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<#GAPDoc Label="intersection">
<Example><![CDATA[
gap> A := Cdd_PolyhedronByInequalities( [ [ 3, 4, 5 ] ], [ 1 ] );;
gap> B := Cdd_PolyhedronByInequalities( [ [ 9, 7, 2 ] ], [ 1 ] );;
gap> C := Cdd_Intersection( A, B );;
gap> Display( Cdd_V_Rep( A ) );
V-representation
linearity 1, [ 2 ]
begin
   2 X 3  rational

   1  -3/4     0
   0    -5     4
end
gap> Display( Cdd_V_Rep( B ) );
V-representation
linearity 1, [ 2 ]
begin
   2 X 3  rational

   1  -9/7     0
   0    -2     7
end
gap> Display( Cdd_V_Rep( C ) );
V-representation
begin
   1 X 3  rational

   1  -13/9    5/9
end
]]></Example>

<#/GAPDoc>
<#GAPDoc Label="Increment">
<Listing Type="Code"><![CDATA[
H-representation
linearity t, [i_1, i_2, ...,i_t]
begin
  m x (d+1) numbertype
  b  A
end
]]></Listing>

<#/GAPDoc>
<#GAPDoc Label="Increment2">
<Listing Type="Code"><![CDATA[
H-representation
linearity 2, [1, 3]
begin
3 x 5 rational
  4 -3  6  0 -5
  1  2 -2 -7  0
  0  0 -3  0  5
end
]]></Listing>

<#/GAPDoc>
<#GAPDoc Label="Increment3">
<Listing Type="Code"><![CDATA[
V-representation
linearity t, [i_1, i_2,...,i_t]
begin
(n+s) x (d+1) numbertype
  1  v_1
  :   :
  1  v_n
  0  r_{n+1}
  :   :
  0  r_{n+s}
end
]]></Listing>

<#/GAPDoc>
<#GAPDoc Label="Increment4">
<Listing Type="Code"><![CDATA[
V-representation
linearity 2, [ 1, 3 ]
begin
4 x 3 rational
1   2   3
1  -1   2
0   1   2
0   1   1
end
]]></Listing>

<#/GAPDoc>
<#GAPDoc Label="Example1">
<Example><![CDATA[
gap> A:= Cdd_PolyhedronByInequalities( [ [ 0, 1, 0 ], [ 0, 1, -1 ] ] );
<Polyhedron given by its H-representation>
gap> Display( A );
H-representation
begin
   2 X 3  rational

   0   1   0
   0   1  -1
end
gap> B:= Cdd_PolyhedronByInequalities( [ [ 0, 1, 0 ], [ 0, 1, -1 ] ], [ 2 ] );
<Polyhedron given by its H-representation>
gap> Display( B );
H-representation
linearity 1, [ 2 ]
begin
   2 X 3  rational

   0   1   0
   0   1  -1
end
]]></Example>

<#/GAPDoc>
<#GAPDoc Label="Example2">
<Example><![CDATA[
gap> A:= Cdd_PolyhedronByGenerators( [ [ 0, 1, 3 ], [ 1, 4, 5 ] ] );
<Polyhedron given by its V-representation>
gap> Display( A );
V-representation
begin
   2 X 3  rational

   0  1  3
   1  4  5
end
gap> B:= Cdd_PolyhedronByGenerators( [ [ 0, 1, 3 ] ], [ 1 ] );
<Polyhedron given by its V-representation>
gap> Display( B );
V-representation
linearity 1, [ 1 ]
begin
   1 X 3  rational

   0  1  3
end
]]></Example>

<#/GAPDoc>
<#GAPDoc Label="Fourier">
To illustrate this projection, Let  <Math>P= \mathrm{conv}( (1,2), (4,5) )</Math> in <Math>\mathbb{Q}^2</Math>.
<Math>\newline</Math>
To find its projection on the subspace <Math>(O, x_1)</Math>, we apply the Fourier elemination to get rid of <Math>x_2</Math>
<Example><![CDATA[
gap> P := Cdd_PolyhedronByGenerators( [ [ 1, 1, 2 ], [ 1, 4, 5 ] ] );
<Polyhedron given by its V-representation>
gap> H := Cdd_H_Rep( P );
<Polyhedron given by its H-representation>
gap> Display( H );
H-representation
linearity 1, [ 3 ]
begin
   3 X 3  rational

