Add positive/Z tests and update code for upstream changes.

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

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 5f058ca9a8c055f972ffffaa216975517e0dc723..a7c16a520cbe19242398e176635ad3250e5c7671 100644 (file)
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -1,388 +1,164 @@
-# Sage doesn't load ~/.sage/init.sage during testing (sage -t), so we
-# have to explicitly mangle our sitedir here so that "mjo.cone"
-# resolves.
-from os.path import abspath
-from site import addsitedir
-addsitedir(abspath('../../'))
-

 from sage.all import *
 
 from sage.all import *
 

-
-def _restrict_to_space(K, W):
+def is_positive_on(L,K):

     r"""
     r"""

-    Restrict this cone a subspace of its ambient space.
-
-    INPUT:
-
-    - ``W`` -- The subspace into which this cone will be restricted.
-
-    OUTPUT:
-
-    A new cone in a sublattice corresponding to ``W``.
+    Determine whether or not ``L`` is positive on ``K``.

 
 

-    EXAMPLES:
+    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``.

 
 

-    When this cone is solid, restricting it into its own span should do
-    nothing::
+    INPUT:

 
 

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

 
 

-    A single ray restricted into its own span gives the same output
-    regardless of the ambient space::
-
-        sage: K2 = Cone([(1,0)])
-        sage: K2_S = _restrict_to_space(K2, K2.span()).rays()
-        sage: K2_S
-        N(1)
-        in 1-d lattice N
-        sage: K3 = Cone([(1,0,0)])
-        sage: K3_S = _restrict_to_space(K3, K3.span()).rays()
-        sage: K3_S
-        N(1)
-        in 1-d lattice N
-        sage: K2_S == K3_S
-        True
+    - ``K`` -- A polyhedral closed convex cone.

 
 

-    TESTS:
+    OUTPUT:

 
 

-    The projected cone should always be solid::
+    ``True`` if it can be proven that ``L`` is positive on ``K``,
+    and ``False`` otherwise.

 
 

-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 8)
-        sage: _restrict_to_space(K, K.span()).is_solid()
-        True
+    .. WARNING::

 
 

-    And the resulting cone should live in a space having the same
-    dimension as the space we restricted it to::
+        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
+        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.

 
 

-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 8)
-        sage: K_P = _restrict_to_space(K, K.dual().span())
-        sage: K_P.lattice_dim() == K.dual().dim()
-        True
+    EXAMPLES:

 
 

-    This function should not affect the dimension of a cone::
+    The identity is always positive in a nontrivial space::

 
         sage: set_random_seed()
 
         sage: set_random_seed()

-        sage: K = random_cone(max_ambient_dim = 8)
-        sage: K.dim() == _restrict_to_space(K,K.span()).dim()
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
+        sage: L = identity_matrix(K.lattice_dim())
+        sage: is_positive_on(L,K)

         True
 
         True
 

-    Nor should it affect the lineality of a cone::
+    As is the "zero" transformation::

 
 

-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 8)
-        sage: K.lineality() == _restrict_to_space(K, K.span()).lineality()
+        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_positive_on(L,K)

         True
 
         True
 

-    No matter which space we restrict to, the lineality should not
-    increase::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 8)
-        sage: S = K.span(); P = K.dual().span()
-        sage: K.lineality() >= _restrict_to_space(K,S).lineality()
-        True
-        sage: K.lineality() >= _restrict_to_space(K,P).lineality()
-        True
+    TESTS:

 
 

-    If we do this according to our paper, then the result is proper::
+    Everything in ``K.positive_operators_gens()`` should be
+    positive on ``K``::

 
 

-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 8)
-        sage: K_S = _restrict_to_space(K, K.span())
-        sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
-        sage: K_SP.is_proper()
-        True
-        sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
-        sage: K_SP.is_proper()
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
+        sage: all([ is_positive_on(L,K)
+        ....:       for L in K.positive_operators_gens() ])

         True
         True

-
-    Test the proposition in our paper concerning the duals and
-    restrictions. Generate a random cone, then create a subcone of
-    it. The operation of dual-taking should then commute with
-    _restrict_to_space::
-
-        sage: set_random_seed()
-        sage: J = random_cone(max_ambient_dim = 8)
-        sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
-        sage: K_W_star = _restrict_to_space(K, J.span()).dual()
-        sage: K_star_W = _restrict_to_space(K.dual(), J.span())
-        sage: _basically_the_same(K_W_star, K_star_W)
+        sage: all([ is_positive_on(L.change_ring(SR),K)
+        ....:       for L in K.positive_operators_gens() ])

