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products/Sources/formale Sprachen/GAP/pkg/nconvex/doc/ (Algebra von RWTH Aachen Version 4.15.1^©) Datei vom 28.11.2024 mit Größe 8 kB

Quellcode-Bibliothek _Chapter_Cones.xml Sprache: XML

<?xml version="1.0" encoding="UTF-8"?>


<Chapter Label="Chapter_Cones">
<Heading>Cones</Heading>

<Section Label="Chapter_Cones_Section_Creating_cones">
<Heading>Creating cones</Heading>

<ManSection>
<Oper Arg="L" Name="ConeByInequalities" Label="for IsList"/>
<Returns>a <C>Cone</C> Object
</Returns>
<Description>
The function takes a list of lists <Math>[L_1, L_2, ...]</Math> where each <Math>L_j</Math> represents
an inequality and returns the cone defined by them.
For example the <Math>j</Math>'th entry

L_j = [a_{j1},a_{j2},...,a_{jn}]

corresponds to the inequality
<Math>\sum_{i=1}^n a_{ji}x_i \geq 0</Math>.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="Eq, Ineq" Name="ConeByEqualitiesAndInequalities" Label="for IsList, IsList"/>
<Returns>a <C>Cone</C> Object
</Returns>
<Description>
The function takes two lists. The first list is the equalities and the second is
the inequalities and returns the cone defined by them.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="L" Name="Cone" Label="for IsList"/>
<Returns>a <C>Cone</C> Object
</Returns>
<Description>
The function takes a list in which every entry represents a ray in the ambient vector space
and returns the cone defined by them.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="cdd_cone" Name="Cone" Label="for IsCddPolyhedron"/>
<Returns>a <C>Cone</C> Object
</Returns>
<Description>
This function takes a cone defined in <Emph>CddInterface</Emph> and converts it to a cone in <Emph>NConvex</Emph>
</Description>
</ManSection>

</Section>

<Section Label="Chapter_Cones_Section_Attributes_of_Cones">
<Heading>Attributes of Cones</Heading>

<ManSection>
  <Attr Arg="C" Name="DefiningInequalities" Label="for IsCone"/>
<Returns>a list
</Returns>
<Description>
Returns the list of the defining inequalities of the cone <C>C</C>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="EqualitiesOfCone" Label="for IsCone"/>
<Returns>a list
</Returns>
<Description>
Returns the list of the equalities in the defining inequalities of the cone <C>C</C>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="DualCone" Label="for IsCone"/>
<Returns>a cone
</Returns>
<Description>
Returns the dual cone of the cone <C>C</C>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="FacesOfCone" Label="for IsCone"/>
<Returns>a list of cones
</Returns>
<Description>
Returns the list of all faces of the cone <C>C</C>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="Facets" Label="for IsCone"/>
<Returns>a list of cones
</Returns>
<Description>
Returns the list of all facets of the cone <C>C</C>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="FVector" Label="for IsCone"/>
<Returns>a list
</Returns>
<Description>
Returns a list whose <Math>i</Math>'th entry is the number of faces of dimension

