Quelle maxplusmat.xml Sprache: XML

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#W maxplusmat.xml
#Y Copyright (C) 2015 James D. Mitchell
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## Licensing information can be found in the README file of this package.
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<#GAPDoc Label="IsXMatrix">
 <ManSection Label = "IsXMatrix">
 <Heading>Matrix filters</Heading>
 <Filt Name = "IsBooleanMat" Arg = "obj" Type = "Category"/>
 <Filt Name = "IsMatrixOverFiniteField" Arg = "obj" Type = "Category"/>
 <Filt Name = "IsMaxPlusMatrix" Arg = "obj" Type = "Category"/>
 <Filt Name = "IsMinPlusMatrix" Arg = "obj" Type = "Category"/>
 <Filt Name = "IsTropicalMatrix" Arg = "obj" Type = "Category"/>
 <Filt Name = "IsTropicalMaxPlusMatrix" Arg = "obj" Type = "Category"/>
 <Filt Name = "IsTropicalMinPlusMatrix" Arg = "obj" Type = "Category"/>
 
 <Filt Name = "IsNTPMatrix" Arg = "obj" Type = "Category"/>
 <Filt Name = "Integers" Arg = "obj" Type = "Category"/>
 <Returns><K>true</K> or <K>false</K>.</Returns>
 <Description>
 Every matrix over a semiring in &SEMIGROUPS; is a member of one of these
 categories, which are subcategory of <Ref Filt = "IsMatrixOverSemiring"/>.
 

 <C>IsTropicalMatrix</C> is a supercategory of
 <C>IsTropicalMaxPlusMatrix</C> and <C>IsTropicalMinPlusMatrix</C>.
 
 

 Basic operations for matrices over semirings include: multiplication via
 \*, <Ref Attr = "DimensionOfMatrixOverSemiring"/>, <Ref Attr="One"
 BookName="ref"/>, the underlying list of lists used to create the
 matrix can be accessed using <Ref Attr = "AsList"/>, the rows of
 <C>mat</C> can be accessed using <C>mat[i]</C> where <C>i</C> is between
 <C>1</C> and the dimension of the matrix, it also possible to loop over
 the rows of a matrix; for tropical matrices <Ref
 Attr = "ThresholdTropicalMatrix"/>; for ntp matrices <Ref
 Attr = "ThresholdNTPMatrix"/> and <Ref Attr = "PeriodNTPMatrix"/>.
 

 For matrices over finite fields see Section
 <Ref Sect = "Matrices over finite fields"/>; for Boolean matrices more
 details can be found in Section <Ref Sect = "Boolean matrices"/>.

 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsXMatrixCollection">
<ManSection Label = "IsXMatrixCollection">
 <Heading>Matrix collection filters</Heading>
 <Filt Name = "IsBooleanMatCollection" Arg = "obj" Type =
 "Category"/>
 <Filt Name = "IsBooleanMatCollColl" Arg = "obj" Type =
 "Category"/>
 <Filt Name = "IsMatrixOverFiniteFieldCollection" Arg = "obj" Type =
 "Category"/>
 <Filt Name = "IsMatrixOverFiniteFieldCollColl" Arg = "obj" Type =
 "Category"/>
 <Filt Name = "IsMaxPlusMatrixCollection" Arg = "obj" Type =
 "Category"/>
 <Filt Name = "IsMaxPlusMatrixCollColl" Arg = "obj" Type =
 "Category"/>
 <Filt Name = "IsMinPlusMatrixCollection" Arg = "obj" Type =
 "Category"/>
 <Filt Name = "IsMinPlusMatrixCollColl" Arg = "obj" Type =
 "Category"/>
 <Filt Name = "IsTropicalMatrixCollection" Arg = "obj" Type =
 "Category"/>
 <Filt Name = "IsTropicalMaxPlusMatrixCollection" Arg = "obj" Type =
 "Category"/>
 <Filt Name = "IsTropicalMaxPlusMatrixCollColl" Arg = "obj" Type =
 "Category"/>
 <Filt Name = "IsTropicalMinPlusMatrixCollection" Arg = "obj" Type =
 "Category"/>
 <Filt Name = "IsTropicalMinPlusMatrixCollColl" Arg = "obj" Type =
 "Category"/>
 
 <Filt Name = "IsNTPMatrixCollection" Arg = "obj" Type =
 "Category"/>
 <Filt Name = "IsNTPMatrixCollColl" Arg = "obj" Type =
 "Category"/>
 <Returns>
 <K>true</K> or <K>false</K>.
 </Returns>
 <Description>
 Every collection of matrices over the same semiring in &SEMIGROUPS;
 belongs to one of the categories above. For example, semigroups of
 boolean matrices belong to <C>IsBooleanMatCollection</C>.
 

