From: Michael Orlitzky <michael@orlitzky.com>
Date: Tue, 13 Oct 2015 00:22:50 +0000 (-0400)
Subject: Remove lyapunov_rank() for inclusion in Sage.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=3cfc8e228ae337aed975118444a8cbad9a5a7ac3;p=sage.d.git

Remove lyapunov_rank() for inclusion in Sage.
---

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 6d7d2d9..32e1386 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -7,430 +7,6 @@ addsitedir(abspath('../../'))
 
 from sage.all import *
 
-
-def _restrict_to_space(K, W):
-    r"""
-    Restrict this cone (up to linear isomorphism) to a vector subspace.
-
-    This operation not only restricts the cone to a subspace of its
-    ambient space, but also represents the rays of the cone in a new
-    (smaller) lattice corresponding to the subspace. The resulting cone
-    will be linearly isomorphic **but not equal** to the desired
-    restriction, since it has likely undergone a change of basis.
-
-    To explain the difficulty, consider the cone ``K = Cone([(1,1,1)])``
-    having a single ray. The span of ``K`` is a one-dimensional subspace
-    containing ``K``, yet we have no way to perform operations like
-    :meth:`dual` in the subspace. To represent ``K`` in the space
-    ``K.span()``, we must perform a change of basis and write its sole
-    ray as ``(1,0,0)``. Now the restricted ``Cone([(1,)])`` is linearly
-    isomorphic (but of course not equal) to ``K`` interpreted as living
-    in ``K.span()``.
-
-    INPUT:
-
-    - ``W`` -- The subspace into which this cone will be restricted.
-
-    OUTPUT:
-
-    A new cone in a sublattice corresponding to ``W``.
-
-    REFERENCES:
-
-    M. Orlitzky. The Lyapunov rank of an improper cone.
-    http://www.optimization-online.org/DB_HTML/2015/10/5135.html
-
-    EXAMPLES:
-
-    Restricting a solid cone to its own span returns a cone linearly
-    isomorphic to the original::
-
-        sage: K = Cone([(1,2,3),(-1,1,0),(9,0,-2)])
-        sage: K.is_solid()
-        True
-        sage: _restrict_to_space(K, K.span()).rays()
-        N(-1,  1,  0),
-        N( 1,  0,  0),
-        N( 9, -6, -1)
-        in 3-d lattice N
-
-    A single ray restricted to its own span has the same representation
-    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([(1,1,1)])
-        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
-
-    Restricting to a trivial space gives the trivial cone::
-
-        sage: K = Cone([(8,3,-1,0),(9,2,2,0),(-4,6,7,0)])
-        sage: trivial_space = K.lattice().vector_space().span([])
-        sage: _restrict_to_space(K, trivial_space)
-        0-d cone in 0-d lattice N
-
-    TESTS:
-
-    Restricting a cone to its own span results in a solid cone::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 8)
-        sage: K_S = _restrict_to_space(K, K.span())
-        sage: K_S.is_solid()
-        True
-
-    Restricting a cone to its own span should not affect the number of
-    rays in the cone::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 8)
-        sage: K_S = _restrict_to_space(K, K.span())
-        sage: K.nrays() == K_S.nrays()
-        True
-
-    Restricting a cone to its own span should not affect its dimension::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 8)
-        sage: K_S = _restrict_to_space(K, K.span())
-        sage: K.dim() == K_S.dim()
-        True
-
-    Restricting a cone to its own span should not affects its lineality::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 8)
-        sage: K_S = _restrict_to_space(K, K.span())
-        sage: K.lineality() == K_S.lineality()
-        True
-
-    Restricting a cone to its own span should not affect the number of
-    facets it has::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 8)
-        sage: K_S = _restrict_to_space(K, K.span())
-        sage: len(K.facets()) == len(K_S.facets())
-        True
-
-    Restricting a solid cone to its own span is a linear isomorphism and
-    should not affect the dimension of its ambient space::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 8, solid = True)
-        sage: K_S = _restrict_to_space(K, K.span())
-        sage: K.lattice_dim() == K_S.lattice_dim()
-        True
-
-    Restricting a solid cone to its own span is a linear isomorphism
-    that establishes a one-to-one correspondence of discrete
-    complementarity sets::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 8, solid = True)
-        sage: K_S = _restrict_to_space(K, K.span())
-        sage: dcs_K   = K.discrete_complementarity_set()
-        sage: dcs_K_S = K_S.discrete_complementarity_set()
-        sage: len(dcs_K) == len(dcs_K_S)
-        True
-
-    Restricting a solid cone to its own span is a linear isomorphism
-    under which the Lyapunov rank (the length of a Lyapunov-like basis)
-    is invariant::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 8, solid = True)
-        sage: K_S = _restrict_to_space(K, K.span())
-        sage: len(K.lyapunov_like_basis()) == len(K_S.lyapunov_like_basis())
-        True
-
-    If we restrict a cone to a subspace of its span, the resulting cone
-    should have the same dimension as the space we restricted it to::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 8)
-        sage: W_basis = random_sublist(K.rays(), 0.5)
-        sage: W = K.lattice().vector_space().span(W_basis)
-        sage: K_W = _restrict_to_space(K, W)
-        sage: K_W.lattice_dim() == W.dimension()
-        True
-
-    Through a series of restrictions, any closed convex cone can be
-    reduced to a cartesian product with a proper factor [Orlitzky]_::
-
-        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, K_S.dual().span())
-        sage: K_SP.is_proper()
-        True
-    """
-    # We want to intersect ``K`` with ``W``. An easy way to do this is
-    # via cone intersection, so we turn the space ``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 W, 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):
-    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.
-
-    OUTPUT:
-
-    A nonnegative integer representing the Lyapunov rank of this cone.
-
-    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]_.
-
-    ALGORITHM:
-
-    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}`.
-
-    REFERENCES:
-
-    .. [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.
-
-    M. Orlitzky. The Lyapunov rank of an improper cone.
-    http://www.optimization-online.org/DB_HTML/2015/10/5135.html
-
-    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.
-
-    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]_::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8)
-        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
-        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::
-
-        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
-        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
-
-    The calculation of the Lyapunov rank of an improper cone can be
-    reduced to that of a proper cone [Orlitzky]_::
-
-        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
-        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()
-        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())
-
-        # 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):
     r"""
     Determine whether or not ``L`` is Lyapunov-like on ``K``.
diff --git a/mjo/cone/rearrangement.py b/mjo/cone/rearrangement.py
index 2cde638..9dde4ec 100644
--- a/mjo/cone/rearrangement.py
+++ b/mjo/cone/rearrangement.py
@@ -6,7 +6,7 @@ from site import addsitedir
 addsitedir(abspath('../../'))
 
