Fix the is_positive_on test and give better examples.

[sage.d.git] / mjo / cone / cone.py

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index c71a24cbee0c9d0857fd3c8d8fe2ecbb0042238c..01ef5f9d5cc56b087344237b9b5c915423ac0921 100644 (file)
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -1,20 +1,12 @@
 from sage.all import *
 
 from sage.all import *
 

-def is_lyapunov_like(L,K):
+def is_positive_on(L,K):

     r"""
     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.
+    Determine whether or not ``L`` is positive on ``K``.

 
 

-    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.
+    We say that ``L`` is positive on ``K`` if `L\left\lparen x
+    \right\rparen` belongs to ``K`` for all `x` in ``K``. This
+    property need only be checked for generators of ``K``.

 
     INPUT:
 
 
     INPUT:
 


@@ -24,31 +16,36 @@ def is_lyapunov_like(L,K):
 
     OUTPUT:
 
 
     OUTPUT:
 

-    ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
+    ``True`` if it can be proven that ``L`` is positive on ``K``,

     and ``False`` otherwise.
 
     .. WARNING::
 
     and ``False`` otherwise.
 
     .. WARNING::
 

-        If this function returns ``True``, then ``L`` is Lyapunov-like
+        If this function returns ``True``, then ``L`` is positive

         on ``K``. However, if ``False`` is returned, that could mean one
         of two things. The first is that ``L`` is definitely not
         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"
+        positive on ``K``. The second is more of an "I don't know"

         answer, returned (for example) if we cannot prove that an inner
         answer, returned (for example) if we cannot prove that an inner

-        product is zero.
+        product is nonnegative.
+
+    EXAMPLES:

 
 

-    REFERENCES:
+    Positive operators on the nonnegative orthant are nonnegative
+    matrices::

 
 

-    M. Orlitzky. The Lyapunov rank of an improper cone.
-    http://www.optimization-online.org/DB_HTML/2015/10/5135.html
+        sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+        sage: L = random_matrix(QQ,3).apply_map(abs)
+        sage: is_positive_on(L,K)
+        True

 
 

-    EXAMPLES:
+    TESTS:

 
 

-    The identity is always Lyapunov-like in a nontrivial space::
+    The identity is always 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: 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)
+        sage: is_positive_on(L,K)

         True
 
     As is the "zero" transformation::
         True
 
     As is the "zero" transformation::


@@ -56,716 +53,274 @@ def is_lyapunov_like(L,K):
         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: 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)
+        sage: is_positive_on(L,K)

         True
 
         True
 

-        Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
-        on ``K``::
+    Everything in ``K.positive_operators_gens()`` should be
+    positive on ``K``::

 
         sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
 
         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() ])
+        sage: all([ is_positive_on(L,K)
+        ....:       for L in K.positive_operators_gens() ])
+        True
+        sage: all([ is_positive_on(L.change_ring(SR),K)
+        ....:       for L in K.positive_operators_gens() ])

         True
 
     """
         True
 
     """

-    return all([(L*x).inner_product(s) == 0
-                for (x,s) in K.discrete_complementarity_set()])
-
-
-def positive_operator_gens(K1, K2 = None):
+    if L.base_ring().is_exact():
+        # This could potentially be extended to other types of ``K``...
+        return all([ L*x in K for x in K ])
+    elif L.base_ring() is SR:
+        # Fall back to inequality-checking when the entries of ``L``
+        # might be symbolic.
+        return all([ s*(L*x) >= 0 for x in K for s in K.dual() ])
+    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_cross_positive_on(L,K):

     r"""
     r"""

-    Compute generators of the cone of positive operators on this cone. A
-    linear operator on a cone is positive if the image of the cone under
-    the operator is a subset of the cone. This concept can be extended
-    to two cones, where the image of the first cone under a positive
-    operator is a subset of the second cone.
+    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 \ge 0` for all pairs
+    `\left\langle x,s \right\rangle` in the complementarity set of
+    ``K``. This property need only be checked for generators of
+    ``K`` and its dual.