 Similarly, every collection of collections of matrices over the same
 semiring in &SEMIGROUPS; belongs to one of the categories above.
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="ThresholdTropicalMatrix">
 <ManSection>
 <Attr Name="ThresholdTropicalMatrix" Arg="mat"/>
 <Returns>A positive integer.</Returns>
 <Description>
 If <A>mat</A> is a tropical matrix (i.e. belongs to the category <Ref
 Filt = "IsTropicalMatrix"/>), then <C>ThresholdTropicalMatrix</C>
 returns the threshold (i.e. the largest integer) of the underlying
 semiring.
 <Example><![CDATA[
gap> mat := Matrix(IsTropicalMaxPlusMatrix,
> [[0, 3, 0, 2],
> [1, 1, 1, 0],
> [-infinity, 1, -infinity, 1],
> [0, -infinity, 2, -infinity]], 10);
Matrix(IsTropicalMaxPlusMatrix, [[0, 3, 0, 2], [1, 1, 1, 0],
 [-infinity, 1, -infinity, 1], [0, -infinity, 2, -infinity]], 10)
gap> ThresholdTropicalMatrix(mat);
10
gap> mat := Matrix(IsTropicalMaxPlusMatrix,
> [[0, 3, 0, 2],
> [1, 1, 1, 0],
> [-infinity, 1, -infinity, 1],
> [0, -infinity, 2, -infinity]], 3);
Matrix(IsTropicalMaxPlusMatrix, [[0, 3, 0, 2], [1, 1, 1, 0],
 [-infinity, 1, -infinity, 1], [0, -infinity, 2, -infinity]], 3)
gap> ThresholdTropicalMatrix(mat);
3]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="ThresholdNTPMatrix">
 <ManSection>
 <Attr Name = "ThresholdNTPMatrix" Arg = "mat"/>
 <Attr Name = "PeriodNTPMatrix" Arg = "mat"/>
 <Returns>A positive integer.</Returns>
 <Description>
 An ntp matrix is a matrix with entries in
 a semiring <M>\mathbb{N}_{t,p} = \{0, 1,
 \ldots, t, t + 1, \ldots, t + p - 1\}</M> for some threshold <M>t</M>
 and period <M>p</M> under addition and multiplication modulo the congruence
 <M>t = t + p</M>.
 

 If <A>mat</A> is a ntp matrix (i.e. belongs to the category <Ref
 Filt = "IsNTPMatrix"/>), then <C>ThresholdNTPMatrix</C> and
 <C>PeriodNTPMatrix</C> return the threshold and period of the underlying
 semiring, respectively.
 <Example><![CDATA[
gap> mat := Matrix(IsNTPMatrix, [[1, 1, 0],
> [2, 1, 0],
> [0, 1, 1]],
> 1, 2);
Matrix(IsNTPMatrix, [[1, 1, 0], [2, 1, 0], [0, 1, 1]], 1, 2)
gap> ThresholdNTPMatrix(mat);
1
gap> PeriodNTPMatrix(mat);
2
gap> mat := Matrix(IsNTPMatrix, [[2, 1, 3],
> [0, 5, 1],
> [4, 1, 0]],
> 3, 4);
Matrix(IsNTPMatrix, [[2, 1, 3], [0, 5, 1], [4, 1, 0]], 3, 4)
gap> ThresholdNTPMatrix(mat);
3
gap> PeriodNTPMatrix(mat);
4]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IntInverseOp">
 <ManSection>
 <Oper Name = "InverseOp" Arg = "mat" Label = "for an integer matrix"/>
 <Returns>An integer matrix.</Returns>
 <Description>
 If <A>mat</A> is an integer matrix (i.e. belongs to the category
 <Ref Filt = "Integers"/>) whose inverse (if it exists) is also an
 integer matrix, then <C>InverseOp</C> returns the inverse of <A>mat</A>.
 

 An integer matrix has an integer matrix inverse if and only if it
 has determinant one.
 <Example><![CDATA[
gap> mat := Matrix(Integers, [[0, 0, -1],
> [0, 1, 0],
> [1, 0, 0]]);
<3x3-matrix over Integers>
gap> InverseOp(mat);
<3x3-matrix over Integers>
gap> mat * InverseOp(mat) = One(mat);
true
]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IntIsTorsion">
 <ManSection>
 <Attr Name = "IsTorsion" Arg = "mat" Label = "for an integer matrix"/>
 <Returns><K>true</K> or <K>false</K></Returns>
 <Description>
 If <A>mat</A> is an integer matrix (i.e. belongs to the
 category <Ref Filt = "Integers"/>), then <C>IsTorsion</C> returns
 <K>true</K> if <A>mat</A> is torsion and <K>false</K> otherwise. 

 An integer matrix <A>mat</A> is torsion if and only if there exists an
 integer <C>n</C> such that <A>mat</A> to the power of <C>n</C> is
 equal to the identity matrix.

