-# 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.geometry.cone import is_Cone
-def rename_lattice(L,s):
+def is_positive_on(L,K):
r"""
- Change all names of the given lattice to ``s``.
- """
- L._name = s
- L._dual_name = s
- L._latex_name = s
- L._latex_dual_name = s
+ Determine whether or not ``L`` is positive on ``K``.
-def span_iso(K):
- r"""
- Return an isomorphism (and its inverse) that will send ``K`` into a
- lower-dimensional space isomorphic to its span (and back).
+ We say that ``L`` is positive on a closed convex cone ``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``.
- EXAMPLES:
+ To reliably check whether or not ``L`` is positive, its base ring
+ must be either exact (for example, the rationals) or ``SR``. An
+ exact ring is more reliable, but in some cases a matrix whose
+ entries contain symbolic constants like ``e`` and ``pi`` will work.
- The inverse composed with the isomorphism should be the identity::
+ INPUT:
- sage: K = random_cone(max_dim=10)
- sage: (phi, phi_inv) = span_iso(K)
- sage: phi_inv(phi(K)) == K
- True
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- The image of ``K`` under the isomorphism should have full dimension::
+ - ``K`` -- A polyhedral closed convex cone.
- sage: K = random_cone(max_dim=10)
- sage: (phi, phi_inv) = span_iso(K)
- sage: phi(K).dim() == phi(K).lattice_dim()
- True
+ OUTPUT:
- The isomorphism should be an inner product space isomorphism, and
- thus it should preserve dual cones (and commute with the "dual"
- operation). But beware the automatic renaming of the dual lattice.
- OH AND YOU HAVE TO SORT THE CONES::
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is positive on ``K``.
- sage: K = random_cone(max_dim=10, strictly_convex=False, solid=True)
- sage: L = K.lattice()
- sage: rename_lattice(L, 'L')
- sage: (phi, phi_inv) = span_iso(K)
- sage: sorted(phi_inv( phi(K).dual() )) == sorted(K.dual())
- True
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- We may need to isomorph twice to make sure we stop moving down to
- smaller spaces. (Once you've done this on a cone and its dual, the
- result should be proper.) OH AND YOU HAVE TO SORT THE CONES::
-
- sage: K = random_cone(max_dim=10, strictly_convex=False, solid=False)
- sage: L = K.lattice()
- sage: rename_lattice(L, 'L')
- sage: (phi, phi_inv) = span_iso(K)
- sage: K_S = phi(K)
- sage: (phi_dual, phi_dual_inv) = span_iso(K_S.dual())
- sage: J_T = phi_dual(K_S.dual()).dual()
- sage: phi_inv(phi_dual_inv(J_T)) == K
- True
+ - ``True`` will be returned if it can be proven that ``L``
+ is positive on ``K``.
+ - ``False`` will be returned if it can be proven that ``L``
+ is not positive on ``K``.
+ - ``False`` will also be returned if we can't decide; specifically
+ if we arrive at a symbolic inequality that cannot be resolved.
- """
- phi_domain = K.sublattice().vector_space()
- phi_codo = VectorSpace(phi_domain.base_field(), phi_domain.dimension())
+ .. SEEALSO::
- # S goes from the new space to the cone space.
- S = linear_transformation(phi_codo, phi_domain, phi_domain.basis())
+ :func:`is_cross_positive_on`,
+ :func:`is_Z_operator_on`,
+ :func:`is_lyapunov_like_on`
- # phi goes from the cone space to the new space.
- def phi(J_orig):
- r"""
- Takes a cone ``J`` and sends it into the new space.
- """
- newrays = map(S.inverse(), J_orig.rays())
- L = None
- if len(newrays) == 0:
- L = ToricLattice(0)
+ EXAMPLES:
- return Cone(newrays, lattice=L)
+ Nonnegative matrices are positive operators on the nonnegative
+ orthant::
- def phi_inverse(J_sub):
- r"""
- The inverse to phi which goes from the new space to the cone space.
- """
- newrays = map(S, J_sub.rays())
- return Cone(newrays, lattice=K.lattice())
+ 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
+ TESTS:
- return (phi, phi_inverse)
+ The identity operator is always positive::
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_positive_on(L,K)
+ True
+ The "zero" operator is always positive::
-def discrete_complementarity_set(K):
- r"""
- Compute the discrete complementarity set of this cone.
