from sage.all import *
-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
-
- """
- return all([(L*x).inner_product(s) == 0
- for (x,s) in K.discrete_complementarity_set()])
-
-
-def positive_operator_gens(K):
- r"""
- Compute generators of the cone of positive operators on this cone.
-
- OUTPUT:
-
- A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
- Each matrix ``P`` in the list should have the property that ``P*x``
- is an element of ``K`` whenever ``x`` is an element of
- ``K``. Moreover, any nonnegative linear combination of these
- matrices shares the same property.
-
- REFERENCES:
-
- .. [Orlitzky-Pi-Z]
- M. Orlitzky.
- Positive and Z-operators on closed convex cones.
-
- .. [Tam]
- B.-S. Tam.
- Some results of polyhedral cones and simplicial cones.
- Linear and Multilinear Algebra, 4:4 (1977) 281--284.
-
- EXAMPLES:
-
- Positive operators on the nonnegative orthant are nonnegative matrices::
-
- sage: K = Cone([(1,)])
- sage: positive_operator_gens(K)
- [[1]]
-
- sage: K = Cone([(1,0),(0,1)])
- sage: positive_operator_gens(K)
- [
- [1 0] [0 1] [0 0] [0 0]
- [0 0], [0 0], [1 0], [0 1]
- ]
-
- The trivial cone in a trivial space has no positive operators::
-
- sage: K = Cone([], ToricLattice(0))
- sage: positive_operator_gens(K)
- []
-
- Every operator is positive on the trivial cone::
-
- sage: K = Cone([(0,)])
- sage: positive_operator_gens(K)
- [[1], [-1]]
-
- sage: K = Cone([(0,0)])
- sage: K.is_trivial()
- True
- sage: positive_operator_gens(K)
- [
- [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
- [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
- ]
-
- Every operator is positive on the ambient vector space::
-
- sage: K = Cone([(1,),(-1,)])
- sage: K.is_full_space()
- True
- sage: positive_operator_gens(K)
- [[1], [-1]]
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: K.is_full_space()
- True
- sage: positive_operator_gens(K)
- [
- [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0]
- [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1]
- ]
-
- A non-obvious application is to find the positive operators on the
- right half-plane::
-
- sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: positive_operator_gens(K)
- [
- [1 0] [0 0] [ 0 0] [0 0] [ 0 0]
- [0 0], [1 0], [-1 0], [0 1], [ 0 -1]
- ]
-
- TESTS:
-
- Each positive operator generator should send the generators of the
- cone into the cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: all([ K.contains(P*x) for P in pi_of_K for x in K ])
- True
-
- Each positive operator generator should send a random element of the
- cone into the cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: all([ K.contains(P*K.random_element(QQ)) for P in pi_of_K ])
- True
-
- A random element of the positive operator cone should send the
- generators of the cone into the cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: pi_cone = Cone([ g.list() for g in pi_of_K ],
- ....: lattice=L,
- ....: check=False)
- sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
- sage: all([ K.contains(P*x) for x in K ])
- True
-
- A random element of the positive operator cone should send a random
- element of the cone into the cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: pi_cone = Cone([ g.list() for g in pi_of_K ],
- ....: lattice=L,
- ....: check=False)
- sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
- sage: K.contains(P*K.random_element(ring=QQ))
- True
-
- The lineality space of the dual of the cone of positive operators
- can be computed from the lineality spaces of the cone and its dual::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: pi_cone = Cone([ g.list() for g in pi_of_K ],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dual().linear_subspace()
- sage: U1 = [ vector((s.tensor_product(x)).list())
- ....: for x in K.lines()
- ....: for s in K.dual() ]
- sage: U2 = [ vector((s.tensor_product(x)).list())
- ....: for x in K
- ....: for s in K.dual().lines() ]
- sage: expected = pi_cone.lattice().vector_space().span(U1 + U2)
- sage: actual == expected
- True
-
- The lineality of the dual of the cone of positive operators
- is known from its lineality space::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: n = K.lattice_dim()
- sage: m = K.dim()
- sage: l = K.lineality()
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(n**2)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dual().lineality()
- sage: expected = l*(m - l) + m*(n - m)
- sage: actual == expected
- True
-
- The dimension of the cone of positive operators is given by the
- corollary in my paper::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: n = K.lattice_dim()
