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<!DOCTYPE html> <html lang="en"> <head> <title>NAGtypes -- Common types used in Numerical Algebraic Geometry</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="Common types used in Numerical Algebraic Geometry" href="index.html">NAGtypes</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="___Abstract__Point.html">next</a> | previous | <a href="___Abstract__Point.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>NAGtypes -- Common types used in Numerical Algebraic Geometry</h1> <div> <h2>Description</h2> <p>The package defines types used by the package <a title="Numerical Algebraic Geometry" href="../../NumericalAlgebraicGeometry/html/index.html">NumericalAlgebraicGeometry</a> as well as other numerical algebraic geometry packages: e.g., interface packages <a title="a package for Polynomial Homotopy Continuation" href="../../PHCpack/html/index.html">PHCpack</a> and <span class="tt">Bertini::Bertini</span>.</p> <p>Datatypes: </p> <ul> <li><a title="a type used to store a point in complex space" href="___Abstract__Point.html">Point</a> -- a numerical approximation of a point in a complex space</li> <li><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system (usually with complex coefficients)</li> <li><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set representing (possibly positive-dimensional) solution components</li> <li><a title="a numerical variety" href="___Numerical__Variety.html">NumericalVariety</a> -- a numerical description of a variety</li> <li><span><a title="a polynomial vector subspace" href="___Poly__Space.html">PolySpace</a> -- a polynomial vector subspace</span></li> <li><span><a title="a dual functional vector subspace" href="___Dual__Space.html">DualSpace</a> -- a dual functional vector subspace</span></li> </ul> <p>See the corresponding documentation nodes for description of provided service functions.</p> <p>We display the objects of all new types showing only partial data. Moreover, if an object is assigned to a global variable, only the name of the variable is shown. Use <a title="examine contents of an object" href="../../Macaulay2Doc/html/_peek.html">peek</a> for more information.</p> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = CC[x,y] o1 = R o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : I = ideal((x^2+y^2+2)*x,(x^2+y^2+2)*y); o2 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : w1 = witnessSet(I , ideal(x-y), {point {{0.999999*ii,0.999999*ii}}, point {{-1.000001*ii,-1.000001*ii}}} ) o3 = w1 o3 : WitnessSet</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : O = point {{0.,0.}} o4 = O o4 : Point</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : numericalVariety {witnessSet(I, ideal R, {O}),w1} o5 = a "numerical variety" with components in dim 0: [dim=0,deg=1]-*may be reducible*- dim 1: w1 o5 : NumericalVariety</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : V = oo o6 = V o6 : NumericalVariety</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : peek V o7 = NumericalVariety{0 => {[dim=0,deg=1]-*may be reducible*-}} 1 => {w1}</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : peek w1 o8 = WitnessSet{cache => CacheTable{...1...} } Equations => {-3} | x3+xy2+2x | {-3} | x2y+y3+2y | Points => {{.999999*ii, .999999*ii}} {{-ii, -ii} } Slice => | 1 -1 0 |</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : peek O o9 = Point{cache => CacheTable{}} Coordinates => {0, 0}</code></pre> </td> </tr> </table> </div> <div> <div> <div> <h2>Author</h2> <ul> <li>Anton Leykin<span> <<a href="mailto:leykin%40math.gatech.edu">leykin@math.gatech.edu</a>></span></li> </ul> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>1.21</b> of NAGtypes, released <b>Nov 2022</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{NAGtypesSource, title = {{NAGtypes: types used in Numerical Algebraic Geometry. Version~1.21}}, author = {Anton Leykin}, howpublished = {A \emph{Macaulay2} package available at "http://people.math.gatech.edu/~aleykin3/NAG4M2"} } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Types <ul> <li><span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></li> <li><span><kbd>Point</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><a title="a dual functional vector subspace" href="___Dual__Space.html">DualSpace</a> -- a dual functional vector