+ sage: K = random_cone(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
- The complementarity set of this cone is the set of all orthogonal
- pairs `(x,s)` such that `x` is in this cone, and `s` is in its
- dual. The discrete complementarity set restricts `x` and `s` to be
- generators of their respective cones.
+ Everything in ``K.positive_operators_gens()`` should be
+ positive on ``K``::
- OUTPUT:
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: all([ is_positive_on(L,K) # long time
+ ....: for L in K.positive_operators_gens() ]) # long time
+ True
+ sage: all([ is_positive_on(L.change_ring(SR),K) # long time
+ ....: for L in K.positive_operators_gens() ]) # long time
+ True
- A list of pairs `(x,s)` such that,
+ Technically we could test this, but for now only closed convex cones
+ are supported as our ``K`` argument::
- * `x` is in this cone.
- * `x` is a generator of this cone.
- * `s` is in this cone's dual.
- * `s` is a generator of this cone's dual.
- * `x` and `s` are orthogonal.
+ sage: K = [ vector([1,2,3]), vector([5,-1,7]) ]
+ sage: L = identity_matrix(3)
+ sage: is_positive_on(L,K)
+ Traceback (most recent call last):
+ ...
+ TypeError: K must be a Cone.
- EXAMPLES:
+ We can't give reliable answers over inexact rings::
- The discrete complementarity set of the nonnegative orthant consists
- of pairs of standard basis vectors::
+ sage: K = Cone([(1,2,3), (4,5,6)])
+ sage: L = identity_matrix(RR,3)
+ sage: is_positive_on(L,K)
+ Traceback (most recent call last):
+ ...
+ ValueError: The base ring of L is neither SR nor exact.
- sage: K = Cone([(1,0),(0,1)])
- sage: discrete_complementarity_set(K)
- [((1, 0), (0, 1)), ((0, 1), (1, 0))]
+ """
- If the cone consists of a single ray, the second components of the
- discrete complementarity set should generate the orthogonal
- complement of that ray::
+ if not is_Cone(K):
+ raise TypeError('K must be a Cone.')
+ if not L.base_ring().is_exact() and not L.base_ring() is SR:
+ raise ValueError('The base ring of L is neither SR nor exact.')
- sage: K = Cone([(1,0)])
- sage: discrete_complementarity_set(K)
- [((1, 0), (0, 1)), ((1, 0), (0, -1))]
- sage: K = Cone([(1,0,0)])
- sage: discrete_complementarity_set(K)
- [((1, 0, 0), (0, 1, 0)),
- ((1, 0, 0), (0, -1, 0)),
- ((1, 0, 0), (0, 0, 1)),
- ((1, 0, 0), (0, 0, -1))]
+ if L.base_ring().is_exact():
+ # This should be way faster than computing the dual and
+ # checking a bunch of inequalities, but it doesn't work if
+ # ``L*x`` is symbolic. For example, ``e in Cone([(1,)])``
+ # is true, but returns ``False``.
+ return all([ L*x in K for x in K ])
+ else:
+ # 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() ])
- When the cone is the entire space, its dual is the trivial cone, so
- the discrete complementarity set is empty::
- sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
- sage: discrete_complementarity_set(K)
- []
+def is_cross_positive_on(L,K):
+ r"""
+ Determine whether or not ``L`` is cross-positive on ``K``.
- TESTS:
+ We say that ``L`` is cross-positive on a closed convex cone``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.
- The complementarity set of the dual can be obtained by switching the
- components of the complementarity set of the original cone::
+ To reliably check whether or not ``L`` is cross-positive, its base
+ ring must be either exact (for example, the rationals) or ``SR``. An
+ exact ring is more reliable, but in some cases a matrix whose
+ entries contain symbolic constants like ``e`` and ``pi`` will work.
- sage: K1 = random_cone(max_dim=10, max_rays=10)
- sage: K2 = K1.dual()
- sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)]
- sage: actual = discrete_complementarity_set(K1)
- sage: actual == expected
- True
+ INPUT:
- """
- V = K.lattice().vector_space()
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- # Convert the rays to vectors so that we can compute inner
- # products.
- xs = [V(x) for x in K.rays()]
- ss = [V(s) for s in K.dual().rays()]
+ - ``K`` -- A polyhedral closed convex cone.
- return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
+ OUTPUT:
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is cross-positive on ``K``.
-def LL(K):
- r"""
- Compute the space `\mathbf{LL}` of all Lyapunov-like transformations
- on this cone.
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- OUTPUT:
+ - ``True`` will be returned if it can be proven that ``L``
+ is cross-positive on ``K``.
