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 8b07f86b329e191e105994e83676f3e30d8c4220..01ef5f9d5cc56b087344237b9b5c915423ac0921 100644 (file)
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -1,14 +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``.
+    Determine whether or not ``L`` is positive 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.
+    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:
 


@@ -18,31 +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.

 
 

-    REFERENCES:
+    EXAMPLES:

 
 

-    M. Orlitzky. The Lyapunov rank of an improper cone.
-    http://www.optimization-online.org/DB_HTML/2015/10/5135.html
+    Positive operators on the nonnegative orthant are nonnegative
+    matrices::

 
 

-    EXAMPLES:
+        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

 
 

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


@@ -50,450 +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()])
+    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"""
+    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:

 
 

-def motzkin_decomposition(K):
-    r"""
-    Return the pair of components in the Motzkin decomposition of this cone.
+    - ``L`` -- A linear transformation or matrix.

 
 

-    Every convex cone is the direct sum of a strictly convex cone and a
-    linear subspace [Stoer-Witzgall]_. Return a pair ``(P,S)`` of cones
-    such that ``P`` is strictly convex, ``S`` is a subspace, and ``K``
-    is the direct sum of ``P`` and ``S``.
+    - ``K`` -- A polyhedral closed convex cone.

 
     OUTPUT:
 
 
     OUTPUT:
 

-    An ordered pair ``(P,S)`` of closed convex polyhedral cones where
-    ``P`` is strictly convex, ``S`` is a subspace, and ``K`` is the
-    direct sum of ``P`` and ``S``.
+    ``True`` if it can be proven that ``L`` is cross-positive on ``K``,
+    and ``False`` otherwise.

 
 

-    REFERENCES:
+    .. WARNING::

 
 

-    .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and
-       Optimization in Finite Dimensions I. Springer-Verlag, New
-       York, 1970.
+        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:
 

-    The nonnegative orthant is strictly convex, so it is its own
-    strictly convex component and its subspace component is trivial::
+    The identity is always cross-positive in a nontrivial space::

 
 

-        sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
-        sage: (P,S) = motzkin_decomposition(K)
-        sage: K.is_equivalent(P)
-        True
-        sage: S.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
 

-    Likewise, full spaces are their own subspace components::
+    As is the "zero" transformation::

 
 

-        sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
-        sage: K.is_full_space()
-        True
-        sage: (P,S) = motzkin_decomposition(K)
-        sage: K.is_equivalent(S)
-        True
-        sage: P.is_trivial()
+        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
 
     TESTS:
 
         True
 
     TESTS:
 

-    A random point in the cone should belong to either the strictly
-    convex component or the subspace component. If the point is nonzero,
-    it cannot be in both::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8)
-        sage: (P,S) = motzkin_decomposition(K)
-        sage: x = K.random_element()
-        sage: P.contains(x) or S.contains(x)
-        True
-        sage: x.is_zero() or (P.contains(x) != S.contains(x))
-        True
-
-    The strictly convex component should always be strictly convex, and
-    the subspace component should always be a subspace::
+    Everything in ``K.cross_positive_operators_gens()`` should be
+    cross-positive on ``K``::

 
 

-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8)
-        sage: (P,S) = motzkin_decomposition(K)
-        sage: P.is_strictly_convex()
+        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

-        sage: S.lineality() == S.dim()
+        sage: all([ is_cross_positive_on(L.change_ring(SR),K)
+        ....:       for L in K.cross_positive_operators_gens() ])

         True
 
         True
 

-    The generators of the components are obtained from orthogonal
-    projections of the original generators [Stoer-Witzgall]_::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8)
-        sage: (P,S) = motzkin_decomposition(K)
-        sage: A = S.linear_subspace().complement().matrix()
-        sage: proj_S_perp = A.transpose() * (A*A.transpose()).inverse() * A
-        sage: expected_P = Cone([ proj_S_perp*g for g in K ], K.lattice())
-        sage: P.is_equivalent(expected_P)
-        True
-        sage: A = S.linear_subspace().matrix()
-        sage: proj_S = A.transpose() * (A*A.transpose()).inverse() * A
-        sage: expected_S = Cone([ proj_S*g for g in K ], K.lattice())
-        sage: S.is_equivalent(expected_S)
-        True

     """
     """

-    # The lines() method only returns one generator per line. For a true
-    # line, we also need a generator pointing in the opposite direction.
-    S_gens = [ direction*gen for direction in [1,-1] for gen in K.lines() ]
-    S = Cone(S_gens, K.lattice())
-
-    # Since ``S`` is a subspace, the rays of its dual generate its
-    # orthogonal complement.
-    S_perp = Cone(S.dual(), K.lattice())
-    P = K.intersection(S_perp)
-
-    return (P,S)
+    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 positive_operator_gens(K):
+def is_Z_on(L,K):

     r"""
     r"""

-    Compute generators of the cone of positive operators on this cone.
+    Determine whether or not ``L`` is a Z-operator on ``K``.

 
 

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

 
 

-    Every operator is positive on the ambient vector space::
+    INPUT:

 
 

-        sage: K = Cone([(1,),(-1,)])
-        sage: K.is_full_space()
-        True
-        sage: positive_operator_gens(K)
-        [[1], [-1]]
+    - ``L`` -- A linear transformation or matrix.

 
 

-        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]
-        ]
+    - ``K`` -- A polyhedral closed convex cone.