         True
 
     """
         True
 
     """

-    # First we want to intersect ``K`` with ``W``. The easiest way to
-    # do this is via cone intersection, so we turn the subspace ``W``
-    # into a cone.
-    W_cone = Cone(W.basis() + [-b for b in W.basis()], lattice=K.lattice())
-    K = K.intersection(W_cone)
-
-    # We've already intersected K with the span of K2, so every
-    # generator of K should belong to W now.
-    K_W_rays = [ W.coordinate_vector(r) for r in K.rays() ]
-
-    L = ToricLattice(W.dimension())
-    return Cone(K_W_rays, lattice=L)
-
-
-def lyapunov_rank(K):
+    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 ])
+    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 the Lyapunov rank of this cone.
-
-    The Lyapunov rank of a cone is the dimension of the space of its
-    Lyapunov-like transformations -- that is, the length of a
-    :meth:`lyapunov_like_basis`. Equivalently, the Lyapunov rank is the
-    dimension of the Lie algebra of the automorphism group of the cone.
+    Determine whether or not ``L`` is cross-positive on ``K``.

 
 

-    OUTPUT:
-
-    A nonnegative integer representing the Lyapunov rank of this cone.
+    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.

 
 

-    If the ambient space is trivial, the Lyapunov rank will be zero.
-    Otherwise, if the dimension of the ambient vector space is `n`, then
-    the resulting Lyapunov rank will be between `1` and `n` inclusive. A
-    Lyapunov rank of `n-1` is not possible [Orlitzky]_.
+    INPUT:

 
 

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

 
 

-    The codimension formula from the second reference is used. We find
-    all pairs `(x,s)` in the complementarity set of `K` such that `x`
-    and `s` are rays of our cone. It is known that these vectors are
-    sufficient to apply the codimension formula. Once we have all such
-    pairs, we "brute force" the codimension formula by finding all
-    linearly-independent `xs^{T}`.
+    - ``K`` -- A polyhedral closed convex cone.

 
 

-    REFERENCES:
+    OUTPUT:

 
 

-    .. [Gowda/Tao] M.S. Gowda and J. Tao. On the bilinearity rank of
-       a proper cone and Lyapunov-like transformations. Mathematical
-       Programming, 147 (2014) 155-170.
+    ``True`` if it can be proven that ``L`` is cross-positive on ``K``,
+    and ``False`` otherwise.

 
 

-    M. Orlitzky. The Lyapunov rank of an improper cone.
-    http://www.optimization-online.org/DB_HTML/2015/10/5135.html
+    .. WARNING::

 
 

-    G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear
-    optimality constraints for the cone of positive polynomials,
-    Mathematical Programming, Series B, 129 (2011) 5-31.
+        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 in `\mathbb{R}^{n}` always has rank `n`
-    [Rudolf]_::
-
-        sage: positives = Cone([(1,)])
-        sage: lyapunov_rank(positives)
-        1
-        sage: quadrant = Cone([(1,0), (0,1)])
-        sage: lyapunov_rank(quadrant)
-        2
-       sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
-        sage: lyapunov_rank(octant)
-        3
-
-    The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}`
-    [Orlitzky]_::
-
-        sage: R5 = VectorSpace(QQ, 5)
-        sage: gs = R5.basis() + [ -r for r in R5.basis() ]
-        sage: K = Cone(gs)
-        sage: lyapunov_rank(K)
-        25
-
-    The `L^{3}_{1}` cone is known to have a Lyapunov rank of one
-    [Rudolf]_::
-
-        sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
-        sage: lyapunov_rank(L31)
-        1
-
-    Likewise for the `L^{3}_{\infty}` cone [Rudolf]_::
-
-        sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
-        sage: lyapunov_rank(L3infty)
-        1
-
-    A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n
-    + 1` [Orlitzky]_::
-
-        sage: K = Cone([(1,0,0,0,0)])
-        sage: lyapunov_rank(K)
-        21
-        sage: K.lattice_dim()**2 - K.lattice_dim() + 1
-        21
-
-    A subspace (of dimension `m`) in `n` dimensions should have a
-    Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky]_::
-
-        sage: e1 = (1,0,0,0,0)
-        sage: neg_e1 = (-1,0,0,0,0)
-        sage: e2 = (0,1,0,0,0)
-        sage: neg_e2 = (0,-1,0,0,0)
-        sage: z = (0,0,0,0,0)
-        sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z])
-        sage: lyapunov_rank(K)
-        19
-        sage: K.lattice_dim()**2 - K.dim()*K.codim()
-        19
-
-    The Lyapunov rank should be additive on a product of proper cones
-    [Rudolf]_::
-
-        sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
-        sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)])
-        sage: K = L31.cartesian_product(octant)
-        sage: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant)
-        True
-
-    Two isomorphic cones should have the same Lyapunov rank [Rudolf]_.
-    The cone ``K`` in the following example is isomorphic to the nonnegative
-    octant in `\mathbb{R}^{3}`::
-
-        sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)])
-        sage: lyapunov_rank(K)
-        3
-
-    The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
-    itself [Rudolf]_::
-
-        sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)])
-        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
-        True
-
-    TESTS:
-
-    The Lyapunov rank should be additive on a product of proper cones
-    [Rudolf]_::
-
-        sage: set_random_seed()
-        sage: K1 = random_cone(max_ambient_dim=8,
-        ....:                  strictly_convex=True,
-        ....:                  solid=True)
-        sage: K2 = random_cone(max_ambient_dim=8,
-        ....:                  strictly_convex=True,
-        ....:                  solid=True)
-        sage: K = K1.cartesian_product(K2)
-        sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
-        True
-
-    The Lyapunov rank is invariant under a linear isomorphism
-    [Orlitzky]_::
-
-        sage: K1 = random_cone(max_ambient_dim = 8)
-        sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
-        sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
-        sage: lyapunov_rank(K1) == lyapunov_rank(K2)
-        True
-
-    The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
-    itself [Rudolf]_::
+    The identity is always cross-positive in a nontrivial space::

