- performs elimination on the chordal network</span></li> <li><span><a title="chordal completion of a graph" href="_chordal__Graph.html">chordalGraph</a> -- chordal completion of a graph</span></li> <li><span><kbd>structure</kbd> -- see <span><a title="a chordal network" href="___Chordal__Net.html">ChordalNet</a> -- a chordal network</span></span></li> <li><span><a title="constructs a chordal network from a polynomial set" href="_chordal__Net.html">chordalNet</a> -- constructs a chordal network from a polynomial set</span></li> <li><span><a title="makes a chordal network triangular" href="_chordal__Tria.html">chordalTria</a> -- makes a chordal network triangular</span></li> <li><span><a title="chromatic ideal of a graph" href="_chromatic__Ideal.html">chromaticIdeal</a> -- chromatic ideal of a graph</span></li> <li><span><a title="codimension counts of the chains of a chordal network" href="_codim__Count.html">codimCount</a> -- codimension counts of the chains of a chordal network</span></li> <li><span><a title="constraint graph of a polynomial set" href="_constraint__Graph.html">constraintGraph</a> -- constraint graph of a polynomial set</span></li> <li><span><a title="displays a chordal network using Graphivz" href="_display__Net.html">displayNet</a> -- displays a chordal network using Graphivz</span></li> <li><span><a title="elimination tree of a chordal graph" href="_elim__Tree.html">elimTree</a> -- elimination tree of a chordal graph</span></li> <li><span><a title="whether a chordal network is triangular" href="_is__Triangular.html">isTriangular</a> -- whether a chordal network is triangular</span></li> <li><span><span class="tt">label</span> (missing documentation)<!--tag: label--> </span></li> <li><span><a title="iterates over the chains of a chordal network" href="_next__Chain.html">nextChain</a> -- iterates over the chains of a chordal network</span></li> <li><span><a title="iterates over ordered partitions of a number" href="_next__Ordered__Partition.html">nextOrderedPartition</a> -- iterates over ordered partitions of a number</span></li> <li><span><a title="list of nodes of a chordal network" href="_nodes.html">nodes</a> -- list of nodes of a chordal network</span></li> <li><span><a title="removes arcs of a chordal network of small dimension" href="_reduce__Dimension.html">reduceDimension</a> -- removes arcs of a chordal network of small dimension</span></li> <li><span><a title="reduces a chordal network" href="_reduce__Net.html">reduceNet</a> -- reduces a chordal network</span></li> <li><span><a title="counts the number of roots of a chordal network" href="_root__Count.html">rootCount</a> -- counts the number of roots of a chordal network</span></li> <li><span><a title="default configuration of a package" href="_set__Default__Configuration.html">setDefaultConfiguration</a> -- default configuration of a package</span></li> <li><span><a title="ideal of subset products" href="_subsets__Products__Ideal.html">subsetsProductsIdeal</a> -- ideal of subset products</span></li> <li><span><a title="suggests a good variable ordering" href="_suggest__Variable__Order.html">suggestVariableOrder</a> -- suggests a good variable ordering</span></li> <li><span><a title="change monomial order to lexicographic" href="_to__Lex.html">toLex</a> -- change monomial order to lexicographic</span></li> <li><span><a title="treewidth of a graph" href="_treewidth.html">treewidth</a> -- treewidth of a graph</span></li> </ul> </li> <li>Methods <ul> <li><span><kbd>chordalElim(ChordalNet)</kbd> -- see <span><a title="performs elimination on the chordal network" href="_chordal__Elim.html">chordalElim</a> -- performs elimination on the chordal network</span></span></li> <li><span><kbd>chromaticNumber(ChordalGraph)</kbd> -- see <span><a title="a chordal graph" href="___Chordal__Graph.html">ChordalGraph</a> -- a chordal graph</span></span></li> <li><span><kbd>cliqueNumber(ChordalGraph)</kbd> -- see <span><a title="a chordal graph" href="___Chordal__Graph.html">ChordalGraph</a> -- a chordal graph</span></span></li> <li><span><kbd>inducedSubgraph(ChordalGraph)</kbd> -- see <span><a title="a chordal graph" href="___Chordal__Graph.html">ChordalGraph</a> -- a chordal graph</span></span></li> <li><span><kbd>isChordal(ChordalGraph)</kbd> -- see <span><a title="a chordal graph" href="___Chordal__Graph.html">ChordalGraph</a> -- a chordal graph</span></span></li> <li><span><kbd>isPerfect(ChordalGraph)</kbd> -- see <span><a title="a chordal graph" href="___Chordal__Graph.html">ChordalGraph</a> -- a chordal graph</span></span></li> <li><span><kbd>net(ChordalGraph)</kbd> -- see <span><a title="a chordal graph" href="___Chordal__Graph.html">ChordalGraph</a> -- a chordal graph</span></span></li> <li><span><kbd>chordalGraph(Digraph)</kbd> -- see <span><a title="chordal completion of a graph" href="_chordal__Graph.html">chordalGraph</a> -- chordal completion of a graph</span></span></li> <li><span><kbd>chordalGraph(Graph)</kbd> -- see <span><a title="chordal completion of a graph" href="_chordal__Graph.html">chordalGraph</a> -- chordal completion of a graph</span></span></li> <li><span><kbd>chordalGraph(Graph,List)</kbd> -- see <span><a title="chordal completion of a graph" href="_chordal__Graph.html">chordalGraph</a> -- chordal completion of a graph</span></span></li> <li><span><kbd>net(ChordalNet)</kbd> -- see <span><a title="a chordal network" href="___Chordal__Net.html">ChordalNet</a> -- a chordal network</span></span></li> <li><span><kbd>ring(ChordalNet)</kbd> -- see <span><a title="a chordal network" href="___Chordal__Net.html">ChordalNet</a> -- a chordal network</span></span></li> <li><span><kbd>chordalNet(Ideal)</kbd> -- see <span><a title="constructs a chordal network from a polynomial set" href="_chordal__Net.html">chordalNet</a> -- constructs a chordal network from a polynomial set</span></span></li> <li><span><kbd>chordalNet(Ideal,List)</kbd> -- see <span><a title="constructs a chordal network from a polynomial set" href="_chordal__Net.html">chordalNet</a> -- constructs a chordal network from a polynomial set</span></span></li> <li><span><kbd>chordalNet(Ideal,String)</kbd> -- see <span><a title="constructs a chordal network from a polynomial set" href="_chordal__Net.html">chordalNet</a> -- constructs a chordal network from a polynomial set</span></span></li> <li><span><a title="construct chordal network from a digraph" href="_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.html">chordalNet(HashTable,HashTable,ElimTree,Digraph)</a> -- construct chordal network from a digraph</span></li> <li><span><kbd>codim(ChordalNetChain)</kbd> -- see <span><a title="a chain of a chordal network" href="___Chordal__Net__Chain.html">ChordalNetChain</a> -- a chain of a chordal network</span></span></li> <li><span><kbd>dim(ChordalNetChain)</kbd> -- see <span><a title="a chain of a chordal network" href="___Chordal__Net__Chain.html">ChordalNetChain</a> -- a chain of a chordal network</span></span></li> <li><span><kbd>triaSystem(ChordalNet,ChordalNetChain)</kbd> -- see <span><a title="a chain of a chordal network" href="___Chordal__Net__Chain.html">ChordalNetChain</a> -- a chain of a chordal network</span></span></li> <li><span><kbd>children(ChordalNetNode)</kbd> -- see <span><a title="a node of a chordal network" href="___Chordal__Net__Node.html">ChordalNetNode</a> -- a node of a chordal network</span></span></li> <li><span><kbd>generators(ChordalNetNode)</kbd> -- see <span><a title="a node of a chordal network" href="___Chordal__Net__Node.html">ChordalNetNode</a> -- a