def voronoi_cell(self, radius=None):
"""
Compute the Voronoi cell of a lattice, returning a Polyhedron.
INPUT:
- ``radius`` -- (default: automatic determination) radius of ball
containing considered vertices
OUTPUT:
The Voronoi cell as a Polyhedron instance.
The result is cached so that subsequent calls to this function
return instantly.
EXAMPLES::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: L = IntegerLattice([[1, 0], [0, 1]])
sage: V = L.voronoi_cell()
sage: V.Vrepresentation()
(A vertex at (1/2, -1/2), A vertex at (1/2, 1/2), A vertex at (-1/2, 1/2), A vertex at (-1/2, -1/2))
The volume of the Voronoi cell is the square root of the
discriminant of the lattice::
sage: L = IntegerLattice(Matrix(ZZ, 4, 4, [[0,0,1,-1],[1,-1,2,1],[-6,0,3,3,],[-6,-24,-6,-5]])); L
Free module of degree 4 and rank 4 over Integer Ring
User basis matrix:
[ 0 0 1 -1]
[ 1 -1 2 1]
[ -6 0 3 3]
[ -6 -24 -6 -5]
sage: V = L.voronoi_cell() # long time
sage: V.volume() # long time
678
sage: sqrt(L.discriminant())
678
Lattices not having full dimension are handled as well::
sage: L = IntegerLattice([[2, 0, 0], [0, 2, 0]])
sage: V = L.voronoi_cell()
sage: V.Hrepresentation()
(An inequality (-1, 0, 0) x + 1 >= 0, An inequality (0, -1, 0) x + 1 >= 0, An inequality (1, 0, 0) x + 1 >= 0, An inequality (0, 1, 0) x + 1 >= 0)
ALGORITHM:
Uses parts of the algorithm from [Vit1996]_.
REFERENCES:
.. [Vit1996] \E. Viterbo, E. Biglieri. *Computing the Voronoi Cell
of a Lattice: The Diamond-Cutting Algorithm*.
IEEE Transactions on Information Theory, 1996.
"""
if not self._basis_is_LLL_reduced:
self.LLL()
B = self.reduced_basis
from diamond_cutting import calculate_voronoi_cell
return calculate_voronoi_cell(B, radius=radius)
def voronoi_cell(self, radius=None):
"""
Compute the Voronoi cell of a lattice, returning a Polyhedron.
INPUT:
- ``radius`` -- (default: automatic determination) radius of ball
containing considered vertices
OUTPUT:
The Voronoi cell as a Polyhedron instance.
The result is cached so that subsequent calls to this function
return instantly.
EXAMPLES::
sage: from sage.modules.free_module_integer import IntegerLattice
sage: L = IntegerLattice([[1, 0], [0, 1]])
sage: V = L.voronoi_cell()
sage: V.Vrepresentation()
(A vertex at (1/2, -1/2), A vertex at (1/2, 1/2), A vertex at (-1/2, 1/2), A vertex at (-1/2, -1/2))
The volume of the Voronoi cell is the square root of the
discriminant of the lattice::
sage: L = IntegerLattice(Matrix(ZZ, 4, 4, [[0,0,1,-1],[1,-1,2,1],[-6,0,3,3,],[-6,-24,-6,-5]])); L
Free module of degree 4 and rank 4 over Integer Ring
User basis matrix:
[ 0 0 1 -1]
[ 1 -1 2 1]
[ -6 0 3 3]
[ -6 -24 -6 -5]
sage: V = L.voronoi_cell() # long time
sage: V.volume() # long time
678
sage: sqrt(L.discriminant())
678
Lattices not having full dimension are handled as well::
sage: L = IntegerLattice([[2, 0, 0], [0, 2, 0]])
sage: V = L.voronoi_cell()
sage: V.Hrepresentation()
(An inequality (-1, 0, 0) x + 1 >= 0, An inequality (0, -1, 0) x + 1 >= 0, An inequality (1, 0, 0) x + 1 >= 0, An inequality (0, 1, 0) x + 1 >= 0)
ALGORITHM:
Uses parts of the algorithm from [Vit1996]_.
REFERENCES:
.. [Vit1996] E. Viterbo, E. Biglieri. *Computing the Voronoi Cell
of a Lattice: The Diamond-Cutting Algorithm*.
IEEE Transactions on Information Theory, 1996.
"""
if not self._basis_is_LLL_reduced:
self.LLL()
B = self.reduced_basis
from diamond_cutting import calculate_voronoi_cell
return calculate_voronoi_cell(B, radius=radius)
def voronoi_cell(self, radius=None):
"""
Compute the Voronoi cell of a lattice, returning a Polyhedron.
INPUT:
- ``radius`` -- radius of ball containing considered vertices
(default: automatic determination).
OUTPUT:
The Voronoi cell as a Polyhedron instance.
The result is cached so that subsequent calls to this function
return instantly.
EXAMPLES::
sage: L = Lattice([[1, 0], [0, 1]])
sage: V = L.voronoi_cell()
sage: V.Vrepresentation()
(A vertex at (1/2, -1/2), A vertex at (1/2, 1/2), A vertex at (-1/2, 1/2), A vertex at (-1/2, -1/2))
The volume of the Voronoi cell is the square root of the discriminant of the lattice::
sage: L = random_lattice(4); L
ZZ-lattice of degree 4 and rank 4
Inner product matrix:
[ 2 1 0 -1]
[ 1 7 3 1]
[ 0 3 54 3]
[ -1 1 3 673]
Basis matrix:
[ 0 0 1 -1]
[ 1 -1 2 1]
[ -6 0 3 3]
[ -6 -24 -6 -5]
sage: V = L.voronoi_cell()
sage: V.convex_volume()
678.0
sage: sqrt(L.discriminant())
678
Lattices not having full dimension are handled as well::
sage: L = Lattice([[2, 0, 0], [0, 2, 0]])
sage: V = L.voronoi_cell()
sage: V.Hrepresentation()
(An inequality (-1, 0, 0) x + 1 >= 0, An inequality (0, -1, 0) x + 1 >= 0, An inequality (1, 0, 0) x + 1 >= 0, An inequality (0, 1, 0) x + 1 >= 0)
"Over-dimensional" lattices are reduced first::
sage: L = Lattice([[1, 0], [2, 0], [0, 2]])
sage: L.voronoi_cell().Vrepresentation()
(A vertex at (1/2, -1), A vertex at (1/2, 1), A vertex at (-1/2, 1), A vertex at (-1/2, -1))
ALGORITHM:
Uses parts of the algorithm from [Vit1996].
REFERENCES:
.. [Vit1996] E. Viterbo, E. Biglieri. Computing the Voronoi Cell
of a Lattice: The Diamond-Cutting Algorithm.
IEEE Transactions on Information Theory, 1996.
"""
if self.__voronoi_cell is None:
basis_matrix = self.reduced_embedded_basis_matrix()
self.__voronoi_cell = calculate_voronoi_cell(basis_matrix, radius=radius)
return self.__voronoi_cell