# The double-import is needed to get the underscore methods.
from mjo.cone.cone import *
-from mjo.cone.cone import _basically_the_same, _rho
+from mjo.cone.cone import _restrict_to_space
#
-# Tests for _rho.
+# Tests for _restrict_to_space.
#
+def _look_isomorphic(K1, K2):
+ r"""
+ Test whether or not ``K1`` and ``K2`` look linearly isomorphic.
+
+ This is a hack to get around the fact that it's difficult to tell
+ when two cones are linearly isomorphic. Instead, we check a list of
+ properties that should be preserved under linear isomorphism.
+
+ OUTPUT:
+
+ ``True`` if ``K1`` and ``K2`` look isomorphic, or ``False``
+ if we can prove that they are not isomorphic.
+
+ EXAMPLES:
+
+ Any proper cone with three generators in `\mathbb{R}^{3}` is
+ isomorphic to the nonnegative orthant::
+
+ sage: K1 = Cone([(1,0,0), (0,1,0), (0,0,1)])
+ sage: K2 = Cone([(1,2,3), (3, 18, 4), (66, 51, 0)])
+ sage: _look_isomorphic(K1, K2)
+ True
+
+ Negating a cone gives you an isomorphic cone::
+
+ sage: K = Cone([(0,2,-5), (-6, 2, 4), (0, 51, 0)])
+ sage: _look_isomorphic(K, -K)
+ True
+
+ TESTS:
+
+ Any cone is isomorphic to itself::
+
+ sage: K = random_cone(max_ambient_dim = 8)
+ sage: _look_isomorphic(K, K)
+ True
+
+ After applying an invertible matrix to the rows of a cone, the
+ result should is isomorphic to the cone we started with::
+
+ sage: K1 = random_cone(max_ambient_dim = 8)
+ sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
+ sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
+ sage: _look_isomorphic(K1, K2)
+ True
+
+ """
+ if K1.lattice_dim() != K2.lattice_dim():
+ return False
+
+ if K1.nrays() != K2.nrays():
+ return False
+
+ if K1.dim() != K2.dim():
+ return False
+
+ if K1.lineality() != K2.lineality():
+ return False
+
+ if K1.is_solid() != K2.is_solid():
+ return False
+
+ if K1.is_strictly_convex() != K2.is_strictly_convex():
+ return False
+
+ if len(K1.lyapunov_like_basis()) != len(K2.lyapunov_like_basis()):
+ return False
+
+ C_of_K1 = K1.discrete_complementarity_set()
+ C_of_K2 = K2.discrete_complementarity_set()
+ if len(C_of_K1) != len(C_of_K2):
+ return False
+
+ if len(K1.facets()) != len(K2.facets()):
+ return False
+
+ return True
+
+
"""
-Apply _rho according to our paper (to obtain our main result). Test all
-four parameter combinations::
+Apply _restrict_to_space according to our paper (to obtain our main
+result). Test all four parameter combinations::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim = 8,
....: strictly_convex=False,
....: solid=False)
- sage: K_S = _rho(K)
- sage: K_SP = _rho(K_S.dual()).dual()
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
sage: K_SP.is_proper()
True
- sage: K_SP = _rho(K_S, K_S.dual())
+ sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
sage: K_SP.is_proper()
True
sage: K = random_cone(max_ambient_dim = 8,
....: strictly_convex=True,
....: solid=False)
- sage: K_S = _rho(K)
- sage: K_SP = _rho(K_S.dual()).dual()
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
sage: K_SP.is_proper()
True
- sage: K_SP = _rho(K_S, K_S.dual())
+ sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
sage: K_SP.is_proper()
True
sage: K = random_cone(max_ambient_dim = 8,
....: strictly_convex=False,
....: solid=True)
- sage: K_S = _rho(K)
- sage: K_SP = _rho(K_S.dual()).dual()
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
sage: K_SP.is_proper()
True
- sage: K_SP = _rho(K_S, K_S.dual())
+ sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
sage: K_SP.is_proper()
True
sage: K = random_cone(max_ambient_dim = 8,
....: strictly_convex=True,
....: solid=True)
- sage: K_S = _rho(K)
- sage: K_SP = _rho(K_S.dual()).dual()
+ sage: K_S = _restrict_to_space(K, K.span())
+ sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
sage: K_SP.is_proper()
True
- sage: K_SP = _rho(K_S, K_S.dual())
+ sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
sage: K_SP.is_proper()
True
....: solid=False,
....: strictly_convex=False)
sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
- sage: K_W_star = _rho(K, J).dual()
- sage: K_star_W = _rho(K.dual(), J)
- sage: _basically_the_same(K_W_star, K_star_W)
+ sage: K_W_star = _restrict_to_space(K, J.span()).dual()
+ sage: K_star_W = _restrict_to_space(K.dual(), J.span())
+ sage: _look_isomorphic(K_W_star, K_star_W)
True
::
....: solid=True,
....: strictly_convex=False)
sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
- sage: K_W_star = _rho(K, J).dual()
- sage: K_star_W = _rho(K.dual(), J)
- sage: _basically_the_same(K_W_star, K_star_W)
+ sage: K_W_star = _restrict_to_space(K, J.span()).dual()
+ sage: K_star_W = _restrict_to_space(K.dual(), J.span())
+ sage: _look_isomorphic(K_W_star, K_star_W)
True
::
....: solid=False,
....: strictly_convex=True)
sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
- sage: K_W_star = _rho(K, J).dual()
- sage: K_star_W = _rho(K.dual(), J)
- sage: _basically_the_same(K_W_star, K_star_W)
+ sage: K_W_star = _restrict_to_space(K, J.span()).dual()
+ sage: K_star_W = _restrict_to_space(K.dual(), J.span())
+ sage: _look_isomorphic(K_W_star, K_star_W)
True
::
....: solid=True,
....: strictly_convex=True)
sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
- sage: K_W_star = _rho(K, J).dual()
- sage: K_star_W = _rho(K.dual(), J)
- sage: _basically_the_same(K_W_star, K_star_W)
+ sage: K_W_star = _restrict_to_space(K, J.span()).dual()
+ sage: K_star_W = _restrict_to_space(K.dual(), J.span())
+ sage: _look_isomorphic(K_W_star, K_star_W)
True
"""
sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
True
-The Lyapunov rank of a cone ``K`` is the dimension of ``LL(K)``. Check
-all combinations of parameters::
+The Lyapunov rank of a cone ``K`` is the dimension of
+``K.lyapunov_like_basis()``. Check all combinations of parameters::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=8,
....: strictly_convex=True,
....: solid=True)
- sage: lyapunov_rank(K) == len(LL(K))
+ sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
True
::
sage: K = random_cone(max_ambient_dim=8,
....: strictly_convex=True,
....: solid=False)
- sage: lyapunov_rank(K) == len(LL(K))
+ sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
True
::
sage: K = random_cone(max_ambient_dim=8,
....: strictly_convex=False,
....: solid=True)
- sage: lyapunov_rank(K) == len(LL(K))
+ sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
True
::
sage: K = random_cone(max_ambient_dim=8,
....: strictly_convex=False,
....: solid=False)
- sage: lyapunov_rank(K) == len(LL(K))
+ sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
True
"""
projects / sage.d.git / blobdiff