node of a chordal network</span></span></li> <li><span><kbd>ineqs(ChordalNetNode)</kbd> -- see <span><a title="a node of a chordal network" href="___Chordal__Net__Node.html">ChordalNetNode</a> -- a node of a chordal network</span></span></li> <li><span><kbd>label(ChordalNetNode)</kbd> -- see <span><a title="a node of a chordal network" href="___Chordal__Net__Node.html">ChordalNetNode</a> -- a node of a chordal network</span></span></li> <li><span><kbd>net(ChordalNetNode)</kbd> -- see <span><a title="a node of a chordal network" href="___Chordal__Net__Node.html">ChordalNetNode</a> -- a node of a chordal network</span></span></li> <li><span><kbd>parents(ChordalNetNode)</kbd> -- see <span><a title="a node of a chordal network" href="___Chordal__Net__Node.html">ChordalNetNode</a> -- a node of a chordal network</span></span></li> <li><span><kbd>rank(ChordalNetNode)</kbd> -- see <span><a title="a node of a chordal network" href="___Chordal__Net__Node.html">ChordalNetNode</a> -- a node of a chordal network</span></span></li> <li><span><kbd>net(ChordalNetRank)</kbd> -- see <span><a title="a rank of a chordal network" href="___Chordal__Net__Rank.html">ChordalNetRank</a> -- a rank of a chordal network</span></span></li> <li><span><kbd>chordalTria(ChordalNet)</kbd> -- see <span><a title="makes a chordal network triangular" href="_chordal__Tria.html">chordalTria</a> -- makes a chordal network triangular</span></span></li> <li><span><kbd>codimCount(ChordalNet)</kbd> -- see <span><a title="codimension counts of the chains of a chordal network" href="_codim__Count.html">codimCount</a> -- codimension counts of the chains of a chordal network</span></span></li> <li><span><kbd>components(ChordalNet)</kbd> -- see <span><a title="components of a chordal network" href="_components_lp__Chordal__Net_cm__Z__Z_rp.html">components(ChordalNet,ZZ)</a> -- components of a chordal network</span></span></li> <li><span><a title="components of a chordal network" href="_components_lp__Chordal__Net_cm__Z__Z_rp.html">components(ChordalNet,ZZ)</a> -- components of a chordal network</span></li> <li><span><kbd>constraintGraph(Ideal)</kbd> -- see <span><a title="constraint graph of a polynomial set" href="_constraint__Graph.html">constraintGraph</a> -- constraint graph of a polynomial set</span></span></li> <li><span><kbd>constraintGraph(Ring,List)</kbd> -- see <span><a title="constraint graph of a polynomial set" href="_constraint__Graph.html">constraintGraph</a> -- constraint graph of a polynomial set</span></span></li> <li><span><a title="digraph associated to a chordal network" href="_digraph_lp__Chordal__Net_rp.html">digraph(ChordalNet)</a> -- digraph associated to a chordal network</span></li> <li><span><kbd>codim(ChordalNet)</kbd> -- see <span><a title="dimension of a chordal network" href="_dim_lp__Chordal__Net_rp.html">dim(ChordalNet)</a> -- dimension of a chordal network</span></span></li> <li><span><a title="dimension of a chordal network" href="_dim_lp__Chordal__Net_rp.html">dim(ChordalNet)</a> -- dimension of a chordal network</span></li> <li><span><kbd>displayGraph(ElimTree)</kbd> -- see <span><a title="displays an elimination tree using Graphviz" href="_display__Graph_lp__String_cm__String_cm__Elim__Tree_rp.html">displayGraph(String,String,ElimTree)</a> -- displays an elimination tree using Graphviz</span></span></li> <li><span><a title="displays an elimination tree using Graphviz" href="_display__Graph_lp__String_cm__String_cm__Elim__Tree_rp.html">displayGraph(String,String,ElimTree)</a> -- displays an elimination tree using Graphviz</span></li> <li><span><kbd>displayNet(ChordalNet)</kbd> -- see <span><a title="displays a chordal network using Graphivz" href="_display__Net.html">displayNet</a> -- displays