subspace</span></li> <li><span><a title="a numerical variety" href="___Numerical__Variety.html">NumericalVariety</a> -- a numerical variety</span></li> <li><span><a title="a polynomial vector subspace" href="___Poly__Space.html">PolySpace</a> -- a polynomial vector subspace</span></li> <li><span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></li> <li><span><a title="a projective witness set" href="___Projective__Witness__Set.html">ProjectiveWitnessSet</a> -- a projective witness set</span></li> <li><span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></li> <li><span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></li> <li><span><kbd>Ambient</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> <li><span><kbd>SlicingVariety</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> <li><span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></li> </ul> </li> <li>Functions and commands <ul> <li><span><kbd>coordinates</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><a title="Union of polynomial spaces" href="_addition.html">addition</a> -- Union of polynomial spaces</span></li> <li><span><a title="determine if solutions are equal" href="_are__Equal.html">areEqual</a> -- determine if solutions are equal</span></li> <li><span><a title="declares a component represented by the witness set irreducible" href="_declare__Irreducible.html">declareIrreducible</a> -- declares a component represented by the witness set irreducible</span></li> <li><span><a title="construct a DualSpace" href="_dual__Space.html">dualSpace</a> -- construct a DualSpace</span></li> <li><span><a title="evaluate a polynomial system or matrix at a point" href="_evaluate.html">evaluate</a> -- evaluate a polynomial system or matrix at a point</span></li> <li><span><a title="random linear combinations of equations/generators" href="_general__Equations.html">generalEquations</a> -- random linear combinations of equations/generators</span></li> <li><span><a title="Is one space contained in the other" href="_is__Contained.html">isContained</a> -- Is one space contained in the other</span></li> <li><span><a title="compare two points" href="_is__G__E__Q.html">isGEQ</a> -- compare two points</span></li> <li><span><a title="determine whether a point is real" href="_is__Real__Point.html">isRealPoint</a> -- determine whether a point is real</span></li> <li><span><a title="affine space as a numerical variety" href="_numerical__Affine__Space.html">numericalAffineSpace</a> -- affine space as a numerical variety</span></li> <li><span><kbd>numericalVariety</kbd> -- see <span><a title="construct a numerical variety" href="_numerical__Variety_lp__List_rp.html">numericalVariety(List)</a> -- construct a numerical variety</span></span></li> <li><span><kbd>projectiveNumericalVariety</kbd> -- see <span><a title="construct a numerical variety" href="_numerical__Variety_lp__List_rp.html">numericalVariety(List)</a> -- construct a numerical variety</span></span></li> <li><span><a title="construct a Point" href="_point.html">point</a> -- construct a Point</span></li> <li><span><a title="construct a PolySpace" href="_poly__Space.html">polySpace</a> -- construct a PolySpace</span></li> <li><span><kbd>parameters</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><a title="construct a polynomial system" href="_poly__System.html">polySystem</a> -- construct a polynomial system</span></li> <li><span><kbd>project</kbd> -- see <span><a title="project a point" href="_project_lp__Abstract__Point_cm__Z__Z_rp.html">project(AbstractPoint,ZZ)</a> -- project a point</span></span></li> <li><span><a title="construct a ProjectiveWitnessSet" href="_projective__Witness__Set.html">projectiveWitnessSet</a> -- construct a ProjectiveWitnessSet</span></li> <li><span><a title="select real points" href="_real__Points.html">realPoints</a> -- select real points</span></li> <li><span><a title="reduce the generators of a space" href="_reduce__Space.html">reduceSpace</a> -- reduce the generators of a space</span></li> <li><span><a title="residual of a polynomial function at a point" href="_residual.html">residual</a> -- residual of a polynomial function at a point</span></li> <li><span><kbd>projectiveSliceEquations</kbd> -- see <span><a title="slicing linear functions" href="_slice__Equations_lp__Matrix_cm__Ring_rp.html">sliceEquations(Matrix,Ring)</a> -- slicing