 from sage.all import *
-from mjo.cone.cone import lyapunov_rank, random_element
+from mjo.cone.cone import random_element
 
 def rearrangement_cone(p,n):
     r"""
@@ -64,11 +64,11 @@ def rearrangement_cone(p,n):
     The Lyapunov rank of the rearrangement cone of order ``p`` in ``n``
     dimensions is ``n`` for ``p == 1`` or ``p == n`` and one otherwise::
 
-        sage: all([ lyapunov_rank(rearrangement_cone(p,n)) == n
+        sage: all([ rearrangement_cone(p,n).lyapunov_rank() == n
         ....:               for n in range(2, 10)
         ....:               for p in [1, n-1] ])
         True
-        sage: all([ lyapunov_rank(rearrangement_cone(p,n)) == 1
+        sage: all([ rearrangement_cone(p,n).lyapunov_rank() == 1
         ....:               for n in range(3, 10)
         ....:               for p in range(2, n-1) ])
         True
diff --git a/mjo/cone/tests.py b/mjo/cone/tests.py
index 550beb3..816215c 100644
--- a/mjo/cone/tests.py
+++ b/mjo/cone/tests.py
@@ -15,7 +15,6 @@ from sage.all import *
 
 # The double-import is needed to get the underscore methods.
 from mjo.cone.cone import *
-from mjo.cone.cone import _restrict_to_space
 
 #
 # Tests for _restrict_to_space.
@@ -107,11 +106,11 @@ result). Test all four parameter combinations::
     sage: K = random_cone(max_ambient_dim = 8,
     ....:                 strictly_convex=False,
     ....:                 solid=False)
-    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_S = K._restrict_to_space(K.span())
+    sage: K_SP = K_S.dual()._restrict_to_space(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 = K_S._restrict_to_space(K_S.dual().span())
     sage: K_SP.is_proper()
     True
 
@@ -121,11 +120,11 @@ result). Test all four parameter combinations::
     sage: K = random_cone(max_ambient_dim = 8,
     ....:                 strictly_convex=True,
     ....:                 solid=False)
-    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_S = K._restrict_to_space(K.span())
+    sage: K_SP = K_S.dual()._restrict_to_space(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 = K_S._restrict_to_space(K_S.dual().span())
     sage: K_SP.is_proper()
     True
 
@@ -135,11 +134,11 @@ result). Test all four parameter combinations::
     sage: K = random_cone(max_ambient_dim = 8,
     ....:                 strictly_convex=False,
     ....:                 solid=True)
-    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_S = K._restrict_to_space(K.span())
+    sage: K_SP = K_S.dual()._restrict_to_space(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 = K_S._restrict_to_space(K_S.dual().span())
     sage: K_SP.is_proper()
     True
 
