from sage.all import *
-def is_cross_positive(L,K):
- r"""
- Determine whether or not ``L`` is cross-positive on ``K``.
-
- We say that ``L`` is cross-positive on ``K`` if `\left\langle
- L\left\lparenx\right\rparen,s\right\rangle >= 0` for all pairs
- `\left\langle x,s \right\rangle` in the complementarity set of
- ``K``. It is known that this property need only be
- checked for generators of ``K`` and its dual.
-
- INPUT:
-
- - ``L`` -- A linear transformation or matrix.
-
- - ``K`` -- A polyhedral closed convex cone.
-
- OUTPUT:
-
- ``True`` if it can be proven that ``L`` is cross-positive on ``K``,
- and ``False`` otherwise.
-
- .. WARNING::
-
- If this function returns ``True``, then ``L`` is cross-positive
- on ``K``. However, if ``False`` is returned, that could mean one
- of two things. The first is that ``L`` is definitely not
- cross-positive on ``K``. The second is more of an "I don't know"
- answer, returned (for example) if we cannot prove that an inner
- product is nonnegative.
-
- EXAMPLES:
-
- The identity is always cross-positive in a nontrivial space::
-
- sage: set_random_seed()
- sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
- sage: L = identity_matrix(K.lattice_dim())
- sage: is_cross_positive(L,K)
- True
-
- As is the "zero" transformation::
-
- sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
- sage: R = K.lattice().vector_space().base_ring()
- sage: L = zero_matrix(R, K.lattice_dim())
- sage: is_cross_positive(L,K)
- True
-
- Everything in ``K.cross_positive_operator_gens()`` should be
- cross-positive on ``K``::
-
- sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
- sage: all([ is_cross_positive(L,K)
- ....: for L in K.cross_positive_operator_gens() ])
- True
-
- """
- if L.base_ring().is_exact() or L.base_ring() is SR:
- return all([ s*(L*x) >= 0
- for (x,s) in K.discrete_complementarity_set() ])
- else:
- # The only inexact ring that we're willing to work with is SR,
- # since it can still be exact when working with symbolic
- # constants like pi and e.
- raise ValueError('base ring of operator L is neither SR nor exact')
-
-
-def is_lyapunov_like(L,K):
- r"""
- Determine whether or not ``L`` is Lyapunov-like on ``K``.
-
- We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle
- L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs
- `\left\langle x,s \right\rangle` in the complementarity set of
- ``K``. It is known [Orlitzky]_ that this property need only be
- checked for generators of ``K`` and its dual.
-
- There are faster ways of checking this property. For example, we
- could compute a `lyapunov_like_basis` of the cone, and then test
- whether or not the given matrix is contained in the span of that
- basis. The value of this function is that it works on symbolic
- matrices.
-
- INPUT:
-
- - ``L`` -- A linear transformation or matrix.
-
- - ``K`` -- A polyhedral closed convex cone.
-
- OUTPUT:
-
- ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
- and ``False`` otherwise.
-
- .. WARNING::
-
- If this function returns ``True``, then ``L`` is Lyapunov-like
- on ``K``. However, if ``False`` is returned, that could mean one
- of two things. The first is that ``L`` is definitely not
- Lyapunov-like on ``K``. The second is more of an "I don't know"
- answer, returned (for example) if we cannot prove that an inner
- product is zero.
-
- REFERENCES:
-
- M. Orlitzky. The Lyapunov rank of an improper cone.
- http://www.optimization-online.org/DB_HTML/2015/10/5135.html
-
- EXAMPLES:
-
- The identity is always Lyapunov-like in a nontrivial space::
-
- sage: set_random_seed()
- sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
- sage: L = identity_matrix(K.lattice_dim())
- sage: is_lyapunov_like(L,K)
- True
-
- As is the "zero" transformation::
-
- sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
- sage: R = K.lattice().vector_space().base_ring()
- sage: L = zero_matrix(R, K.lattice_dim())
- sage: is_lyapunov_like(L,K)
- True
-
- Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
- on ``K``::
-
- sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
- sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
- True
-
- """
- if L.base_ring().is_exact() or L.base_ring() is SR:
- V = VectorSpace(K.lattice().base_field(), K.lattice_dim()**2)
- LL_of_K = V.span([ V(m.list()) for m in K.lyapunov_like_basis() ])
- return V(L.list()) in LL_of_K
- else:
- # The only inexact ring that we're willing to work with is SR,
- # since it can still be exact when working with symbolic
- # constants like pi and e.
- raise ValueError('base ring of operator L is neither SR nor exact')
-
def LL_cone(K):
gens = K.lyapunov_like_basis()
L = ToricLattice(K.lattice_dim()**2)
- return Cone([ g.list() for g in gens ], lattice=L, check=False)
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
def Sigma_cone(K):
- gens = K.cross_positive_operator_gens()
+ gens = K.cross_positive_operators_gens()
L = ToricLattice(K.lattice_dim()**2)
- return Cone([ g.list() for g in gens ], lattice=L, check=False)
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
def Z_cone(K):
- gens = K.Z_operator_gens()
+ gens = K.Z_operators_gens()
L = ToricLattice(K.lattice_dim()**2)
- return Cone([ g.list() for g in gens ], lattice=L, check=False)
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
def pi_cone(K1, K2=None):
if K2 is None:
K2 = K1
- gens = K1.positive_operator_gens(K2)
+ gens = K1.positive_operators_gens(K2)
L = ToricLattice(K1.lattice_dim()*K2.lattice_dim())
- return Cone([ g.list() for g in gens ], lattice=L, check=False)
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
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