from sage.all import *
+from sage.geometry.cone import is_Cone
def is_positive_on(L,K):
r"""
Determine whether or not ``L`` is positive on ``K``.
- 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``.
+ 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``.
+
+ 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.
INPUT:
- - ``L`` -- A linear transformation or matrix.
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- ``K`` -- A polyhedral closed convex cone.
OUTPUT:
- ``True`` if it can be proven that ``L`` is positive on ``K``,
- and ``False`` otherwise. If ``L`` is over an exact ring (the
- rationals, for example), then you can trust the answer. Only
- for symbolic ``L`` might there be difficulty in proving
- positivity.
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is positive on ``K``.
- .. WARNING::
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- 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.
+ - ``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.
EXAMPLES:
True
"""
+ 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('base ring of operator L is neither SR nor exact')
+
if L.base_ring().is_exact():
- # This could potentially be extended to other types of ``K``...
+ # 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 ])
- elif L.base_ring() is SR:
+ 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() ])
- 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
+ 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.
+ 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.
+
INPUT:
- - ``L`` -- A linear transformation or matrix.
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- ``K`` -- A polyhedral closed convex cone.
OUTPUT:
- ``True`` if it can be proven that ``L`` is cross-positive on ``K``,
- and ``False`` otherwise.
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is cross-positive on ``K``.
- .. WARNING::
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- 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.
+ - ``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.
EXAMPLES:
True
"""
- if L.base_ring().is_exact() or L.base_ring() is SR:
- return all([ s*(L*x) >= 0
- for (x,s) in K.discrete_complementarity_set() ])
- else:
- # The only inexact ring that we're willing to work with is SR,
- # since it can still be exact when working with symbolic
- # constants like pi and e.
+ 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('base ring of operator L is neither SR nor exact')
+ return all([ s*(L*x) >= 0
+ for (x,s) in K.discrete_complementarity_set() ])
def is_Z_on(L,K):
r"""
Determine whether or not ``L`` is a Z-operator on ``K``.
- 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.
+ 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.
A matrix is a Z-operator on ``K`` if and only if its negation is a
cross-positive operator on ``K``.
+ 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.
+
INPUT:
- - ``L`` -- A linear transformation or matrix.
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- ``K`` -- A polyhedral closed convex cone.
OUTPUT:
- ``True`` if it can be proven that ``L`` is a Z-operator on ``K``,
- and ``False`` otherwise.
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is a Z-operator on ``K``.
- .. WARNING::
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- 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.
+ - ``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.
EXAMPLES:
r"""
Determine whether or not ``L`` is Lyapunov-like on ``K``.
- We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle
- L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs
- `\left\langle x,s \right\rangle` in the complementarity set of
- ``K``. This property need only be checked for generators of
+ 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.
+ 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.
+
INPUT:
- - ``L`` -- A linear transformation or matrix.
+ - ``L`` -- A matrix over either an exact ring or ``SR``.
- ``K`` -- A polyhedral closed convex cone.
OUTPUT:
- ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
- and ``False`` otherwise.
+ If the base ring of ``L`` is exact, then ``True`` will be returned if
+ and only if ``L`` is Lyapunov-like on ``K``.
- .. WARNING::
+ If the base ring of ``L`` is ``SR``, then the situation is more
+ complicated:
- 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.
+ - ``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.
EXAMPLES:
True
"""
- if L.base_ring().is_exact() or L.base_ring() is SR:
- return all([ s*(L*x) == 0
- for (x,s) in K.discrete_complementarity_set() ])
- else:
- # The only inexact ring that we're willing to work with is SR,
- # since it can still be exact when working with symbolic
- # constants like pi and e.
+ 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('base ring of operator 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)
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