X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=ff7d195d134c15943dbb75c1f26b741bb4a0afba;hb=10142e85f34c47fa35df002f519d1d58a79a74f4;hp=421fb3c8f04abb23fd420271ed10e0a7e65815d4;hpb=f8e779ee533992a940e1014d00960fc58c3c4b79;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 421fb3c..ff7d195 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -60,6 +60,160 @@ def project_span(K):
 
 
 
+def lineality(K):
+    r"""
+    Compute the lineality of this cone.
+
+    The lineality of a cone is the dimension of the largest linear
+    subspace contained in that cone.
+
+    OUTPUT:
+
+    A nonnegative integer; the dimension of the largest subspace
+    contained within this cone.
+
+    REFERENCES:
+
+    .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton
+       University Press, Princeton, 1970.
+
+    EXAMPLES:
+
+    The lineality of the nonnegative orthant is zero, since it clearly
+    contains no lines::
+
+        sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
+        sage: lineality(K)
+        0
+
+    However, if we add another ray so that the entire `x`-axis belongs
+    to the cone, then the resulting cone will have lineality one::
+
+        sage: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)])
+        sage: lineality(K)
+        1
+
+    If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal
+    to the dimension of the ambient space (i.e. two)::
+
+        sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
+        sage: lineality(K)
+        2
+
+    Per the definition, the lineality of the trivial cone in a trivial
+    space is zero::
+
+        sage: K = Cone([], lattice=ToricLattice(0))
+        sage: lineality(K)
+        0
+
+    TESTS:
+
+    The lineality of a cone should be an integer between zero and the
+    dimension of the ambient space, inclusive::
+
+        sage: K = random_cone(max_dim = 10)
+        sage: l = lineality(K)
+        sage: l in ZZ
+        True
+        sage: (0 <= l) and (l <= K.lattice_dim())
+        True
+
+    A strictly convex cone should have lineality zero::
+
+        sage: K = random_cone(max_dim = 10, strictly_convex = True)
+        sage: lineality(K)
+        0
+
+    """
+    return K.linear_subspace().dimension()
+
+
+def codim(K):
+    r"""
+    Compute the codimension of this cone.
+
+    The codimension of a cone is the dimension of the space of all
+    elements perpendicular to every element of the cone. In other words,
+    the codimension is the difference between the dimension of the
+    ambient space and the dimension of the cone itself.
+
+    OUTPUT:
+
+    A nonnegative integer representing the dimension of the space of all
+    elements perpendicular to this cone.
+
+    .. seealso::
+
+        :meth:`dim`, :meth:`lattice_dim`
+
+    EXAMPLES:
+
+    The codimension of the nonnegative orthant is zero, since the span of
+    its generators equals the entire ambient space::
+
+        sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
+        sage: codim(K)
+        0
+
+    However, if we remove a ray so that the entire cone is contained
+    within the `x-y`-plane, then the resulting cone will have
+    codimension one, because the `z`-axis is perpendicular to every
+    element of the cone::
+
+        sage: K = Cone([(1,0,0), (0,1,0)])
+        sage: codim(K)
+        1
+
+    If our cone is all of `\mathbb{R}^{2}`, then its codimension is zero::
+
+        sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
+        sage: codim(K)
+        0
+
+    And if the cone is trivial in any space, then its codimension is
+    equal to the dimension of the ambient space::
+
+        sage: K = Cone([], lattice=ToricLattice(0))
+        sage: codim(K)
+        0
+
+        sage: K = Cone([(0,)])
+        sage: codim(K)
+        1
+
+        sage: K = Cone([(0,0)])
+        sage: codim(K)
+        2
+
+    TESTS:
+
+    The codimension of a cone should be an integer between zero and
+    the dimension of the ambient space, inclusive::
+
+        sage: K = random_cone(max_dim = 10)
+        sage: c = codim(K)
+        sage: c in ZZ
+        True
+        sage: (0 <= c) and (c <= K.lattice_dim())
+        True
+
+    A solid cone should have codimension zero::
+
+        sage: K = random_cone(max_dim = 10, solid = True)
+        sage: codim(K)
+        0
+
+    The codimension of a cone is equal to the lineality of its dual::
+
+        sage: K = random_cone(max_dim = 10, solid = True)
+        sage: codim(K) == lineality(K.dual())
+        True
+
+    """
+    return (K.lattice_dim() - K.dim())
+
+
 def discrete_complementarity_set(K):
     r"""
     Compute the discrete complementarity set of this cone.
@@ -349,7 +503,7 @@ def lyapunov_rank(K):
         sage: K = Cone([e1, neg_e1, e2, neg_e2, zero, zero, zero])
         sage: lyapunov_rank(K)
         19
-        sage: K.lattice_dim()**2 - K.dim()*(K.lattice_dim() - K.dim())
+        sage: K.lattice_dim()**2 - K.dim()*codim(K)
         19
 
     The Lyapunov rank should be additive on a product of proper cones
@@ -424,9 +578,9 @@ def lyapunov_rank(K):
         sage: actual = lyapunov_rank(K)
         sage: K_S = project_span(K)
         sage: P = project_span(K_S.dual()).dual()
-        sage: l = K.linear_subspace().dimension()
-        sage: codim = K.lattice_dim() - K.dim()
-        sage: expected = lyapunov_rank(P) + K.dim()*(l + codim) + codim**2
+        sage: l = lineality(K)
+        sage: c = codim(K)
+        sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2
         sage: actual == expected
         True
 
@@ -441,7 +595,7 @@ def lyapunov_rank(K):
 
     m = K.dim()
     n = K.lattice_dim()
-    l = K.linear_subspace().dimension()
+    l = lineality(K)
 
     if m < n:
         # K is not solid, project onto its span.