i

.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="RelativeInteriorRay" Label="for IsCone"/>
<Returns>a list
</Returns>
<Description>
Returns a relative interior point (or ray) in the cone <C>C</C>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="HilbertBasis" Label="for IsCone"/>
<Returns>a list
</Returns>
<Description>
Returns the Hilbert basis of the cone <C>C</C>
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="HilbertBasisOfDualCone" Label="for IsCone"/>
<Returns>a list
</Returns>
<Description>
Returns the Hilbert basis of the dual cone of the cone <C>C</C>
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="LinealitySpaceGenerators" Label="for IsCone"/>
<Returns>a list
</Returns>
<Description>
Returns a basis of the lineality space of the cone <C>C</C>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="ExternalCddCone" Label="for IsCone"/>
<Returns>a cdd object
</Returns>
<Description>
Converts the cone to a cdd object. The operations of CddInterface can then be applied
on this convex object.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="ExternalNmzCone" Label="for IsCone"/>
<Returns>an normaliz object
</Returns>
<Description>
Converts the cone to a normaliz object. The operations of NormalizInterface can then be applied
on this convex object.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="AmbientSpaceDimension" Label="for IsCone"/>
<Returns>an integer
</Returns>
<Description>
The dimension of the ambient space of the cone, i.e., the space that contains the cone.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="LatticePointsGenerators" Label="for IsCone"/>
<Returns>a list
</Returns>
<Description>
See <C>LatticePointsGenerators</C> for polyhedrons. Please note that any cone is a polyhedron.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="GridGeneratedByCone" Label="for IsCone"/>
<Returns>a homalg module
</Returns>
<Description>
Returns the homalg <Math>\mathbb{Z}</Math>-module that is generated by the ray generators of the cone.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="FactorGrid" Label="for IsCone"/>
<Returns>a homalg module
</Returns>
<Description>
Returns the homalg <Math>\mathbb{Z}</Math>-module that is presented by the matrix whose raws are the ray generators of the cone.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="FactorGridMorphism" Label="for IsCone"/>
<Returns>a homalg morphism
</Returns>
<Description>
Returns an epimorphism from a free <Math>\mathbb{Z}</Math>-module to <C>FactorGrid(C)</C>.
</Description>
</ManSection>

<ManSection>
  <Attr Arg="C" Name="GridGeneratedByOrthogonalCone" Label="for IsCone"/>
<Returns>a homalg module
</Returns>
<Description>
Returns the homalg <Math>\mathbb{Z}</Math>-module that is by generated the ray generators of the orthogonal cone on <C>C</C>.
</Description>
</ManSection>

</Section>

<Section Label="Chapter_Cones_Section_Properties_of_Cones">
<Heading>Properties of Cones</Heading>

<ManSection>
  <Prop Arg="C" Name="IsRegularCone" Label="for IsCone"/>
<Returns>true or false
</Returns>
<Description>
Returns if the cone <C>C</C> is regular or not.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="C" Name="IsRay" Label="for IsCone"/>
<Returns>true or false
</Returns>
<Description>
Returns if the cone <C>C</C> is ray or not.
</Description>
</ManSection>

<ManSection>
  <Prop Arg="C" Name="IsZero" Label="for IsCone"/>
<Returns>true or false
</Returns>
<Description>
Returns whether the cone is the zero cone or not.
</Description>
</ManSection>

</Section>

<Section Label="Chapter_Cones_Section_Operations_on_cones">
<Heading>Operations on cones</Heading>

<ManSection>
  <Oper Arg="C, m" Name="FourierProjection" Label="for IsCone, IsInt"/>
<Returns>a cone
</Returns>
<Description>
Returns the projection of the cone on the space (O, <Math>x_1,...,x_{m-1}, x_{m+1},...,x_n</Math> ).
</Description>
</ManSection>

<ManSection>
  <Oper Arg="C1, C2" Name="IntersectionOfCones" Label="for IsCone, IsCone"/>
<Returns>a cone
</Returns>
<Description>
Returns the intersection.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="L" Name="IntersectionOfCones" Label="for IsList"/>
<Returns>a cone
</Returns>
<Description>
The input is a list of cones and the output is their intersection.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="C1, C2" Name="Contains" Label="for IsCone, IsCone"/>
<Returns>a true or false
</Returns>
<Description>
Returns if the cone <C>C1</C> contains the cone <C>C2</C>.
</Description>
</ManSection>

<ManSection>
  <Oper Arg="L, C" Name="IsRelativeInteriorRay" Label="for IsList, IsCone"/>
<Returns>a true or false
</Returns>
<Description>
Checks whether the input point (or ray) <C>L</C> is in the relative interior of the cone <C>C</C>.
</Description>
</ManSection>

<#Include Label="example1">

<ManSection>
  <Oper Arg="C" Name="NonReducedInequalities" Label="for IsCone"/>
<Returns>a list
</Returns>
<Description>
It returns a list of inequalities that define the cone.
</Description>
</ManSection>

</Section>

</Chapter>

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