+ - ``False`` will be returned if it can be proven that ``L``
+ is not cross-positive on ``K``.
+ - ``False`` will also be returned if we can't decide; specifically
+ if we arrive at a symbolic inequality that cannot be resolved.
- A list of matrices forming a basis for the space of all
- Lyapunov-like transformations on the given cone.
+ .. SEEALSO::
+
+ :func:`is_positive_on`,
+ :func:`is_Z_operator_on`,
+ :func:`is_lyapunov_like_on`
EXAMPLES:
- The trivial cone has no Lyapunov-like transformations::
+ The identity operator is always cross-positive::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_cross_positive_on(L,K)
+ True
+
+ The "zero" operator is always cross-positive::
- sage: L = ToricLattice(0)
- sage: K = Cone([], lattice=L)
- sage: LL(K)
- []
+ sage: K = random_cone(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
- The Lyapunov-like transformations on the nonnegative orthant are
- simply diagonal matrices::
+ TESTS:
- sage: K = Cone([(1,)])
- sage: LL(K)
- [[1]]
+ Everything in ``K.cross_positive_operators_gens()`` should be
+ cross-positive on ``K``::
- sage: K = Cone([(1,0),(0,1)])
- sage: LL(K)
- [
- [1 0] [0 0]
- [0 0], [0 1]
- ]
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: all([ is_cross_positive_on(L,K) # long time
+ ....: for L in K.cross_positive_operators_gens() ]) # long time
+ True
+ sage: all([ is_cross_positive_on(L.change_ring(SR),K) # long time
+ ....: for L in K.cross_positive_operators_gens() ]) # long time
+ True
- sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
- sage: LL(K)
- [
- [1 0 0] [0 0 0] [0 0 0]
- [0 0 0] [0 1 0] [0 0 0]
- [0 0 0], [0 0 0], [0 0 1]
- ]
-
- Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and
- `L^{3}_{\infty}` cones [Rudolf et al.]_::
-
- sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
- sage: LL(L31)
- [
- [1 0 0]
- [0 1 0]
- [0 0 1]
- ]
-
- sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
- sage: LL(L3infty)
- [
- [1 0 0]
- [0 1 0]
- [0 0 1]
- ]
+ Technically we could test this, but for now only closed convex cones
+ are supported as our ``K`` argument::
- TESTS:
+ sage: L = identity_matrix(3)
+ sage: K = [ vector([8,2,-8]), vector([5,-5,7]) ]
+ sage: is_cross_positive_on(L,K)
+ Traceback (most recent call last):
+ ...
+ TypeError: K must be a Cone.
- The inner product `\left< L\left(x\right), s \right>` is zero for
- every pair `\left( x,s \right)` in the discrete complementarity set
- of the cone::
+ We can't give reliable answers over inexact rings::
- sage: K = random_cone(max_dim=8, max_rays=10)
- sage: C_of_K = discrete_complementarity_set(K)
- sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ]
- sage: sum(map(abs, l))
- 0
+ sage: K = Cone([(1,2,3), (4,5,6)])
+ sage: L = identity_matrix(RR,3)
+ sage: is_cross_positive_on(L,K)
+ Traceback (most recent call last):
+ ...
+ ValueError: The base ring of L is neither SR nor exact.
"""
- V = K.lattice().vector_space()
+ if not is_Cone(K):
+ raise TypeError('K must be a Cone.')
+ if not L.base_ring().is_exact() and not L.base_ring() is SR:
+ raise ValueError('The base ring of L is neither SR nor exact.')
- C_of_K = discrete_complementarity_set(K)
+ return all([ s*(L*x) >= 0
+ for (x,s) in K.discrete_complementarity_set() ])
- tensor_products = [s.tensor_product(x) for (x,s) in C_of_K]
+def is_Z_operator_on(L,K):
+ r"""
+ Determine whether or not ``L`` is a Z-operator on ``K``.
- # 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.
- W = VectorSpace(V.base_ring(), V.dimension()**2)
+ We say that ``L`` is a Z-operator on a closed convex cone``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.
- # Turn our matrices into long vectors...
- vectors = [ W(m.list()) for m in tensor_products ]
+ A matrix is a Z-operator on ``K`` if and only if its negation is a
+ cross-positive operator on ``K``.
- # Vector space representation of Lyapunov-like matrices
- # (i.e. vec(L) where L is Luapunov-like).