linear functions</span></span></li> <li><span><kbd>sliceEquations</kbd> -- see <span><a title="slicing linear functions" href="_slice__Equations_lp__Matrix_cm__Ring_rp.html">sliceEquations(Matrix,Ring)</a> -- slicing linear functions</span></span></li> <li><span><a title="replaces clusters of approximately equal points by single points with multiplicity" href="_solutions__With__Multiplicity.html">solutionsWithMultiplicity</a> -- replaces clusters of approximately equal points by single points with multiplicity</span></li> <li><span><kbd>sortSolutions</kbd> -- see <span><a title="sort the list of solutions" href="_sort__Solutions_lp__List_rp.html">sortSolutions(List)</a> -- sort the list of solutions</span></span></li> <li><span><kbd>evaluateJacobian</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><kbd>numFunctions</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><kbd>numParameters</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><kbd>numVariables</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><kbd>toAffineChart</kbd> -- see <span><a title="coordinates of a point in the projective space in an affine chart" href="_to__Affine__Chart_lp__Z__Z_cm__List_rp.html">toAffineChart(ZZ,List)</a> -- coordinates of a point in the projective space in an affine chart</span></span></li> <li><span><kbd>equations</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>points</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>slice</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><a title="construct a WitnessSet" href="_witness__Set.html">witnessSet</a> -- construct a WitnessSet</span></li> </ul> </li> <li>Methods <ul> <li><span><kbd>coordinates(AbstractPoint)</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>coordinates(Point)</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>matrix(AbstractPoint)</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>net(AbstractPoint)</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>status(AbstractPoint)</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>addition(PolySpace,PolySpace)</kbd> -- see <span><a title="Union of polynomial spaces" href="_addition.html">addition</a> -- Union of polynomial spaces</span></span></li> <li><span><kbd>AbstractPoint == AbstractPoint</kbd> -- see <span><a title="determine if solutions are equal" href="_are__Equal.html">areEqual</a> -- determine if solutions are equal</span></span></li> <li><span><kbd>areEqual(AbstractPoint,AbstractPoint)</kbd> -- see <span><a title="determine if solutions are equal" href="_are__Equal.html">areEqual</a> -- determine if solutions are equal</span></span></li> <li><span><kbd>areEqual(BasicList,BasicList)</kbd> -- see <span><a title="determine if solutions are equal" href="_are__Equal.html">areEqual</a> -- determine if solutions are equal</span></span></li> <li><span><kbd>areEqual(CC,CC)</kbd> -- see <span><a title="determine if solutions are equal" href="_are__Equal.html">areEqual</a> -- determine if solutions are equal</span></span></li> <li><span><kbd>areEqual(List,List)</kbd> -- see <span><a title="determine if solutions are equal" href="_are__Equal.html">areEqual</a> -- determine if solutions are equal</span></span></li> <li><span><kbd>areEqual(Matrix,Matrix)</kbd> -- see <span><a title="determine if solutions are equal" href="_are__Equal.html">areEqual</a> -- determine if solutions are equal</span></span></li> <li><span><kbd>areEqual(MutableMatrix,MutableMatrix)</kbd> -- see <span><a title="determine if solutions are equal" href="_are__Equal.html">areEqual</a> -- determine if solutions are equal</span></span></li> <li><span><kbd>areEqual(Number,Number)</kbd> -- see <span><a title="determine if solutions are equal" href="_are__Equal.html">areEqual</a> -- determine if solutions are equal</span></span></li> <li><span><a title="approximate equality of dual spaces" href="_are__Equal_lp__Dual__Space_cm__Dual__Space_rp.html">areEqual(DualSpace,DualSpace)</a> -- approximate equality of dual spaces</span></li> <li><span><a title="approximate equality of subspaces spanned by polynomials" href="_are__Equal_lp__Poly__Space_cm__Poly__Space_rp.html">areEqual(PolySpace,PolySpace)</a> -- approximate equality of subspaces spanned by polynomials</span></li> <li><span><a title="list components of a numerical variety" href="_components_lp__Numerical__Variety_rp.html">components(NumericalVariety)</a> -- list