- sage: m = K.dim()
- sage: l = K.lineality()
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(n**2)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dim()
- sage: expected = n**2 - l*(m - l) - (n - m)*m
- sage: actual == expected
- True
-
- The trivial cone, full space, and half-plane all give rise to the
- expected dimensions::
-
- sage: n = ZZ.random_element().abs()
- sage: K = Cone([[0] * n], ToricLattice(n))
- sage: K.is_trivial()
- True
- sage: L = ToricLattice(n^2)
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dim()
- sage: actual == n^2
- True
- sage: K = K.dual()
- sage: K.is_full_space()
- True
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.dim()
- sage: actual == n^2
- True
- sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: pi_of_K = positive_operator_gens(K)
- sage: actual = Cone([p.list() for p in pi_of_K], check=False).dim()
- sage: actual == 3
- True
-
- The lineality of the cone of positive operators follows from the
- description of its generators::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: n = K.lattice_dim()
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(n**2)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.lineality()
- sage: expected = n**2 - K.dim()*K.dual().dim()
- sage: actual == expected
- True
-
- The trivial cone, full space, and half-plane all give rise to the
- expected linealities::
-
- sage: n = ZZ.random_element().abs()
- sage: K = Cone([[0] * n], ToricLattice(n))
- sage: K.is_trivial()
- True
- sage: L = ToricLattice(n^2)
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = pi_cone.lineality()
- sage: actual == n^2
- True
- sage: K = K.dual()
- sage: K.is_full_space()
- True
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
- sage: pi_cone.lineality() == n^2
- True
- sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_cone = Cone([p.list() for p in pi_of_K], check=False)
- sage: actual = pi_cone.lineality()
- sage: actual == 2
- True
-
- A cone is proper if and only if its cone of positive operators
- is proper::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: K.is_proper() == pi_cone.is_proper()
- True
-
- The positive operators of a permuted cone can be obtained by
- conjugation::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix()
- sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False)
- sage: pi_of_pK = positive_operator_gens(pK)
- sage: actual = Cone([t.list() for t in pi_of_pK],
- ....: lattice=L,
- ....: check=False)
- sage: pi_of_K = positive_operator_gens(K)
- sage: expected = Cone([(p*t*p.inverse()).list() for t in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual.is_equivalent(expected)
- True
-
- A transformation is positive on a cone if and only if its adjoint is
- positive on the dual of that cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: F = K.lattice().vector_space().base_field()
- sage: n = K.lattice_dim()
- sage: L = ToricLattice(n**2)
- sage: W = VectorSpace(F, n**2)
- sage: pi_of_K = positive_operator_gens(K)
- sage: pi_of_K_star = positive_operator_gens(K.dual())
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: pi_star = Cone([p.list() for p in pi_of_K_star],
- ....: lattice=L,
- ....: check=False)
- sage: M = MatrixSpace(F, n)
- sage: L = M(pi_cone.random_element(ring=QQ).list())
- sage: pi_star.contains(W(L.transpose().list()))
- True
-
- sage: L = W.random_element()
- sage: L_star = W(M(L.list()).transpose().list())
- sage: pi_cone.contains(L) == pi_star.contains(L_star)
- True
- """
- # Matrices are not vectors in Sage, so we have to convert them
- # to vectors explicitly before we can find a basis. We need these
- # two values to construct the appropriate "long vector" space.
- F = K.lattice().base_field()
- n = K.lattice_dim()
-
- tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
-
- # Convert those tensor products to long vectors.
- W = VectorSpace(F, n**2)
- vectors = [ W(tp.list()) for tp in tensor_products ]
-
- check = True
- if K.is_proper():
- # All of the generators involved are extreme vectors and
- # therefore minimal [Tam]_. If this cone is neither solid nor
- # strictly convex, then the tensor product of ``s`` and ``x``
- # is the same as that of ``-s`` and ``-x``. However, as a
- # /set/, ``tensor_products`` may still be minimal.
- check = False
-
- # Create the dual cone of the positive operators, expressed as
- # long vectors.
- pi_dual = Cone(vectors, ToricLattice(W.dimension()), check=check)
-
- # Now compute the desired cone from its dual...
- pi_cone = pi_dual.dual()
-
- # And finally convert its rays back to matrix representations.
