def induced_out_partition(self, trans, ieq):
r"""
Returns the output partition obtained as the induction of the given
transformation on the domain given by an inequality.
Note: the output partition corresponds to the arrival partition in
the domain, not the initial one.
INPUT:
- ``trans`` -- a function: polyhedron -> polyhedron
- ``ieq`` -- list, an inequality. An entry equal to "[-1,7,3,4]"
represents the inequality 7x_1+3x_2+4x_3>= 1.
OUTPUT:
dict of polyhedron partitions with keys giving the return time
EXAMPLES::
sage: from slabbe import PolyhedronPartition, rotation_mod
sage: h = 1/3
sage: p = Polyhedron([(0,h),(0,1),(h,1)])
sage: q = Polyhedron([(0,0), (0,h), (h,1), (h,0)])
sage: r = Polyhedron([(h,1), (1,1), (1,h), (h,0)])
sage: s = Polyhedron([(h,0), (1,0), (1,h)])
sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s})
sage: u = rotation_mod(0, 1/3, 1, QQ)
sage: ieq = [h, -1, 0] # x0 <= h
sage: P.induced_out_partition(u, ieq)
{3: Polyhedron partition of 4 atoms with 4 letters}
::
sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s})
sage: ieq2 = [1/2, -1, 0] # x0 <= 1/2
sage: d = P.induced_out_partition(u, ieq2)
sage: d
{1: Polyhedron partition of 2 atoms with 2 letters,
2: Polyhedron partition of 3 atoms with 3 letters,
3: Polyhedron partition of 4 atoms with 4 letters}
sage: Q = PolyhedronPartition(d[1].atoms()+d[2].atoms()+d[3].atoms())
sage: Q.is_pairwise_disjoint()
True
::
sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s})
sage: ieq3 = [-1/2, 1, 0] # x0 >= 1/2
sage: P.induced_out_partition(u, ieq3)
{2: Polyhedron partition of 3 atoms with 3 letters,
3: Polyhedron partition of 4 atoms with 4 letters}
It is an error if the induced region is empty::
sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s})
sage: ieq4 = [-1/2, -1, 0] # x0 <= -1/2
sage: P.induced_out_partition(u, ieq4)
Traceback (most recent call last):
...
ValueError: Inequality An inequality (-2, 0) x - 1 >= 0 does
not intersect P (=Polyhedron partition of 4 atoms with 4
letters)
The whole domain::
sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s})
sage: ieq5 = [1/2, 1, 0] # x0 >= -1/2
sage: P.induced_out_partition(u, ieq5)
{1: Polyhedron partition of 4 atoms with 4 letters}
An irrational rotation::
sage: z = polygen(QQ, 'z') #z = QQ['z'].0 # same as
sage: K = NumberField(z**2-z-1, 'phi', embedding=RR(1.6))
sage: phi = K.gen()
sage: h = 1/phi^2
sage: p = Polyhedron([(0,h),(0,1),(h,1)])
sage: q = Polyhedron([(0,0), (0,h), (h,1), (h,0)])
sage: r = Polyhedron([(h,1), (1,1), (1,h), (h,0)])
sage: s = Polyhedron([(h,0), (1,0), (1,h)])
sage: P = PolyhedronPartition({0:p, 1:q, 2:r, 3:s}, base_ring=K)
sage: u = rotation_mod(0, 1/phi, 1, K)
sage: ieq = [phi^-4, -1, 0] # x0 <= phi^-4
sage: d = P.induced_out_partition(u, ieq)
sage: d
{5: Polyhedron partition of 6 atoms with 6 letters,
8: Polyhedron partition of 9 atoms with 9 letters}
"""
# good side of the hyperplane
half = Polyhedron(ieqs=[ieq])
half_part = PolyhedronPartition([half])
# the other side of the hyperplane
other_half = Polyhedron(ieqs=[[-a for a in ieq]])
other_half_part = PolyhedronPartition([other_half])
# initial refinement
P = self.refinement(half_part)
if len(P) == 0:
raise ValueError("Inequality {} does not intersect P "
"(={})".format(half.inequalities()[0], self))
level = 1
ans = {}
P = P.apply_transformation(trans)
while len(P):
P_returned = P.refinement(half_part)
if P_returned:
ans[level] = P_returned
# for what is remaining we do:
P = P.refinement(other_half_part)
P = P.refinement(self)
P = P.apply_transformation(trans)
level += 1
return ans