 <Example><![CDATA[
gap> mat := Matrix(Integers, [[0, 0, -1],
> [0, 1, 0],
> [1, 0, 0]]);
<3x3-matrix over Integers>
gap> IsTorsion(mat);
true
gap> mat := Matrix(Integers, [[0, 0, -1, 0],
> [0, -1, 0, 0],
> [4, 4, 2, -1],
> [1, 1, 0, 3]]);
<4x4-matrix over Integers>
gap> IsTorsion(mat);
false
]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="IntOrder">
 <ManSection>
 <Attr Name = "Order" Arg = "mat"/>
 <Returns>An integer or <C>infinity</C>.</Returns>
 <Description>
 If <A>mat</A> is an integer matrix, then <C>InverseOp</C> returns the
 order of <A>mat</A>. The order of <A>mat</A> is the smallest integer
 power of <A>mat</A> equal to the identity. If no such integer exists, the
 order is equal to <C>infinity</C>.
 <Example><![CDATA[
gap> mat := Matrix(Integers, [[0, 0, -1, 0],
> [0, -1, 0, 0],
> [4, 4, 2, -1],
> [1, 1, 0, 3]]);;
gap> Order(mat);
infinity
gap> mat := Matrix(Integers, [[0, 0, -1],
> [0, 1, 0],
> [1, 0, 0]]);;
gap> Order(mat);
4
]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="SpectralRadius">
 <ManSection>
 <Oper Name = "SpectralRadius" Arg = "mat"/>
 <Returns>A rational number.</Returns>
 <Description>
 If <A>mat</A> is a max-plus matrix (i.e. belongs to the category <Ref
 Filt = "IsMaxPlusMatrix"/>), then <C>SpectralRadius</C>
 returns the supremum of the set of eigenvalues of <A>mat</A>.
 This method is described in <Cite Key="Bacelli1992aa"/>.
 <Example><![CDATA[
gap> SpectralRadius(Matrix(IsMaxPlusMatrix, [[-infinity, 1, -4],
> [2, -infinity, 0],
> [0, 1, 1]]));
3/2]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="InverseOp">
 <ManSection>
 <Oper Name = "InverseOp" Arg = "mat"/>
 <Returns>A max-plus matrix.</Returns>
 <Description>
 If <A>mat</A> is an invertible max-plus matrix (i.e. belongs to the
 category <Ref Filt = "IsMaxPlusMatrix"/> and is a row permutation
 applied to the identity), then <C>InverseOp</C> returns the inverse
 of <A>mat</A>. This method is described in <Cite Key="Kacie2009aa"/>.
 <Example><![CDATA[
gap> InverseOp(Matrix(IsMaxPlusMatrix, [[-infinity, -infinity, 0],
> [0, -infinity, -infinity],
> [-infinity, 0, -infinity]]));
Matrix(IsMaxPlusMatrix, [[-infinity, 0, -infinity],
 [-infinity, -infinity, 0], [0, -infinity, -infinity]])
]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="RadialEigenvector">
 <ManSection>
 <Oper Name = "RadialEigenvector" Arg = "mat"/>
 <Returns>A vector.</Returns>
 <Description>
 If <A>mat</A> is a <C>n</C> by <C>n</C> max-plus matrix (i.e. belongs to
 the category <Ref Filt = "IsMaxPlusMatrix"/>), then
 <C>RadialEigenvector</C> returns an eigenvector corresponding to the
 eigenvalue equal to the spectral radius of <A>mat</A>. This method
 is described in <Cite Key="Kacie2009aa"/>.
 <Example><![CDATA[
gap> RadialEigenvector(Matrix(IsMaxPlusMatrix, [[0, -3], [-2, -10]]));
[ 0, -2 ]
]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

<#GAPDoc Label="UnweightedPrecedenceDigraph">
 <ManSection>
 <Oper Name = "UnweightedPrecedenceDigraph" Arg = "mat"/>
 <Returns>A digraph.</Returns>
 <Description>
 If <A>mat</A> is a max-plus matrix (i.e. belongs to the category <Ref
 Filt = "IsMaxPlusMatrix"/>), then <C>UnweightedPrecedenceDigraph</C>
 returns the unweighted precedence digraph corresponding to <A>mat</A>.
 

 For an <C>n</C> by <C>n</C> matrix <A>mat</A>, the unweighted precedence
 digraph is defined as the digraph with <C>n</C> vertices where vertex
 <C>i</C> is adjacent to vertex <C>j</C> if and only if
 <A>mat</A><C>[i][j]</C> is not equal to <K>-infinity</K>.
 <Example><![CDATA[
gap> UnweightedPrecedenceDigraph(Matrix(IsMaxPlusMatrix, [[2, -2, 0],
> [-infinity, 10, -2], [-infinity, 2, 1]]));
<immutable digraph with 3 vertices, 7 edges>
]]></Example>
 </Description>
 </ManSection>
<#/GAPDoc>

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