- LL_vector = W.span(vectors).complement()
+ To reliably check whether or not ``L`` is a Z operator, its base
+ ring must be either exact (for example, the rationals) or ``SR``. An
+ exact ring is more reliable, but in some cases a matrix whose
+ entries contain symbolic constants like ``e`` and ``pi`` will work.
- # Now construct an ambient MatrixSpace in which to stick our
- # transformations.
- M = MatrixSpace(V.base_ring(), V.dimension())
+ INPUT:
- matrix_basis = [ M(v.list()) for v in LL_vector.basis() ]
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- return matrix_basis
+ - ``K`` -- A polyhedral closed convex cone.
+ OUTPUT:
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is a Z-operator on ``K``.
-def lyapunov_rank(K):
- r"""
- Compute the Lyapunov (or bilinearity) rank of this cone.
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- The Lyapunov rank of a cone can be thought of in (mainly) two ways:
+ - ``True`` will be returned if it can be proven that ``L``
+ is a Z-operator on ``K``.
+ - ``False`` will be returned if it can be proven that ``L``
+ is not a Z-operator on ``K``.
+ - ``False`` will also be returned if we can't decide; specifically
+ if we arrive at a symbolic inequality that cannot be resolved.
- 1. The dimension of the Lie algebra of the automorphism group of the
- cone.
+ .. SEEALSO::
- 2. The dimension of the linear space of all Lyapunov-like
- transformations on the cone.
+ :func:`is_positive_on`,
+ :func:`is_cross_positive_on`,
+ :func:`is_lyapunov_like_on`
- INPUT:
+ EXAMPLES:
- A closed, convex polyhedral cone.
+ The identity operator is always a Z-operator::
- OUTPUT:
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_Z_operator_on(L,K)
+ True
- An integer representing the Lyapunov rank of the cone. If the
- dimension of the ambient vector space is `n`, then the Lyapunov rank
- will be between `1` and `n` inclusive; however a rank of `n-1` is
- not possible (see the first reference).
+ The "zero" operator is always a Z-operator::
- .. note::
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: R = K.lattice().vector_space().base_ring()
+ sage: L = zero_matrix(R, K.lattice_dim())
+ sage: is_Z_operator_on(L,K)
+ True
- In the references, the cones are always assumed to be proper. We
- do not impose this restriction.
+ TESTS:
- .. seealso::
+ Everything in ``K.Z_operators_gens()`` should be a Z-operator
+ on ``K``::
- :meth:`is_proper`
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: all([ is_Z_operator_on(L,K) # long time
+ ....: for L in K.Z_operators_gens() ]) # long time
+ True
+ sage: all([ is_Z_operator_on(L.change_ring(SR),K) # long time
+ ....: for L in K.Z_operators_gens() ]) # long time
+ True
- ALGORITHM:
+ Technically we could test this, but for now only closed convex cones
+ are supported as our ``K`` argument::
- 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}`.
+ sage: L = identity_matrix(3)
+ sage: K = [ vector([-4,20,3]), vector([1,-5,2]) ]
+ sage: is_Z_operator_on(L,K)
+ Traceback (most recent call last):
+ ...
+ TypeError: K must be a Cone.
- 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.
+ We can't give reliable answers over inexact rings::
- .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an
- Improper Cone. Work in-progress.
+ sage: K = Cone([(1,2,3), (4,5,6)])
+ sage: L = identity_matrix(RR,3)
+ sage: is_Z_operator_on(L,K)
+ Traceback (most recent call last):
+ ...
+ ValueError: The base ring of L is neither SR nor exact.
- .. [Rudolf et al.] 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.
+ """
+ return is_cross_positive_on(-L,K)
- EXAMPLES:
- The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`
- [Rudolf et al.]_::
+def is_lyapunov_like_on(L,K):
+ r"""
+ Determine whether or not ``L`` is Lyapunov-like on ``K``.
- 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
+ We say that ``L`` is Lyapunov-like on a closed convex cone ``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.
- The `L^{3}_{1}` cone is known to have a Lyapunov rank of one
- [Rudolf et al.]_::
+ An operator is Lyapunov-like on ``K`` if and only if both the
+ operator itself and its negation are cross-positive on ``K``.
- sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
- sage: lyapunov_rank(L31)
- 1
+ To reliably check whether or not ``L`` is Lyapunov-like, its base
+ ring must be either exact (for example, the rationals) or ``SR``. An
+ exact ring is more reliable, but in some cases a matrix whose
+ entries contain symbolic constants like ``e`` and ``pi`` will work.