components of a numerical variety</span></li> <li><span><kbd>components(NumericalVariety,ZZ)</kbd> -- see <span><a title="list components of a numerical variety" href="_components_lp__Numerical__Variety_rp.html">components(NumericalVariety)</a> -- list components of a numerical variety</span></span></li> <li><span><kbd>components(NumericalVariety,ZZ,InfiniteNumber)</kbd> -- see <span><a title="list components of a numerical variety" href="_components_lp__Numerical__Variety_rp.html">components(NumericalVariety)</a> -- list components of a numerical variety</span></span></li> <li><span><kbd>components(NumericalVariety,ZZ,ZZ)</kbd> -- see <span><a title="list components of a numerical variety" href="_components_lp__Numerical__Variety_rp.html">components(NumericalVariety)</a> -- list components of a numerical variety</span></span></li> <li><span><kbd>declareIrreducible(WitnessSet)</kbd> -- see <span><a title="declares a component represented by the witness set irreducible" href="_declare__Irreducible.html">declareIrreducible</a> -- declares a component represented by the witness set irreducible</span></span></li> <li><span><kbd>dim(DualSpace)</kbd> -- see <span><a title="a dual functional vector subspace" href="___Dual__Space.html">DualSpace</a> -- a dual functional vector subspace</span></span></li> <li><span><kbd>generators(DualSpace)</kbd> -- see <span><a title="a dual functional vector subspace" href="___Dual__Space.html">DualSpace</a> -- a dual functional vector subspace</span></span></li> <li><span><kbd>net(DualSpace)</kbd> -- see <span><a title="a dual functional vector subspace" href="___Dual__Space.html">DualSpace</a> -- a dual functional vector subspace</span></span></li> <li><span><kbd>point(DualSpace)</kbd> -- see <span><a title="a dual functional vector subspace" href="___Dual__Space.html">DualSpace</a> -- a dual functional vector subspace</span></span></li> <li><span><kbd>ring(DualSpace)</kbd> -- see <span><a title="a dual functional vector subspace" href="___Dual__Space.html">DualSpace</a> -- a dual functional vector subspace</span></span></li> <li><span><kbd>dualSpace(DualSpace)</kbd> -- see <span><a title="construct a DualSpace" href="_dual__Space.html">dualSpace</a> -- construct a DualSpace</span></span></li> <li><span><kbd>dualSpace(Matrix,AbstractPoint)</kbd> -- see <span><a title="construct a DualSpace" href="_dual__Space.html">dualSpace</a> -- construct a DualSpace</span></span></li> <li><span><kbd>dualSpace(PolySpace,AbstractPoint)</kbd> -- see <span><a title="construct a DualSpace" href="_dual__Space.html">dualSpace</a> -- construct a DualSpace</span></span></li> <li><span><kbd>evaluate(Matrix,AbstractPoint)</kbd> -- see <span><a title="evaluate a polynomial system or matrix at a point" href="_evaluate.html">evaluate</a> -- evaluate a polynomial system or matrix at a point</span></span></li> <li><span><kbd>evaluate(Matrix,Matrix)</kbd> -- see <span><a title="evaluate a polynomial system or matrix at a point" href="_evaluate.html">evaluate</a> -- evaluate a polynomial system or matrix at a point</span></span></li> <li><span><kbd>evaluate(PolySystem,Matrix)</kbd> -- see <span><a title="evaluate a polynomial system or matrix at a point" href="_evaluate.html">evaluate</a> -- evaluate a polynomial system or matrix at a point</span></span></li> <li><span><kbd>evaluateJacobian(PolySystem,AbstractPoint)</kbd> -- see <span><a title="evaluate a polynomial system or matrix at a point" href="_evaluate.html">evaluate</a> -- evaluate a polynomial system or matrix at a point</span></span></li> <li><span><kbd>generalEquations(WitnessSet)</kbd> -- see <span><a title="random linear combinations of equations/generators" href="_general__Equations.html">generalEquations</a> -- random linear combinations of equations/generators</span></span></li> <li><span><kbd>generalEquations(ZZ,Ideal)</kbd> -- see <span><a title="random linear combinations of equations/generators" href="_general__Equations.html">generalEquations</a> -- random linear combinations of equations/generators</span></span></li> <li><span><kbd>generalEquations(ZZ,List)</kbd> -- see <span><a title="random linear combinations of equations/generators" href="_general__Equations.html">generalEquations</a> -- random linear combinations of equations/generators</span></span></li> <li><span><a