a chordal network using Graphivz</span></span></li> <li><span><kbd>displayNet(Function,ChordalNet)</kbd> -- see <span><a title="displays a chordal network using Graphivz" href="_display__Net.html">displayNet</a> -- displays a chordal network using Graphivz</span></span></li> <li><span><kbd>displayNet(String,String,Function,ChordalNet)</kbd> -- see <span><a title="displays a chordal network using Graphivz" href="_display__Net.html">displayNet</a> -- displays a chordal network using Graphivz</span></span></li> <li><span><kbd>chordalGraph(ElimTree)</kbd> -- see <span><a title="the elimination tree of a chordal graph" href="___Elim__Tree.html">ElimTree</a> -- the elimination tree of a chordal graph</span></span></li> <li><span><kbd>elimTree(ChordalNet)</kbd> -- see <span><a title="the elimination tree of a chordal graph" href="___Elim__Tree.html">ElimTree</a> -- the elimination tree of a chordal graph</span></span></li> <li><span><kbd>net(ElimTree)</kbd> -- see <span><a title="the elimination tree of a chordal graph" href="___Elim__Tree.html">ElimTree</a> -- the elimination tree of a chordal graph</span></span></li> <li><span><kbd>elimTree(ChordalGraph)</kbd> -- see <span><a title="elimination tree of a chordal graph" href="_elim__Tree.html">elimTree</a> -- elimination tree of a chordal graph</span></span></li> <li><span><span class="tt">elimTree(List,HashTable,HashTable)</span> (missing documentation)<!--tag: (elimTree,List,HashTable,HashTable)--> </span></li> <li><span><a title="simple primality test of a chordal network" href="_is__Prime__Simple_lp__Chordal__Net_rp.html">isPrimeSimple(ChordalNet)</a> -- simple primality test of a chordal network</span></li> <li><span><kbd>isTriangular(ChordalNet)</kbd> -- see <span><a title="whether a chordal network is triangular" href="_is__Triangular.html">isTriangular</a> -- whether a chordal network is triangular</span></span></li> <li><span><a title="leaves of an elimination tree" href="_leaves_lp__Elim__Tree_rp.html">leaves(ElimTree)</a> -- leaves of an elimination tree</span></li> <li><span><kbd>nextChain(ChordalNet)</kbd> -- see <span><a title="iterates over the chains of a chordal network" href="_next__Chain.html">nextChain</a> -- iterates over the chains of a chordal network</span></span></li> <li><span><kbd>nextChain(ChordalNetChain,ChordalNet)</kbd> -- see <span><a title="iterates over the chains of a chordal network" href="_next__Chain.html">nextChain</a> -- iterates over the chains of a chordal network</span></span></li> <li><span><kbd>nextChain(ChordalNetChain,Sequence,ZZ,ChordalNet)</kbd> -- see <span><a title="iterates over the chains of a chordal network" href="_next__Chain.html">nextChain</a> -- iterates over the chains of a chordal network</span></span></li> <li><span><kbd>nextChain(ZZ,ChordalNet)</kbd> -- see <span><a title="iterates over the chains of a chordal network" href="_next__Chain.html">nextChain</a> -- iterates over the chains of a chordal network</span></span></li> <li><span><kbd>nextOrderedPartition(List,ZZ,List)</kbd> -- see <span><a title="iterates over ordered partitions of a number" href="_next__Ordered__Partition.html">nextOrderedPartition</a> -- iterates over ordered partitions of a number</span></span></li> <li><span><kbd>nextOrderedPartition(ZZ,List)</kbd> -- see <span><a title="iterates over ordered partitions of a number" href="_next__Ordered__Partition.html">nextOrderedPartition</a> -- iterates over ordered partitions of a number</span></span></li> <li><span><kbd>nodes(ChordalNet)</kbd> -- see <span><a title="list of nodes of a chordal network" href="_nodes.html">nodes</a> -- list of nodes of a chordal network</span></span></li> <li><span><kbd>nodes(ChordalNet,RingElement)</kbd> -- see <span><a title="list of nodes of a chordal