 
         sage: set_random_seed()
 
         sage: set_random_seed()

-        sage: K = random_cone(max_ambient_dim=8)
-        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+        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
 

-    The Lyapunov rank of a proper polyhedral cone in `n` dimensions can
-    be any number between `1` and `n` inclusive, excluding `n-1`
-    [Gowda/Tao]_. By accident, the `n-1` restriction will hold for the
-    trivial cone in a trivial space as well. However, in zero dimensions,
-    the Lyapunov rank of the trivial cone will be zero::
+    As is the "zero" transformation::

 
 

-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8,
-        ....:                 strictly_convex=True,
-        ....:                 solid=True)
-        sage: b = lyapunov_rank(K)
-        sage: n = K.lattice_dim()
-        sage: (n == 0 or 1 <= b) and b <= n
+        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: b == n-1
-        False
-
-    In fact [Orlitzky]_, no closed convex polyhedral cone can have
-    Lyapunov rank `n-1` in `n` dimensions::

 
 

-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8)
-        sage: b = lyapunov_rank(K)
-        sage: n = K.lattice_dim()
-        sage: b == n-1
-        False
+    TESTS:

 
 

-    The calculation of the Lyapunov rank of an improper cone can be
-    reduced to that of a proper cone [Orlitzky]_::
+    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: actual = lyapunov_rank(K)
-        sage: K_S = _restrict_to_space(K, K.span())
-        sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
-        sage: l = K.lineality()
-        sage: c = K.codim()
-        sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2
-        sage: actual == expected
+        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

-
-    The Lyapunov rank of a cone is the size of a :meth:`lyapunov_like_basis`::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8)
-        sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
-        True
-
-    We can make an imperfect cone perfect by adding a slack variable
-    (a Theorem in [Orlitzky]_)::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8,
-        ....:                 strictly_convex=True,
-        ....:                 solid=True)
-        sage: L = ToricLattice(K.lattice_dim() + 1)
-        sage: K = Cone([ r.list() + [0] for r in K.rays() ], lattice=L)
-        sage: lyapunov_rank(K) >= K.lattice_dim()
+        sage: all([ is_cross_positive_on(L.change_ring(SR),K)
+        ....:       for L in K.cross_positive_operators_gens() ])

         True
 
     """
         True
 
     """

-    beta = 0 # running tally of the Lyapunov rank
-
-    m = K.dim()
-    n = K.lattice_dim()
-    l = K.lineality()
-
-    if m < n:
-        # K is not solid, restrict to its span.
-        K = _restrict_to_space(K, K.span())
+    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')

 
 

-        # Non-solid reduction lemma.
-        beta += (n - m)*n

 
 