title="homogenize a polynomial system" href="_homogenize_lp__Poly__System_cm__Ring_cm__Ring__Element_rp.html">homogenize(PolySystem,Ring,RingElement)</a> -- homogenize a polynomial system</span></li> <li><span><a title="Intersection of polynomial spaces" href="_intersect_lp__Poly__Space_cm__Poly__Space_rp.html">intersect(PolySpace,PolySpace)</a> -- Intersection of polynomial spaces</span></li> <li><span><kbd>isContained(DualSpace,DualSpace)</kbd> -- see <span><a title="Is one space contained in the other" href="_is__Contained.html">isContained</a> -- Is one space contained in the other</span></span></li> <li><span><kbd>isContained(PolySpace,PolySpace)</kbd> -- see <span><a title="Is one space contained in the other" href="_is__Contained.html">isContained</a> -- Is one space contained in the other</span></span></li> <li><span><kbd>isGEQ(AbstractPoint,AbstractPoint)</kbd> -- see <span><a title="compare two points" href="_is__G__E__Q.html">isGEQ</a> -- compare two points</span></span></li> <li><span><kbd>isGEQ(List,List)</kbd> -- see <span><a title="compare two points" href="_is__G__E__Q.html">isGEQ</a> -- compare two points</span></span></li> <li><span><kbd>isRealPoint(AbstractPoint)</kbd> -- see <span><a title="determine whether a point is real" href="_is__Real__Point.html">isRealPoint</a> -- determine whether a point is real</span></span></li> <li><span><span class="tt">net(Point)</span> (missing documentation)<!--tag: (net,Point)--> </span></li> <li><span><a title="p-norm of the point" href="_norm_lp__Thing_cm__Abstract__Point_rp.html">norm(Thing,AbstractPoint)</a> -- p-norm of the point</span></li> <li><span><kbd>numericalAffineSpace(PolynomialRing)</kbd> -- see <span><a title="affine space as a numerical variety" href="_numerical__Affine__Space.html">numericalAffineSpace</a> -- affine space as a numerical variety</span></span></li> <li><span><kbd>check(NumericalVariety)</kbd> -- see <span><a title="a numerical variety" href="___Numerical__Variety.html">NumericalVariety</a> -- a numerical variety</span></span></li> <li><span><kbd>degree(NumericalVariety)</kbd> -- see <span><a title="a numerical variety" href="___Numerical__Variety.html">NumericalVariety</a> -- a numerical variety</span></span></li> <li><span><kbd>dim(NumericalVariety)</kbd> -- see <span><a title="a numerical variety" href="___Numerical__Variety.html">NumericalVariety</a> -- a numerical variety</span></span></li> <li><span><kbd>net(NumericalVariety)</kbd> -- see <span><a title="a numerical variety" href="___Numerical__Variety.html">NumericalVariety</a> -- a numerical variety</span></span></li> <li><span><a title="construct a numerical variety" href="_numerical__Variety_lp__List_rp.html">numericalVariety(List)</a> -- construct a numerical variety</span></li> <li><span><kbd>projectiveNumericalVariety(List)</kbd> -- see <span><a title="construct a numerical variety" href="_numerical__Variety_lp__List_rp.html">numericalVariety(List)</a> -- construct a numerical variety</span></span></li> <li><span><kbd>point(AbstractPoint)</kbd> -- see <span><a title="construct a Point" href="_point.html">point</a> -- construct a Point</span></span></li> <li><span><kbd>point(List)</kbd> -- see <span><a title="construct a Point" href="_point.html">point</a> -- construct a Point</span></span></li> <li><span><kbd>point(List,CacheTable)</kbd> -- see <span><a title="construct a Point" href="_point.html">point</a> -- construct a Point</span></span></li> <li><span><kbd>point(Matrix)</kbd> -- see <span><a title="construct a Point" href="_point.html">point</a> -- construct a Point</span></span></li> <li><span><kbd>dim(PolySpace)</kbd> -- see <span><a title="a polynomial vector subspace" href="___Poly__Space.html">PolySpace</a> -- a polynomial vector subspace</span></span></li> <li><span><kbd>generators(PolySpace)</kbd> -- see <span><a title="a polynomial vector subspace" href="___Poly__Space.html">PolySpace</a> -- a polynomial vector subspace</span></span></li> <li><span><kbd>net(PolySpace)</kbd> -- see <span><a title="a polynomial vector subspace" href="___Poly__Space.html">PolySpace</a> -- a polynomial vector subspace</span></span></li> <li><span><kbd>ring(PolySpace)</kbd> -- see <span><a title="a polynomial vector subspace" href="___Poly__Space.html">PolySpace</a> -- a polynomial vector subspace</span></span></li> <li><span><kbd>polySpace(Matrix)</kbd> -- see <span><a title="construct a PolySpace" href="_poly__Space.html">polySpace</a> -- construct a PolySpace</span></span></li> <li><span><kbd>polySpace(PolySpace)</kbd> -- see <span><a title="construct a PolySpace" href="_poly__Space.html">polySpace</a> -- construct a PolySpace</span></span></li> <li><span><kbd>equations(PolySystem)</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><kbd>ideal(PolySystem)</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><kbd>isHomogeneous(PolySystem)</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><kbd>jacobian(PolySystem)</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><kbd>net(PolySystem)</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><kbd>numFunctions(PolySystem)</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><kbd>numParameters(PolySystem)</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><kbd>numVariables(PolySystem)</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><kbd>parameters(PolySystem)</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><kbd>ring(PolySystem)</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><kbd>polySystem(Ideal)</kbd> -- see <span><a title="construct a polynomial system" href="_poly__System.html">polySystem</a> -- construct a polynomial system</span></span></li> <li><span><kbd>polySystem(List)</kbd> -- see <span><a title="construct a polynomial system" href="_poly__System.html">polySystem</a> -- construct a polynomial system</span></span></li> <li><span><kbd>polySystem(Matrix)</kbd> -- see <span><a title="construct a polynomial system" href="_poly__System.html">polySystem</a> -- construct a polynomial system</span></span></li> <li><span><kbd>polySystem(PolySystem)</kbd> -- see <span><a title="construct a polynomial system" href="_poly__System.html">polySystem</a> -- construct a polynomial system</span></span></li> <li><span><a title="project a point" href="_project_lp__Abstract__Point_cm__Z__Z_rp.html">project(AbstractPoint,ZZ)</a> -- project a point</span></li> <li><span><kbd>project(Point,ZZ)</kbd> -- see <span><a title="project a point" href="_project_lp__Abstract__Point_cm__Z__Z_rp.html">project(AbstractPoint,ZZ)</a> -- project a point</span></span></li> <li><span><kbd>projectiveWitnessSet(Ideal,Matrix,Matrix,List)</kbd> -- see <span><a title="construct a ProjectiveWitnessSet" href="_projective__Witness__Set.html">projectiveWitnessSet</a> -- construct a ProjectiveWitnessSet</span></span></li> <li><span><kbd>random(DualSpace)</kbd> -- see <span><a title="random element of a subspace" href="_random_lp__Poly__Space_rp.html">random(PolySpace)</a> -- random element of a subspace</span></span></li> <li><span><a title="random element of a subspace" href="_random_lp__Poly__Space_rp.html">random(PolySpace)</a> -- random element of a subspace</span></li> <li><span><kbd>random(ZZ,DualSpace)</kbd> -- see <span><a title="random element of a subspace" href="_random_lp__Poly__Space_rp.html">random(PolySpace)</a> -- random element of a subspace</span></span></li> <li><span><kbd>random(ZZ,PolySpace)</kbd> -- see <span><a title="random element of a subspace" href="_random_lp__Poly__Space_rp.html">random(PolySpace)</a> -- random element of a subspace</span></span></li> <li><span><kbd>realPoints(List)</kbd> -- see <span><a title="select real points" href="_real__Points.html">realPoints</a> -- select real points</span></span></li> <li><span><kbd>reduceSpace(DualSpace)</kbd> -- see <span><a title="reduce the generators of a space" href="_reduce__Space.html">reduceSpace</a> -- reduce the generators of a space</span></span></li> <li><span><kbd>reduceSpace(PolySpace)</kbd> -- see <span><a title="reduce the generators of a space" href="_reduce__Space.html">reduceSpace</a> -- reduce the generators of a space</span></span></li> <li><span><kbd>residual(List,Point)</kbd> -- see <span><a title="residual of a polynomial function at a point" href="_residual.html">residual</a> -- residual of a polynomial function at a point</span></span></li> <li><span><kbd>residual(Matrix,Matrix)</kbd> -- see <span><a title="residual of a polynomial function at a point" href="_residual.html">residual</a> -- residual of a polynomial function at a point</span></span></li> <li><span><kbd>residual(System,AbstractPoint)</kbd> -- see <span><a title="residual of a polynomial function at a