 
     INPUT:
 
 
     INPUT:
 

-    - ``K2`` -- (default: ``K1``) the codomain cone; the image of this
-                cone under the returned operators is a subset of ``K2``.
+    - ``L`` -- A linear transformation or matrix.

 
 

-    OUTPUT:
+    - ``K`` -- A polyhedral closed convex cone.

 
 

-    A list of `m`-by-``n`` matrices where ``m == K2.lattice_dim()`` and
-    ``n == K1.lattice_dim()``. Each matrix ``P`` in the list should have
-    the property that ``P*x`` is an element of ``K2`` whenever ``x`` is
-    an element of ``K1``. Moreover, any nonnegative linear combination of
-    these matrices shares the same property.
+    OUTPUT:

 
 

-    REFERENCES:
+    ``True`` if it can be proven that ``L`` is cross-positive on ``K``,
+    and ``False`` otherwise.

 
 

-    .. [Orlitzky-Pi-Z]
-       M. Orlitzky.
-       Positive and Z-operators on closed convex cones.
+    .. WARNING::

 
 

-    .. [Tam]
-       B.-S. Tam.
-       Some results of polyhedral cones and simplicial cones.
-       Linear and Multilinear Algebra, 4:4 (1977) 281--284.
+        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:
 
 
     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::
+    The identity is always cross-positive in a nontrivial space::

 
 

-        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()
+        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_on(L,K)

         True
         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]]
+    As is the "zero" transformation::

 
 

-        sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
-        sage: K.is_full_space()
+        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_on(L,K)

         True
         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:
 
 
     TESTS:
 

-    Each positive operator generator should send the generators of one
-    cone into the other cone::
-
-        sage: set_random_seed()
-        sage: K1 = random_cone(max_ambient_dim=4)
-        sage: K2 = random_cone(max_ambient_dim=4)
-        sage: pi_K1_K2 = positive_operator_gens(K1,K2)
-        sage: all([ K2.contains(P*x) for P in pi_K1_K2 for x in K1 ])
-        True
-
-    Each positive operator generator should send a random element of one
-    cone into the other cone::
+    Everything in ``K.cross_positive_operators_gens()`` should be
+    cross-positive on ``K``::

 
 

-        sage: set_random_seed()
-        sage: K1 = random_cone(max_ambient_dim=4)
-        sage: K2 = random_cone(max_ambient_dim=4)
-        sage: pi_K1_K2 = positive_operator_gens(K1,K2)
-        sage: all([ K2.contains(P*K1.random_element(QQ)) for P in pi_K1_K2 ])
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
+        sage: all([ is_cross_positive_on(L,K)
+        ....:       for L in K.cross_positive_operators_gens() ])

         True
         True

-
-    A random element of the positive operator cone should send the
-    generators of one cone into the other cone::
-
-        sage: set_random_seed()
-        sage: K1 = random_cone(max_ambient_dim=4)
-        sage: K2 = random_cone(max_ambient_dim=4)
-        sage: pi_K1_K2 = positive_operator_gens(K1,K2)
-        sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim())
-        sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ],
-        ....:                lattice=L,
-        ....:                check=False)
-        sage: P = matrix(K2.lattice_dim(),
-        ....:            K1.lattice_dim(),
-        ....:            pi_cone.random_element(QQ).list())
-        sage: all([ K2.contains(P*x) for x in K1 ])
+        sage: all([ is_cross_positive_on(L.change_ring(SR),K)
+        ....:       for L in K.cross_positive_operators_gens() ])

         True
 
         True
 

-    A random element of the positive operator cone should send a random
-    element of one cone into the other cone::
-
-        sage: set_random_seed()
-        sage: K1 = random_cone(max_ambient_dim=4)
-        sage: K2 = random_cone(max_ambient_dim=4)
-        sage: pi_K1_K2 = positive_operator_gens(K1,K2)
-        sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim())
-        sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ],
-        ....:                lattice=L,
-        ....:                check=False)
-        sage: P = matrix(K2.lattice_dim(),
-        ....:            K1.lattice_dim(),
-        ....:            pi_cone.random_element(QQ).list())
-        sage: K2.contains(P*K1.random_element(ring=QQ))
-        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')

 
 

-    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
+def is_Z_on(L,K):
+    r"""
+    Determine whether or not ``L`` is a Z-operator on ``K``.