-    if l > 0:
-        # K is not pointed, restrict to the span of its dual. Uses a
-        # proposition from our paper, i.e. this is equivalent to K =
-        # _rho(K.dual()).dual().
-        K = _restrict_to_space(K, K.dual().span())
-
-        # Non-pointed reduction lemma.
-        beta += l * m
-
-    beta += len(K.lyapunov_like_basis())
-    return beta
-
-
-
-def is_lyapunov_like(L,K):
+def is_Z_on(L,K):

     r"""
     r"""

-    Determine whether or not ``L`` is Lyapunov-like on ``K``.
+    Determine whether or not ``L`` is a Z-operator 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
+    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
     `\left\langle x,s \right\rangle` in the complementarity set of

-    ``K``. It is known [Orlitzky]_ that this property need only be
+    ``K``. It is known that this property need only be

     checked for generators of ``K`` and its dual.
 
     checked for generators of ``K`` and its dual.
 

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

     INPUT:
 
     - ``L`` -- A linear transformation or matrix.
     INPUT:
 
     - ``L`` -- A linear transformation or matrix.


@@ -391,296 +167,144 @@ 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 a Z-operator 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 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
         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"
+        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
         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
+        product is nonnegative.

 
     EXAMPLES:
 
 
     EXAMPLES:
 

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

 
         sage: set_random_seed()
 
         sage: set_random_seed()

-        sage: K = random_cone(min_ambient_dim = 1, max_rays = 8)
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)

         sage: L = identity_matrix(K.lattice_dim())
         sage: L = identity_matrix(K.lattice_dim())

-        sage: is_lyapunov_like(L,K)
+        sage: is_Z_on(L,K)

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

-        sage: K = random_cone(min_ambient_dim = 1, max_rays = 5)
+        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: R = K.lattice().vector_space().base_ring()
         sage: L = zero_matrix(R, K.lattice_dim())

-        sage: is_lyapunov_like(L,K)
+        sage: is_Z_on(L,K)

         True
 
         True
 

-        Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
-        on ``K``::
-
-        sage: K = random_cone(min_ambient_dim = 1, max_rays = 5)
-        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 random_element(K):
-    r"""
-    Return a random element of ``K`` from its ambient vector space.
-
-    ALGORITHM:
-
-    The cone ``K`` is specified in terms of its generators, so that
-    ``K`` is equal to the convex conic combination of those generators.
-    To choose a random element of ``K``, we assign random nonnegative
-    coefficients to each generator of ``K`` and construct a new vector
-    from the scaled rays.
-
-    A vector, rather than a ray, is returned so that the element may
-    have non-integer coordinates. Thus the element may have an
-    arbitrarily small norm.
-
-    EXAMPLES:
-
-    A random element of the trivial cone is zero::
-
-        sage: set_random_seed()
-        sage: K = Cone([], ToricLattice(0))
-        sage: random_element(K)
-        ()
-        sage: K = Cone([(0,)])
-        sage: random_element(K)
-        (0)
-        sage: K = Cone([(0,0)])
-        sage: random_element(K)
-        (0, 0)
-        sage: K = Cone([(0,0,0)])
-        sage: random_element(K)
-        (0, 0, 0)
-

     TESTS:
 
     TESTS:
 

-    Any cone should contain an element of itself::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_rays = 8)
-        sage: K.contains(random_element(K))
-        True
-
-    """
-    V = K.lattice().vector_space()
-    F = V.base_ring()
-    coefficients = [ F.random_element().abs() for i in range(K.nrays()) ]
-    vector_gens  = map(V, K.rays())
-    scaled_gens  = [ coefficients[i]*vector_gens[i]
-                         for i in range(len(vector_gens)) ]
-
-    # Make sure we return a vector. Without the coercion, we might
-    # return ``0`` when ``K`` has no rays.
-    v = V(sum(scaled_gens))
-    return v
-
-
-def positive_operators(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.
-
-    EXAMPLES:
-
-    The trivial cone in a trivial space has no positive operators::
-
-        sage: K = Cone([], ToricLattice(0))
-        sage: positive_operators(K)
-        []
-
-    Positive operators on the nonnegative orthant are nonnegative matrices::
-
-        sage: K = Cone([(1,)])
-        sage: positive_operators(K)
-        [[1]]
-
-        sage: K = Cone([(1,0),(0,1)])
-        sage: positive_operators(K)
-        [
-        [1 0]  [0 1]  [0 0]  [0 0]
-        [0 0], [0 0], [1 0], [0 1]
-        ]
-
-    Every operator is positive on the ambient vector space::
+    Everything in ``K.Z_operators_gens()`` should be a Z-operator
+    on ``K``::