    4  -1   0
   -1   1   0
   -1  -1   1
end
gap> P_x1 := Cdd_FourierProjection( H, 2);
<Polyhedron given by its H-representation>
gap> Display( P_x1 );
H-representation
linearity 1, [ 3 ]
begin
   3 X 3  rational

    4  -1   0
   -1   1   0
    0   0   1
end
gap> Display( Cdd_V_Rep( P_x1 ) );
V-representation
begin
   2 X 3  rational

   1  1  0
   1  4  0
end
]]></Example>

Let again <Math>Q= Conv( (2,3,4), (2,4,5) )+ nonneg( (1,1,1) )</Math>, and let us compute its projection on <Math>(O,x_2,x_3)</Math>
<Example><![CDATA[
gap> Q := Cdd_PolyhedronByGenerators( [ [ 1, 2, 3, 4 ],[ 1, 2, 4, 5 ], [ 0, 1, 1, 1 ] ] );
<Polyhedron given by its V-representation>
gap> R := Cdd_H_Rep( Q );
<Polyhedron given by its H-representation>
gap> Display( R );
H-representation
linearity 1, [ 4 ]
begin
   4 X 4  rational

    2   1  -1   0
   -2   1   0   0
   -1  -1   1   0
   -1   0  -1   1
end
gap> P_x2_x3 := Cdd_FourierProjection( R, 1);
<Polyhedron given by its H-representation>
gap> Display( P_x2_x3 );
H-representation
linearity 2, [ 1, 3 ]
begin
   3 X 4  rational

   -1   0  -1   1
   -3   0   1   0
    0   1   0   0
end
gap> Display( Cdd_V_Rep( last ) ) ;
V-representation
begin
   2 X 4  rational

   0  0  1  1
   1  0  3  4
end
]]></Example>

<#/GAPDoc>
<#GAPDoc Label="comparing_polyhedrons">
<Example><![CDATA[
gap> A := Cdd_PolyhedronByInequalities( [ [ 10, -1, 1, 0 ],
> [ -24, 9, 2, 0 ], [ 1, 1, -1, 0 ], [ -23, -12, 1, 11 ] ], [ 4 ] );
<Polyhedron given by its H-representation>
gap> B := Cdd_PolyhedronByInequalities( [ [ 1, 0, 0, 0 ],
> [ -4, 1, 0, 0 ], [ 10, -1, 1, 0 ], [ -3, -1, 0, 1 ] ], [ 3, 4 ] );
<Polyhedron given by its H-representation>
gap> Cdd_IsContained( B, A );
true
gap> Display( Cdd_V_Rep( A ) );
V-representation
begin
   3 X 4  rational

   1   2   3   4
   1   4  -6   7
   0   1   1   1
end
gap> Display( Cdd_V_Rep( B ) );
V-representation
begin
   2 X 4  rational

   1   4  -6   7
   0   1   1   1
end
]]></Example>

<#/GAPDoc>
<#GAPDoc Label="minkuwski">
<Example><![CDATA[
gap> P := Cdd_PolyhedronByGenerators( [ [ 1, 2, 5 ], [ 0, 1, 2 ] ] );
< Polyhedron given by its V-representation >
gap> Q := Cdd_PolyhedronByGenerators( [ [ 1, 4, 6 ], [ 1, 3, 7 ], [ 0, 3, 1 ] ] );
< Polyhedron given by its V-representation >
gap> S := P+Q;
< Polyhedron given by its H-representation >
gap> V := Cdd_V_Rep( S );
< Polyhedron given by its V-representation >
gap> Display( V );
V-representation
begin
   4 X 3  rational