 
 

-    The lineality of the dual of the cone of positive operators
-    is known from its lineality space::
+    We say that ``L`` is a Z-operator on ``K`` if `\left\langle
+    L\left\lparenx\right\rparen,s\right\rangle \le 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.

 
 

-        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
+    A matrix is a Z-operator on ``K`` if and only if its negation is a
+    cross-positive operator on ``K``.

 
 

-    The dimension of the cone of positive operators is given by the
-    corollary in my paper::
+    INPUT:

 
 

-        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
+    - ``L`` -- A linear transformation or matrix.

 
 

-    The trivial cone, full space, and half-plane all give rise to the
-    expected dimensions::
+    - ``K`` -- A polyhedral closed convex cone.

 
 

-        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
+    OUTPUT:

 
 

-    The lineality of the cone of positive operators follows from the
-    description of its generators::
+    ``True`` if it can be proven that ``L`` is a Z-operator on ``K``,
+    and ``False`` otherwise.

 
 

-        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
+    .. WARNING::

 
 

-    The trivial cone, full space, and half-plane all give rise to the
-    expected linealities::
+        If this function returns ``True``, then ``L`` is a Z-operator
+        on ``K``. However, if ``False`` is returned, that could mean one
+        of two things. The first is that ``L`` is definitely not
+        a Z-operator 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.

 
 

-        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
+    EXAMPLES:

 
 

-    A cone is proper if and only if its cone of positive operators
-    is proper::
+    The identity is always a Z-operator in a nontrivial space::

 
         sage: set_random_seed()
 
         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()
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+        sage: L = identity_matrix(K.lattice_dim())
+        sage: is_Z_on(L,K)

         True
 
         True
 

-    The positive operators of a permuted cone can be obtained by
-    conjugation::
+    As is the "zero" transformation::

 
 

-        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)
+        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_Z_on(L,K)

         True
 
         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
+    TESTS:

 
 

-        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
+    Everything in ``K.Z_operators_gens()`` should be a Z-operator
+    on ``K``::

 
 

-    The Lyapunov rank of the positive operator cone is the product of
-    the Lyapunov ranks of the associated cones if they're all proper::
-
-        sage: K1 = random_cone(max_ambient_dim=4,
-        ....:                  strictly_convex=True,
-        ....:                  solid=True)
-        sage: K2 = random_cone(max_ambient_dim=4,
-        ....:                  strictly_convex=True,
-        ....:                  solid=True)
-        sage: pi_K1_K2 = positive_operator_gens(K1,K2)
-        sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim())
-        sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ],
-        ....:                lattice=L,
-        ....:                check=False)
-        sage: beta1 = K1.lyapunov_rank()
-        sage: beta2 = K2.lyapunov_rank()
-        sage: pi_cone.lyapunov_rank() == beta1*beta2
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
+        sage: all([ is_Z_on(L,K)
+        ....:       for L in K.Z_operators_gens() ])

         True
         True

-
-    The Lyapunov-like operators on a proper polyhedral positive operator
-    cone can be computed from the Lyapunov-like operators on the cones
-    with respect to which the operators are positive::
-
-        sage: K1 = random_cone(max_ambient_dim=4,
-        ....:                  strictly_convex=True,
-        ....:                  solid=True)
-        sage: K2 = random_cone(max_ambient_dim=4,
-        ....:                  strictly_convex=True,
-        ....:                  solid=True)
-        sage: pi_K1_K2 = positive_operator_gens(K1,K2)
-        sage: F = K1.lattice().base_field()
-        sage: m = K1.lattice_dim()
-        sage: n = K2.lattice_dim()
-        sage: L = ToricLattice(m*n)
-        sage: M1 = MatrixSpace(F, m, m)
-        sage: M2 = MatrixSpace(F, n, n)
-        sage: LL_K1 = [ M1(x.list()) for x in K1.dual().lyapunov_like_basis() ]
-        sage: LL_K2 = [ M2(x.list()) for x in K2.lyapunov_like_basis() ]
-        sage: tps = [ s.tensor_product(x) for x in LL_K1 for s in LL_K2 ]
-        sage: W = VectorSpace(F, (m**2)*(n**2))
-        sage: expected = span(F, [ W(x.list()) for x in tps ])
-        sage: pi_cone = Cone([p.list() for p in pi_K1_K2],
-        ....:                 lattice=L,
-        ....:                 check=False)
-        sage: LL_pi = pi_cone.lyapunov_like_basis()
-        sage: actual = span(F, [ W(x.list()) for x in LL_pi ])
-        sage: actual == expected
+        sage: all([ is_Z_on(L.change_ring(SR),K)
+        ....:       for L in K.Z_operators_gens() ])