 
 

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

-        sage: positive_operators(K)
-        [[1], [-1]]
-
-        sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
-        sage: K.is_full_space()
-        True
-        sage: positive_operators(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]
-        ]
-
-    TESTS:
-
-    A positive operator on a cone should send its generators into the cone::
-
-        sage: K = random_cone(max_ambient_dim = 6)
-        sage: pi_of_K = positive_operators(K)
-        sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
+        sage: all([ is_Z_on(L.change_ring(SR),K)
+        ....:       for L in K.Z_operators_gens() ])

         True
 
     """
         True
 
     """

-    # Sage doesn't think matrices are vectors, so we have to convert
-    # our matrices to vectors explicitly before we can figure out how
-    # many are linearly-indepenedent.
-    #
-    # The space W has the same base ring as V, but dimension
-    # dim(V)^2. So it has the same dimension as the space of linear
-    # transformations on V. In other words, it's just the right size
-    # to create an isomorphism between it and our matrices.
-    V = K.lattice().vector_space()
-    W = VectorSpace(V.base_ring(), V.dimension()**2)
+    return is_cross_positive_on(-L,K)

 
 

-    tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]

 
 

-    # Turn our matrices into long vectors...
-    vectors = [ W(m.list()) for m in tensor_products ]
+def is_lyapunov_like_on(L,K):
+    r"""
+    Determine whether or not ``L`` is Lyapunov-like on ``K``.

 
 

-    # Create the *dual* cone of the positive operators, expressed as
-    # long vectors..
-    L = ToricLattice(W.dimension())
-    pi_dual = Cone(vectors, lattice=L)
+    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.

 
 

-    # Now compute the desired cone from its dual...
-    pi_cone = pi_dual.dual()
+    INPUT:

 
 

-    # And finally convert its rays back to matrix representations.
-    M = MatrixSpace(V.base_ring(), V.dimension())
+    - ``L`` -- A linear transformation or matrix.

 
 

-    return [ M(v.list()) for v in pi_cone.rays() ]
+    - ``K`` -- A polyhedral closed convex cone.

 
 

+    OUTPUT:

 
 

-def Z_transformations(K):
-    r"""
-    Compute generators of the cone of Z-transformations on this cone.
+    ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
+    and ``False`` otherwise.

 
 

-    OUTPUT:
+    .. WARNING::

 
 

-    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.
+        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-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_transformations(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_transformations(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-transformations::
+    The identity is always Lyapunov-like in a nontrivial space::

 
 

-        sage: K = Cone([], ToricLattice(0))
-        sage: Z_transformations(K)
-        []
+        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_on(L,K)
+        True

 
 

-    Z-transformations on a subspace are Lyapunov-like and vice-versa::
+    As is the "zero" transformation::

 
 

-        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_transformations(K) ])
-        sage: zs == 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
 
     TESTS:
 
         True
 
     TESTS:
 

-    The Z-property is possessed by every Z-transformation::
+    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_transformations(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=6)
+        sage: all([ is_lyapunov_like_on(L,K)
+        ....:       for L in K.lyapunov_like_basis() ])

         True
         True

-
-    The lineality space of Z is LL::
-
-        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_transformations(K) ])
-        sage: z_cone.linear_subspace() == lls
+        sage: all([ is_lyapunov_like_on(L.change_ring(SR),K)
+        ....:       for L in K.lyapunov_like_basis() ])

         True
 
     """
         True
 
     """

-    # Sage doesn't think matrices are vectors, so we have to convert
-    # our matrices to vectors explicitly before we can figure out how
-    # many are linearly-indepenedent.
-    #
-    # The space W has the same base ring as V, but dimension
-    # dim(V)^2. So it has the same dimension as the space of linear
-    # transformations on V. In other words, it's just the right size
-    # to create an isomorphism between it and our matrices.
-    V = K.lattice().vector_space()
-    W = VectorSpace(V.base_ring(), V.dimension()**2)
-
-    C_of_K = K.discrete_complementarity_set()
-    tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ]
-
-    # Turn our matrices into long vectors...
-    vectors = [ W(m.list()) for m in tensor_products ]
-
-    # Create the *dual* cone of the cross-positive operators,
-    # expressed as long vectors..
-    L = ToricLattice(W.dimension())
-    Sigma_dual = Cone(vectors, lattice=L)
-
-    # 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(V.base_ring(), V.dimension())
-
-    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):
+    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)