   0   3   1
   1   6  11
   1   5  12
   0   1   2
end
gap> Cdd_GeneratingVertices( P );
[ [ 2, 5 ] ]
gap> Cdd_GeneratingVertices( Q );
[ [ 3, 7 ], [ 4, 6 ] ]
gap> Cdd_GeneratingVertices( S );
[ [ 5, 12 ], [ 6, 11 ] ]
gap> Cdd_GeneratingRays( P );
[ [ 1, 2 ] ]
gap> Cdd_GeneratingRays( Q );
[ [ 3, 1 ] ]
gap> Cdd_GeneratingRays( S );
[ [ 1, 2 ], [ 3, 1 ] ]
]]></Example>

<#/GAPDoc>
<#GAPDoc Label="Example5">
To illustrate the using of these functions, let us solve the linear program given by:
<Display>\textbf{Maximize}\;\;P(x,y)= 1-2x+5y,\;\mathrm{with}</Display>
<Display>100\leq x \leq 200,80\leq y\leq 170,y \geq -x+200.</Display>
We bring the inequalities to the form <Math>b+AX\geq 0</Math> and get:
<Display>-100+x\geq 0, 200-x \geq 0, -80+y \geq 0, 170 -y \geq 0,-200 +x+y \geq 0.</Display>
<Example><![CDATA[
gap> A:= Cdd_PolyhedronByInequalities( [ [ -100, 1, 0 ], [ 200, -1, 0 ],
> [ -80, 0, 1 ], [ 170, 0, -1 ], [ -200, 1, 1 ] ] );
<Polyhedron given by its H-representation>
gap> lp1:= Cdd_LinearProgram( A, "max", [1, -2, 5 ] );
<Linear program>
gap> Display( lp1 );
Linear program given by:
H-representation
begin
   5 X 3  rational

   -100     1     0
    200    -1     0
    -80     0     1
    170     0    -1
   -200     1     1
end
max  [ 1, -2, 5 ]
gap> Cdd_SolveLinearProgram( lp1 );
[ [ 100, 170 ], 651 ]
gap> lp2:= Cdd_LinearProgram( A, "min", [ 1, -2, 5 ] );
<Linear program>
gap> Display( lp2 );
Linear program given by:
H-representation
begin
   5 X 3  rational

   -100     1     0
    200    -1     0
    -80     0     1
    170     0    -1
   -200     1     1
end
min  [ 1, -2, 5 ]
gap> Cdd_SolveLinearProgram( lp2 );
[ [ 200, 80 ], 1 ]
gap> B:= Cdd_V_Rep( A );
<Polyhedron given by its V-representation>
gap> Display( B );
V-representation
begin
   5 X 3  rational

   1  100  170
   1  100  100
   1  120   80
   1  200   80
   1  200  170
end
]]></Example>

So the optimal solution for <Math>\texttt{lp1}</Math> is <Math>(x=100,y=170)</Math> with optimal value <Math>p=1-2(100)+5(170)=651</Math> and for <Math>\texttt{lp2}</Math> is
<Math>(x=200,y=80)</Math> with optimal value <Math>p=1-2(200)+5(80)=1</Math>.

<#/GAPDoc>
<#GAPDoc Label="Example3">
<Example><![CDATA[
gap> A:= Cdd_PolyhedronByInequalities( [ [ 0, 2, 6 ], [ 0, 1, 3 ], [1, 4, 10 ] ] );
<Polyhedron given by its H-representation>
gap> B:= Cdd_Canonicalize( A );
<Polyhedron given by its H-representation>
gap> Display( B );
H-representation
begin
   2 X 3  rational

   0   1   3
   1   4  10
end
]]></Example>

<#/GAPDoc>
<#GAPDoc Label="Example4">
<Example><![CDATA[
gap> A:= Cdd_PolyhedronByInequalities( [ [ 0, 1, 1 ], [ 0, 5, 5 ] ] );
<Polyhedron given by its H-representation>
gap> B:= Cdd_V_Rep( A );
<Polyhedron given by its V-representation>
gap> Display( B );
V-representation
linearity 1, [ 2 ]
begin
   2 X 3  rational