         True
 
     """
         True
 
     """

-    if K2 is None:
-        K2 = K1
-
-    # 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 = K1.lattice().base_field()
-    n = K1.lattice_dim()
-    m = K2.lattice_dim()
-
-    tensor_products = [ s.tensor_product(x) for x in K1 for s in K2.dual() ]
-
-    # Convert those tensor products to long vectors.
-    W = VectorSpace(F, n*m)
-    vectors = [ W(tp.list()) for tp in tensor_products ]
+    return is_cross_positive_on(-L,K)

 
 

-    check = True
-    if K1.is_proper() and K2.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)
+def is_lyapunov_like_on(L,K):
+    r"""
+    Determine whether or not ``L`` is Lyapunov-like on ``K``.

 
 

-    # Now compute the desired cone from its dual...
-    pi_cone = pi_dual.dual()
+    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``. This property need only be checked for generators of
+    ``K`` and its dual.

 
 

-    # And finally convert its rays back to matrix representations.
-    M = MatrixSpace(F, m, n)
-    return [ M(v.list()) for v in pi_cone ]
+    INPUT:

 
 

+    - ``L`` -- A linear transformation or matrix.

 
 

-def Z_operator_gens(K):
-    r"""
-    Compute generators of the cone of Z-operators on this cone.
+    - ``K`` -- A polyhedral closed convex cone.

 
     OUTPUT:
 
 
     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.
+    ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
+    and ``False`` otherwise.

 
 

-    REFERENCES:
+    .. WARNING::

 
 

-    M. Orlitzky.
-    Positive and Z-operators on closed convex cones.
+        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.

 
     EXAMPLES:
 
 
     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)
-        []
+    Lyapunov-like operators on the nonnegative orthant are diagonal
+    matrices::

 
 

-    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
+        sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+        sage: L = diagonal_matrix(random_vector(QQ,3))
+        sage: is_lyapunov_like_on(L,K)

         True
 
     TESTS:
 
         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::
+    The identity is always Lyapunov-like in a nontrivial space::

 
         sage: set_random_seed()
 
         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()
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+        sage: L = identity_matrix(K.lattice_dim())
+        sage: is_lyapunov_like_on(L,K)

         True
 
         True
 

-    The trivial cone, full space, and half-plane all give rise to the
-    expected dimensions::
+    As is the "zero" transformation::

 
 

-        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
+        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_on(L,K)

         True
 
         True
 

-    The Z-operators of a permuted cone can be obtained by conjugation::
+    Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
+    on ``K``::

 
 

-        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)
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
+        sage: all([ is_lyapunov_like_on(L,K)
+        ....:       for L in K.lyapunov_like_basis() ])

         True
         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()))
+        sage: all([ is_lyapunov_like_on(L.change_ring(SR),K)
+        ....:       for L in K.lyapunov_like_basis() ])

         True
 
         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 ]
-
+    if L.base_ring().is_exact() or L.base_ring() is SR:
+        # The "fast method" of creating a vector space based on a
+        # ``lyapunov_like_basis`` is actually slower than this.
+        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 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 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)
 
     L = ToricLattice(K.lattice_dim()**2)
     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)
     L = ToricLattice(K.lattice_dim()**2)
     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)