point" href="_residual.html">residual</a> -- residual of a polynomial function at a point</span></span></li> <li><span><kbd>residual(System,Point)</kbd> -- see <span><a title="residual of a polynomial function at a point" href="_residual.html">residual</a> -- residual of a polynomial function at a point</span></span></li> <li><span><kbd>projectiveSliceEquations(Matrix,Ring)</kbd> -- see <span><a title="slicing linear functions" href="_slice__Equations_lp__Matrix_cm__Ring_rp.html">sliceEquations(Matrix,Ring)</a> -- slicing linear functions</span></span></li> <li><span><a title="slicing linear functions" href="_slice__Equations_lp__Matrix_cm__Ring_rp.html">sliceEquations(Matrix,Ring)</a> -- slicing linear functions</span></li> <li><span><kbd>solutionsWithMultiplicity(List)</kbd> -- see <span><a title="replaces clusters of approximately equal points by single points with multiplicity" href="_solutions__With__Multiplicity.html">solutionsWithMultiplicity</a> -- replaces clusters of approximately equal points by single points with multiplicity</span></span></li> <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> <li><span><span class="tt">status(Point)</span> (missing documentation)<!--tag: (status,Point)--> </span></li> <li><span><a title="substitute a ring in a polynomial system" href="_substitute_lp__Poly__System_cm__Ring_rp.html">substitute(PolySystem,Ring)</a> -- substitute a ring in a polynomial system</span></li> <li><span><kbd>evaluate(System,AbstractPoint)</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><kbd>evaluate(System,AbstractPoint,AbstractPoint)</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><kbd>evaluate(System,Matrix)</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><kbd>evaluate(System,Matrix,Matrix)</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><kbd>evaluateJacobian(System,AbstractPoint)</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><kbd>evaluateJacobian(System,AbstractPoint,AbstractPoint)</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><kbd>evaluateJacobian(System,Matrix)</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><kbd>evaluateJacobian(System,Matrix,Matrix)</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><kbd>numFunctions(System)</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><kbd>numParameters(System)</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><kbd>numVariables(System)</kbd> -- see <span><a title="a system of functions" href="___System.html">System</a> -- a system of functions</span></span></li> <li><span><a title="coordinates of a point in the projective space in an affine chart" href="_to__Affine__Chart_lp__Z__Z_cm__List_rp.html">toAffineChart(ZZ,List)</a> -- coordinates of a point in the projective space in an affine chart</span></li> <li><span><kbd>codim(WitnessSet)</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>degree(WitnessSet)</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>dim(WitnessSet)</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>equations(WitnessSet)</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>ideal(WitnessSet)</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>net(WitnessSet)</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>points(WitnessSet)</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>ring(WitnessSet)</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>slice(WitnessSet)</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>witnessSet(Ideal,Ideal,List)</kbd> -- see <span><a title="construct a WitnessSet" href="_witness__Set.html">witnessSet</a> -- construct a WitnessSet</span></span></li> <li><span><kbd>witnessSet(Ideal,Matrix,List)</kbd> -- see <span><a title="construct a WitnessSet" href="_witness__Set.html">witnessSet</a> -- construct a WitnessSet</span></span></li> <li><span><kbd>witnessSet(PolySystem,Matrix,List)</kbd> -- see <span><a title="construct a WitnessSet" href="_witness__Set.html">witnessSet</a> -- construct a WitnessSet</span></span></li> <li><span><kbd>witnessSet(PolySystem,PolySystem,List)</kbd> -- see <span><a title="construct a WitnessSet" href="_witness__Set.html">witnessSet</a> -- construct a