network" href="_nodes.html">nodes</a> -- list of nodes of a chordal network</span></span></li> <li><span><kbd>reduceNet(ChordalNet)</kbd> -- see <span><a title="reduces a chordal network" href="_reduce__Net.html">reduceNet</a> -- reduces a chordal network</span></span></li> <li><span><kbd>pseudoRemainder(RingElement,ChordalNet)</kbd> -- see <span><a title="ideal membership test" href="___Ring__Element_sp_pc_sp__Chordal__Net.html">RingElement % ChordalNet</a> -- ideal membership test</span></span></li> <li><span><a title="ideal membership test" href="___Ring__Element_sp_pc_sp__Chordal__Net.html">RingElement % ChordalNet</a> -- ideal membership test</span></li> <li><span><a title="apply ring map to a chordal network" href="___Ring__Map_sp__Chordal__Net.html">RingMap ChordalNet</a> -- apply ring map to a chordal network</span></li> <li><span><kbd>RingMap ElimTree</kbd> -- see <span><a title="apply ring map to a chordal network" href="___Ring__Map_sp__Chordal__Net.html">RingMap ChordalNet</a> -- apply ring map to a chordal network</span></span></li> <li><span><kbd>rootCount(ChordalNet)</kbd> -- see <span><a title="counts the number of roots of a chordal network" href="_root__Count.html">rootCount</a> -- counts the number of roots of a chordal network</span></span></li> <li><span><kbd>setDefaultConfiguration(Package,String,String)</kbd> -- see <span><a title="default configuration of a package" href="_set__Default__Configuration.html">setDefaultConfiguration</a> -- default configuration of a package</span></span></li> <li><span><kbd>setDefaultConfiguration(String,String,String)</kbd> -- see <span><a title="default configuration of a package" href="_set__Default__Configuration.html">setDefaultConfiguration</a> -- default configuration of a package</span></span></li> <li><span><a title="size of a chordal network" href="_size_lp__Chordal__Net_rp.html">size(ChordalNet)</a> -- size of a chordal network</span></li> <li><span><kbd>suggestVariableOrder(Ideal)</kbd> -- see <span><a title="suggests a good variable ordering" href="_suggest__Variable__Order.html">suggestVariableOrder</a> -- suggests a good variable ordering</span></span></li> <li><span><kbd>treewidth(ChordalGraph)</kbd> -- see <span><a title="treewidth of a graph" href="_treewidth.html">treewidth</a> -- treewidth of a graph</span></span></li> <li><span><kbd>treewidth(ElimTree)</kbd> -- see <span><a title="treewidth of a graph" href="_treewidth.html">treewidth</a> -- treewidth of a graph</span></span></li> <li><span><kbd>writeDotFile(String,ElimTree)</kbd> -- see <span><a title="writes a chordal network to a dot file" href="_write__Dot__File_lp__String_cm__Function_cm__Chordal__Net_rp.html">writeDotFile(String,Function,ChordalNet)</a> -- writes a chordal network to a dot file</span></span></li> <li><span><a title="writes a chordal network to a dot file" href="_write__Dot__File_lp__String_cm__Function_cm__Chordal__Net_rp.html">writeDotFile(String,Function,ChordalNet)</a> -- writes a chordal network to a dot file</span></li> </ul> </li> <li>Symbols <ul> <li><span><kbd>cliques</kbd> -- see <span><a title="the elimination tree of a chordal graph" href="___Elim__Tree.html">ElimTree</a> -- the elimination tree of a chordal graph</span></span></li> <li><span><a title="get dynamic programming table" href="___Get__Table.html">GetTable</a> -- get dynamic programming table</span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="exploiting chordal structure in polynomial ideals" href="index.html">Chordal</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">Chordal.m2</span>, with auxiliary files in <span class="tt">Chordal/</span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Chordal/ChordalDoc.m2:121:0</span>.</p> </div> </div> </div> </body> </html>