 
 

-    TESTS:
+    OUTPUT:

 
 

-    Each positive operator generator should send the generators of the
-    cone into the cone::
+    ``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=5)
-        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
+    .. WARNING::

 
 

-    Each positive operator generator should send a random element of the
-    cone into the cone::
+        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: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=5)
-        sage: pi_of_K = positive_operator_gens(K)
-        sage: all([ K.contains(P*K.random_element()) for P in pi_of_K ])
-        True
+    EXAMPLES:

 
 

-    A random element of the positive operator cone should send the
-    generators of the cone into the cone::
+    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=5)
-        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)
-        sage: P = matrix(K.lattice_dim(), pi_cone.random_element().list())
-        sage: all([ K.contains(P*x) for x in K ])
+        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
 

-    A random element of the positive operator cone should send a random
-    element of the cone into the cone::
+    As is the "zero" transformation::

 
 

-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=5)
-        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)
-        sage: P = matrix(K.lattice_dim(), pi_cone.random_element().list())
-        sage: K.contains(P*K.random_element())
+        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
 

-    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=5)
-        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)
-        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
+    TESTS:

 
 

-    The dimension of the cone of positive operators is given by the
-    corollary in my paper::
+    Everything in ``K.Z_operators_gens()`` should be a Z-operator
+    on ``K``::

 
 

-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=5)
-        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: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
-        sage: expected = n**2 - l*(m - l) - (n - m)*m
-        sage: actual == expected
+        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 lineality 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=5)
-        sage: n = K.lattice_dim()
-        sage: pi_of_K = positive_operator_gens(K)
-        sage: L = ToricLattice(n**2)
-        sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
-        sage: expected = n**2 - K.dim()*K.dual().dim()
-        sage: actual == expected
+        sage: all([ is_Z_on(L.change_ring(SR),K)
+        ....:       for L in K.Z_operators_gens() ])

         True
 
         True
 

-    The cone ``K`` is proper if and only if the cone of positive
-    operators on ``K`` is proper::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=5)
-        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)
-        sage: K.is_proper() == pi_cone.is_proper()
-        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()
+    return is_cross_positive_on(-L,K)

 
 

-    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 ]
-
-    # Create the *dual* cone of the positive operators, expressed as
-    # long vectors..
-    pi_dual = Cone(vectors, ToricLattice(W.dimension()))
+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, n)
-    return [ M(v.list()) for v in pi_cone.rays() ]
+    INPUT:

 
 

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

 
 

-def Z_transformation_gens(K):
-    r"""
-    Compute generators of the cone of Z-transformations 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 the
-    discrete complementarity set of ``K``. Moreover, any nonnegative
-    linear combination of these matrices shares the same property.
-
-    EXAMPLES:
+    ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
+    and ``False`` otherwise.

 
 

-    Z-transformations 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_transformation_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_transformation_gens(K)
-        ....:                    for i in range(z.nrows())
-        ....:                    for j in range(z.ncols())
-        ....:                    if i != j ])
-        True
+    .. WARNING::

 
 

-    The trivial cone in a trivial space has no Z-transformations::
+        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.

 
 

-        sage: K = Cone([], ToricLattice(0))
-        sage: Z_transformation_gens(K)
-        []
+    EXAMPLES:

 
 

-    Z-transformations on a subspace are Lyapunov-like and vice-versa::
+    Lyapunov-like operators on the nonnegative orthant are diagonal
+    matrices::

 
 

-        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_transformation_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-transformation::
+    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=6)
-        sage: Z_of_K = Z_transformation_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])
+        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 lineality space of Z is LL::
+    As is the "zero" transformation::

 
 

-        sage: set_random_seed()
-        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
-        sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
-        sage: z_cone  = Cone([ z.list() for z in Z_transformation_gens(K) ])
-        sage: z_cone.linear_subspace() == lls
+        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
 

-    And thus, the lineality of Z is the Lyapunov rank::
+    Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
+    on ``K``::

 
 

-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=6)
-        sage: Z_of_K = Z_transformation_gens(K)
-        sage: L = ToricLattice(K.lattice_dim()**2)
-        sage: z_cone  = Cone([ z.list() for z in Z_of_K ], lattice=L)
-        sage: z_cone.lineality() == K.lyapunov_rank()
+        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

-
-    The lineality spaces of pi-star and Z-star are equal:
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=5)
-        sage: pi_of_K = positive_operator_gens(K)
-        sage: Z_of_K = Z_transformation_gens(K)
-        sage: L = ToricLattice(K.lattice_dim()**2)
-        sage: pi_star = Cone([p.list() for p in pi_of_K], lattice=L).dual()
-        sage: z_star  = Cone([ z.list() for z in Z_of_K], lattice=L).dual()
-        sage: pi_star.linear_subspace() == z_star.linear_subspace()
+        sage: all([ is_lyapunov_like_on(L.change_ring(SR),K)
+        ....:       for L in K.lyapunov_like_basis() ])

         True
         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 transformations.
-    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 ]
-
-    # Create the *dual* cone of the cross-positive operators,
-    # expressed as long vectors..
-    Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()))
-
-    # 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-transformations and
-    # not cross-positive ones.
-    M = MatrixSpace(F, n)
-    return [ -M(v.list()) for v in Sigma_cone.rays() ]

 
 

+    """
+    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 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)

 
 def Z_cone(K):
 
 def Z_cone(K):

-    gens = Z_transformation_gens(K)
-    L = None
-    if len(gens) == 0:
-        L = ToricLattice(0)
-    return Cone([ g.list() for g in gens ], lattice=L)
-
-def pi_cone(K):
-    gens = positive_operator_gens(K)
-    L = None
-    if len(gens) == 0:
-        L = ToricLattice(0)
-    return Cone([ g.list() for g in gens ], lattice=L)
+    gens = K.Z_operators_gens()
+    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)