WitnessSet</span></span></li> <li><span><kbd>ambient(SlicingVariety)</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> <li><span><kbd>ambient(WSet)</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> <li><span><kbd>codim(SlicingVariety)</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> <li><span><kbd>codim(WSet)</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> <li><span><kbd>degree(WSet)</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> <li><span><kbd>dim(Ambient)</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> <li><span><kbd>dim(SlicingVariety)</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> <li><span><kbd>dim(WSet)</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> <li><span><kbd>map(SlicingVariety)</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> <li><span><kbd>net(Ambient)</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> <li><span><kbd>net(SlicingVariety)</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> <li><span><kbd>net(WSet)</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> <li><span><kbd>points(WSet)</kbd> -- see <span><a title="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</a> -- (under construction!) new types and methods needed to generalize WitnessSet</span></span></li> </ul> </li> <li>Symbols <ul> <li><span><kbd>ConditionNumber</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>Coordinates</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>DecreasePrecision</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>DeflationNumber</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>ErrorBoundEstimate</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>IncreasePrecision</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>Infinity</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>LastT</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>MaxPrecision</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>MinStepFailure</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>Multiplicity</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>NumberOfSteps</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>NumericalRankFailure</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>Origin</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>RefinementFailure</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>Regular</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>Singular</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>SolutionStatus</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><kbd>WindingNumber</kbd> -- see <span><a title="a type used to store a point in complex space" href="___Abstract__Point.html">AbstractPoint</a> -- a type used to store a point in complex space</span></span></li> <li><span><a title="p in the p-norm" href="___Norm.html">Norm</a> -- p in the p-norm</span></li> <li><span><kbd>Reduced</kbd> -- see <span><a title="construct a PolySpace" href="_poly__Space.html">polySpace</a> -- construct a PolySpace</span></span></li> <li><span><kbd>ContinuationParameter</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><kbd>NumberOfPolys</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><kbd>NumberOfVariables</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><kbd>PolyMap</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><kbd>SpecializationRing</kbd> -- see <span><a title="a polynomial system" href="___Poly__System.html">PolySystem</a> -- a polynomial system</span></span></li> <li><span><a title="a projective numerical variety" href="___Projective__Numerical__Variety.html">ProjectiveNumericalVariety</a> -- a projective numerical variety</span></li> <li><span><kbd>AffineChart</kbd> -- see <span><a title="a projective witness set" href="___Projective__Witness__Set.html">ProjectiveWitnessSet</a> -- a projective witness set</span></span></li> <li><span><kbd>Equations</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>IsIrreducible</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>Points</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>ProjectionDimension</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> <li><span><kbd>Slice</kbd> -- see <span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Common types used in Numerical Algebraic Geometry" href="index.html">NAGtypes</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">NAGtypes.m2</span>, with auxiliary files in <span class="tt">NAGtypes/</span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">NAGtypes/doc-NAGtypes.m2:36:0</span>.</p> </div> </div> </div> </body> </html>