- M = MatrixSpace(F, n)
- return [ M(v.list()) for v in pi_cone ]
-
-
-def Z_operator_gens(K):
- r"""
- Compute generators of the cone of Z-operators on this cone.
-
- OUTPUT:
-
- A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
- Each matrix ``L`` in the list should have the property that
- ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element of
- this cone's :meth:`discrete_complementarity_set`. Moreover, any
- conic (nonnegative linear) combination of these matrices shares the
- same property.
-
- REFERENCES:
-
- M. Orlitzky.
- Positive and Z-operators on closed convex cones.
-
- EXAMPLES:
-
- Z-operators on the nonnegative orthant are just Z-matrices.
- That is, matrices whose off-diagonal elements are nonnegative::
-
- sage: K = Cone([(1,0),(0,1)])
- sage: Z_operator_gens(K)
- [
- [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0]
- [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1]
- ]
- sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
- sage: all([ z[i][j] <= 0 for z in Z_operator_gens(K)
- ....: for i in range(z.nrows())
- ....: for j in range(z.ncols())
- ....: if i != j ])
- True
-
- The trivial cone in a trivial space has no Z-operators::
-
- sage: K = Cone([], ToricLattice(0))
- sage: Z_operator_gens(K)
- []
-
- Every operator is a Z-operator on the ambient vector space::
-
- sage: K = Cone([(1,),(-1,)])
- sage: K.is_full_space()
- True
- sage: Z_operator_gens(K)
- [[-1], [1]]
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: K.is_full_space()
- True
- sage: Z_operator_gens(K)
- [
- [-1 0] [1 0] [ 0 -1] [0 1] [ 0 0] [0 0] [ 0 0] [0 0]
- [ 0 0], [0 0], [ 0 0], [0 0], [-1 0], [1 0], [ 0 -1], [0 1]
- ]
-
- A non-obvious application is to find the Z-operators on the
- right half-plane::
-
- sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: Z_operator_gens(K)
- [
- [-1 0] [1 0] [ 0 0] [0 0] [ 0 0] [0 0]
- [ 0 0], [0 0], [-1 0], [1 0], [ 0 -1], [0 1]
- ]
-
- Z-operators on a subspace are Lyapunov-like and vice-versa::
-
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: K.is_full_space()
- True
- sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
- sage: zs = span([ vector(z.list()) for z in Z_operator_gens(K) ])
- sage: zs == lls
- True
-
- TESTS:
-
- The Z-property is possessed by every Z-operator::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: Z_of_K = Z_operator_gens(K)
- sage: dcs = K.discrete_complementarity_set()
- sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
- ....: for (x,s) in dcs])
- True
-
- The lineality space of the cone of Z-operators is the space of
- Lyapunov-like operators::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: Z_cone = Cone([ z.list() for z in Z_operator_gens(K) ],
- ....: lattice=L,
- ....: check=False)
- sage: ll_basis = [ vector(l.list()) for l in K.lyapunov_like_basis() ]
- sage: lls = L.vector_space().span(ll_basis)
- sage: Z_cone.linear_subspace() == lls
- True
-
- The lineality of the Z-operators on a cone is the Lyapunov
- rank of that cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: Z_of_K = Z_operator_gens(K)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: Z_cone = Cone([ z.list() for z in Z_of_K ],
- ....: lattice=L,
- ....: check=False)
- sage: Z_cone.lineality() == K.lyapunov_rank()
- True
-
- The lineality spaces of the duals of the positive and Z-operator
- cones are equal. From this it follows that the dimensions of the
- Z-operator cone and positive operator cone are equal::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: pi_of_K = positive_operator_gens(K)
- sage: Z_of_K = Z_operator_gens(K)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: pi_cone = Cone([p.list() for p in pi_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: Z_cone = Cone([ z.list() for z in Z_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: pi_cone.dim() == Z_cone.dim()
- True
- sage: pi_star = pi_cone.dual()
- sage: z_star = Z_cone.dual()
- sage: pi_star.linear_subspace() == z_star.linear_subspace()
- True
-
- The trivial cone, full space, and half-plane all give rise to the
- expected dimensions::