   0   1   0
   0  -1   1
end
gap> C:= Cdd_H_Rep( B );
<Polyhedron given by its H-representation>
gap> Display( C );
H-representation
begin
   1 X 3  rational

   0  1  1
end
gap> D:= Cdd_PolyhedronByInequalities( [ [ 0, 1, 1, 34, 22, 43 ],
> [ 11, 2, 2, 54, 53, 221 ], [33, 23, 45, 2, 40, 11 ] ] );
<Polyhedron given by its H-representation>
gap> Cdd_V_Rep( D );
<Polyhedron given by its V-representation>
gap> Display( last );
V-representation
linearity 2, [ 5, 6 ]
begin
   6 X 6  rational

   1  -743/14   369/14    11/14        0        0
   0    -1213      619       22        0        0
   0       -1        1        0        0        0
   0      764     -390      -11        0        0
   0   -13526     6772       99      154        0
   0  -116608    59496     1485        0      154
end
]]></Example>

<#/GAPDoc>
<#GAPDoc Label="demo">
<Example><![CDATA[
gap> poly:= Cdd_PolyhedronByInequalities( [ [ 1, 3, 4, 5, 7 ], [ 1, 3, 5, 12, 34 ],
> [ 9, 3, 0, 2, 13 ]  ], [ 1 ] );
<Polyhedron given by its H-representation>
gap> Cdd_InteriorPoint( poly );
[ -194/75, 46/25, -3/25, 0 ]
gap> Cdd_FacesWithInteriorPoints( poly );
[ [ 3, [ 1 ], [ -194/75, 46/25, -3/25, 0 ] ], [ 2, [ 1, 2 ],
[ -62/25, 49/25, -7/25, 0 ] ], [ 1, [ 1, 2, 3 ],
[ -209/75, 56/25, -8/25, 0 ] ], [ 2, [ 1, 3 ], [ -217/75, 53/25, -4/25, 0 ] ] ]
gap> Cdd_Dimension( poly );
3
gap> Cdd_IsPointed( poly );
false
gap> Cdd_IsEmpty( poly );
false
gap> Cdd_Faces( poly );
[ [ 3, [ 1 ] ], [ 2, [ 1, 2 ] ], [ 1, [ 1, 2, 3 ] ], [ 2, [ 1, 3  ] ] ]
gap> poly1 := Cdd_ExtendLinearity( poly, [ 1, 2, 3 ] );
<Polyhedron given by its H-representation>
gap> Display( poly1 );
H-representation
linearity 3, [ 1, 2, 3 ]
begin
   3 X 5  rational

   1   3   4   5   7
   1   3   5  12  34
   9   3   0   2  13
end
gap> Cdd_Dimension( poly1 );
1
gap> Cdd_Facets( poly );
[ [ 1, 2 ], [ 1, 3 ] ]
gap> Cdd_GeneratingVertices( poly );
[ [ -209/75, 56/25, -8/25, 0 ] ]
gap> Cdd_GeneratingRays( poly );
[ [ -97, 369, -342, 75 ], [ -8, -9, 12, 0 ],
[ 23, -21, 3, 0 ], [ 97, -369, 342, -75 ] ]
gap> Cdd_Inequalities( poly );
[ [ 1, 3, 5, 12, 34 ], [ 9, 3, 0, 2, 13 ] ]
gap> Cdd_Equalities( poly );
[ [ 1, 3, 4, 5, 7 ] ]
gap> P := Cdd_FourierProjection( poly, 2);
<Polyhedron given by its H-representation>
gap> Display( P );
H-representation
linearity 1, [ 3 ]
begin
   3 X 5  rational

    9    3    0    2   13
   -1   -3    0   23  101
    0    0    1    0    0
end
]]></Example>

<#/GAPDoc>

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