- Likewise for the `L^{3}_{\infty}` cone [Rudolf et al.]_::
+ INPUT:
- sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
- sage: lyapunov_rank(L3infty)
- 1
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- The Lyapunov rank should be additive on a product of cones
- [Rudolf et al.]_::
+ - ``K`` -- A polyhedral closed convex cone.
- 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
+ OUTPUT:
+
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is Lyapunov-like on ``K``.
- Two isomorphic cones should have the same Lyapunov rank [Rudolf et al.]_.
- The cone ``K`` in the following example is isomorphic to the nonnegative
- octant in `\mathbb{R}^{3}`::
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)])
- sage: lyapunov_rank(K)
- 3
+ - ``True`` will be returned if it can be proven that ``L``
+ is Lyapunov-like on ``K``.
+ - ``False`` will be returned if it can be proven that ``L``
+ is not Lyapunov-like on ``K``.
+ - ``False`` will also be returned if we can't decide; specifically
+ if we arrive at a symbolic inequality that cannot be resolved.
- The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself [Rudolf et al.]_::
+ .. SEEALSO::
- sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)])
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ :func:`is_positive_on`,
+ :func:`is_cross_positive_on`,
+ :func:`is_Z_operator_on`
+
+ EXAMPLES:
+
+ Diagonal matrices are Lyapunov-like operators on the nonnegative
+ orthant::
+
+ 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:
- The Lyapunov rank should be additive on a product of cones
- [Rudolf et al.]_::
+ The identity operator is always Lyapunov-like::
- sage: K1 = random_cone(max_dim=10, max_rays=10)
- sage: K2 = random_cone(max_dim=10, max_rays=10)
- sage: K = K1.cartesian_product(K2)
- sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=8)
+ sage: L = identity_matrix(K.lattice_dim())
+ sage: is_lyapunov_like_on(L,K)
True
- The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
- itself [Rudolf et al.]_::
+ The "zero" operator is always Lyapunov-like::
- sage: K = random_cone(max_dim=10, max_rays=10)
- sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+ sage: K = random_cone(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
- 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::
+ Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
+ on ``K``::
- sage: K = random_cone(max_dim=10, 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/Gowda]_, no closed convex polyhedral cone can have
- Lyapunov rank `n-1` in `n` dimensions::
-
- sage: K = random_cone(max_dim=10)
- 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/Gowda]_::
-
- sage: K = random_cone(max_dim=15, solid=False, strictly_convex=False)
- sage: actual = lyapunov_rank(K)
- sage: (phi1, phi1_inv) = span_iso(K)
- sage: K_S = phi1(K)
- sage: (phi2, phi2_inv) = span_iso(K_S.dual())
- sage: J_T = phi2(K_S.dual()).dual()
- sage: phi1_inv(phi2_inv(J_T)) == K
+ sage: K = random_cone(max_ambient_dim=5)
+ sage: all([ is_lyapunov_like_on(L,K) # long time
+ ....: for L in K.lyapunov_like_basis() ]) # long time
True
- sage: l = K.linear_subspace().dimension()
- sage: codim = K.lattice_dim() - K.dim()
- sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
- sage: actual == expected
+ sage: all([ is_lyapunov_like_on(L.change_ring(SR),K) # long time
+ ....: for L in K.lyapunov_like_basis() ]) # long time
True
+ Technically we could test this, but for now only closed convex cones
+ are supported as our ``K`` argument::
+
+ sage: L = identity_matrix(3)
+ sage: K = [ vector([2,2,-1]), vector([5,4,-3]) ]
+ sage: is_lyapunov_like_on(L,K)
+ Traceback (most recent call last):
+ ...
+ TypeError: K must be a Cone.
+
+ We can't give reliable answers over inexact rings::
+
+ sage: K = Cone([(1,2,3), (4,5,6)])
+ sage: L = identity_matrix(RR,3)
+ sage: is_lyapunov_like_on(L,K)
+ Traceback (most recent call last):
+ ...
+ ValueError: The base ring of L is neither SR nor exact.
+
"""
- return len(LL(K))
+ if not is_Cone(K):
+ raise TypeError('K must be a Cone.')
+ if not L.base_ring().is_exact() and not L.base_ring() is SR:
+ raise ValueError('The base ring of L is neither SR nor exact.')
+
+ return all([ s*(L*x) == 0
+ for (x,s) in K.discrete_complementarity_set() ])
+
+
+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)
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