-
- sage: n = ZZ.random_element().abs()
- sage: K = Cone([[0] * n], ToricLattice(n))
- sage: K.is_trivial()
- True
- sage: L = ToricLattice(n^2)
- sage: Z_of_K = Z_operator_gens(K)
- sage: Z_cone = Cone([z.list() for z in Z_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = Z_cone.dim()
- sage: actual == n^2
- True
- sage: K = K.dual()
- sage: K.is_full_space()
- True
- sage: Z_of_K = Z_operator_gens(K)
- sage: Z_cone = Cone([z.list() for z in Z_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual = Z_cone.dim()
- sage: actual == n^2
- True
- sage: K = Cone([(1,0),(0,1),(0,-1)])
- sage: Z_of_K = Z_operator_gens(K)
- sage: Z_cone = Cone([z.list() for z in Z_of_K], check=False)
- sage: Z_cone.dim() == 3
- True
-
- The Z-operators of a permuted cone can be obtained by conjugation::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: L = ToricLattice(K.lattice_dim()**2)
- sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix()
- sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False)
- sage: Z_of_pK = Z_operator_gens(pK)
- sage: actual = Cone([t.list() for t in Z_of_pK],
- ....: lattice=L,
- ....: check=False)
- sage: Z_of_K = Z_operator_gens(K)
- sage: expected = Cone([(p*t*p.inverse()).list() for t in Z_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: actual.is_equivalent(expected)
- True
-
- An operator is a Z-operator on a cone if and only if its
- adjoint is a Z-operator on the dual of that cone::
-
- sage: set_random_seed()
- sage: K = random_cone(max_ambient_dim=4)
- sage: F = K.lattice().vector_space().base_field()
- sage: n = K.lattice_dim()
- sage: L = ToricLattice(n**2)
- sage: W = VectorSpace(F, n**2)
- sage: Z_of_K = Z_operator_gens(K)
- sage: Z_of_K_star = Z_operator_gens(K.dual())
- sage: Z_cone = Cone([p.list() for p in Z_of_K],
- ....: lattice=L,
- ....: check=False)
- sage: Z_star = Cone([p.list() for p in Z_of_K_star],
- ....: lattice=L,
- ....: check=False)
- sage: M = MatrixSpace(F, n)
- sage: L = M(Z_cone.random_element(ring=QQ).list())
- sage: Z_star.contains(W(L.transpose().list()))
- True
-
- sage: L = W.random_element()
- sage: L_star = W(M(L.list()).transpose().list())
- sage: Z_cone.contains(L) == Z_star.contains(L_star)
- True
- """
- # Matrices are not vectors in Sage, so we have to convert them
- # to vectors explicitly before we can find a basis. We need these
- # two values to construct the appropriate "long vector" space.
- F = K.lattice().base_field()
- n = K.lattice_dim()
-
- # These tensor products contain generators for the dual cone of
- # the cross-positive operators.
- tensor_products = [ s.tensor_product(x)
- for (x,s) in K.discrete_complementarity_set() ]
-
- # Turn our matrices into long vectors...
- W = VectorSpace(F, n**2)
- vectors = [ W(m.list()) for m in tensor_products ]
-
- check = True
- if K.is_proper():
- # All of the generators involved are extreme vectors and
- # therefore minimal. If this cone is neither solid nor
- # strictly convex, then the tensor product of ``s`` and ``x``
- # is the same as that of ``-s`` and ``-x``. However, as a
- # /set/, ``tensor_products`` may still be minimal.
- check = False
-
- # Create the dual cone of the cross-positive operators,
- # expressed as long vectors.
- Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()), check=check)
-
- # Now compute the desired cone from its dual...
- Sigma_cone = Sigma_dual.dual()
-
- # And finally convert its rays back to matrix representations.
- # But first, make them negative, so we get Z-operators and
- # not cross-positive ones.
- M = MatrixSpace(F, n)
- return [ -M(v.list()) for v in Sigma_cone ]
-
+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)
-def Z_cone(K):
- gens = Z_operator_gens(K)
+def Sigma_cone(K):
+ 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 pi_cone(K):
- gens = positive_operator_gens(K)
+def Z_cone(K):
+ 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_operators_gens(K2)
+ L = ToricLattice(K1.lattice_dim()*K2.lattice_dim())
+ return Cone(( g.list() for g in gens ), lattice=L, check=False)
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