@@ -149,11 +148,11 @@ result). Test all four parameter combinations::
     sage: K = random_cone(max_ambient_dim = 8,
     ....:                 strictly_convex=True,
     ....:                 solid=True)
-    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_S = K._restrict_to_space(K.span())
+    sage: K_SP = K_S.dual()._restrict_to_space(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 = K_S._restrict_to_space(K_S.dual().span())
     sage: K_SP.is_proper()
     True
 
@@ -168,8 +167,8 @@ all parameter combinations::
     ....:                 solid=False,
     ....:                 strictly_convex=False)
     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: K_W_star = K._restrict_to_space(J.span()).dual()
+    sage: K_star_W = K.dual()._restrict_to_space(J.span())
     sage: _look_isomorphic(K_W_star, K_star_W)
     True
 
@@ -180,8 +179,8 @@ all parameter combinations::
     ....:                 solid=True,
     ....:                 strictly_convex=False)
     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: K_W_star = K._restrict_to_space(J.span()).dual()
+    sage: K_star_W = K.dual()._restrict_to_space(J.span())
     sage: _look_isomorphic(K_W_star, K_star_W)
     True
 
@@ -192,8 +191,8 @@ all parameter combinations::
     ....:                 solid=False,
     ....:                 strictly_convex=True)
     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: K_W_star = K._restrict_to_space(J.span()).dual()
+    sage: K_star_W = K.dual()._restrict_to_space(J.span())
     sage: _look_isomorphic(K_W_star, K_star_W)
     True
 
@@ -204,8 +203,8 @@ all parameter combinations::
     ....:                 solid=True,
     ....:                 strictly_convex=True)
     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: K_W_star = K._restrict_to_space(J.span()).dual()
+    sage: K_star_W = K.dual()._restrict_to_space(J.span())
     sage: _look_isomorphic(K_W_star, K_star_W)
     True
 
@@ -225,7 +224,7 @@ combinations of parameters::
     ....:                  solid=True)
     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)
+    sage: K1.lyapunov_rank() == K2.lyapunov_rank()
     True
 
 ::
@@ -235,7 +234,7 @@ combinations of parameters::
     ....:                  solid=False)
     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)
+    sage: K1.lyapunov_rank() == K2.lyapunov_rank()
     True
 
 ::
@@ -245,7 +244,7 @@ combinations of parameters::
     ....:                  solid=True)
     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)
+    sage: K1.lyapunov_rank() == K2.lyapunov_rank()
     True
 
 ::
@@ -255,7 +254,7 @@ combinations of parameters::
     ....:                  solid=False)
     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)
+    sage: K1.lyapunov_rank() == K2.lyapunov_rank()
     True
 
 The Lyapunov rank of a dual cone should be the same as the original
@@ -265,7 +264,7 @@ cone. Check all combinations of parameters::
     sage: K = random_cone(max_ambient_dim=8,
     ....:                 strictly_convex=False,
     ....:                 solid=False)
-    sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+    sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
     True
 
 ::
@@ -274,7 +273,7 @@ cone. Check all combinations of parameters::
     sage: K = random_cone(max_ambient_dim=8,
     ....:                 strictly_convex=False,
     ....:                 solid=True)
-    sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+    sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
     True
 
 ::
@@ -283,7 +282,7 @@ cone. Check all combinations of parameters::
     sage: K = random_cone(max_ambient_dim=8,
     ....:                 strictly_convex=True,
     ....:                 solid=False)
-    sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+    sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
     True
 
 ::
@@ -292,7 +291,7 @@ cone. Check all combinations of parameters::
     sage: K = random_cone(max_ambient_dim=8,
     ....:                 strictly_convex=True,
     ....:                 solid=True)
-    sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+    sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
     True
 
 The Lyapunov rank of a cone ``K`` is the dimension of
@@ -302,7 +301,7 @@ The Lyapunov rank of a cone ``K`` is the dimension of
     sage: K = random_cone(max_ambient_dim=8,
     ....:                 strictly_convex=True,
     ....:                 solid=True)
-    sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
+    sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
     True
 
 ::
@@ -311,7 +310,7 @@ The Lyapunov rank of a cone ``K`` is the dimension of
     sage: K = random_cone(max_ambient_dim=8,
     ....:                 strictly_convex=True,
     ....:                 solid=False)
-    sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
+    sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
     True
 
 ::
@@ -320,7 +319,7 @@ The Lyapunov rank of a cone ``K`` is the dimension of
     sage: K = random_cone(max_ambient_dim=8,
     ....:                 strictly_convex=False,
     ....:                 solid=True)
-    sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
+    sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
     True
 
 ::
@@ -329,7 +328,7 @@ The Lyapunov rank of a cone ``K`` is the dimension of
     sage: K = random_cone(max_ambient_dim=8,
     ....:                 strictly_convex=False,
     ....:                 solid=False)
-    sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
+    sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
     True
 
 """