From e041595c10751828f196db2cda86bd0f15a81191 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 12 Jun 2015 19:13:40 -0400 Subject: [PATCH] Revert "Remove unused codim() function." This reverts commit 4418c497a443fb1f5cb068ced5a2ddd5a9a0ad05. --- mjo/cone/cone.py | 94 ++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 94 insertions(+) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index b9e930e..ba5f51e 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -313,6 +313,100 @@ def lineality(K): 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: K.lattice_dim() + 0 + sage: codim(K) + 0 + + sage: K = Cone([(0,)]) + sage: K.lattice_dim() + 1 + sage: codim(K) + 1 + + sage: K = Cone([(0,0)]) + sage: K.lattice_dim() + 2 + 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: set_random_seed() + sage: K = random_cone(max_dim = 8) + 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: set_random_seed() + sage: K = random_cone(max_dim = 8, solid = True) + sage: codim(K) + 0 + + The codimension of a cone is equal to the lineality of its dual:: + + sage: set_random_seed() + sage: K = random_cone(max_dim = 8, 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. -- 2.44.2