From: Michael Orlitzky <michael@orlitzky.com>
Date: Tue, 12 Jan 2016 01:59:35 +0000 (-0500)
Subject: Remove the motzkin_decomposition() method, now Trac #19867.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=b439372b31ba6d3df3847719ca7aefd307dc4ecd;p=sage.d.git

Remove the motzkin_decomposition() method, now Trac #19867.
---

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index a70b016..dd28f2f 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -65,120 +65,6 @@ def is_lyapunov_like(L,K):
                 for (x,s) in K.discrete_complementarity_set()])
 
 
-def motzkin_decomposition(K):
-    r"""
-    Return the pair of components in the Motzkin decomposition of this cone.
-
-    Every convex cone is the direct sum of a strictly convex cone and a
-    linear subspace [Stoer-Witzgall]_. Return a pair ``(P,S)`` of cones
-    such that ``P`` is strictly convex, ``S`` is a subspace, and ``K``
-    is the direct sum of ``P`` and ``S``.
-
-    .. NOTE::
-
-        The name "Motzkin decomposition" is not standard. The result
-        is usually stated as the "decomposition theorem", or "cone
-        decomposition theorem."
-
-    OUTPUT:
-
-    An ordered pair ``(P,S)`` of closed convex polyhedral cones where
-    ``P`` is strictly convex, ``S`` is a subspace, and ``K`` is the
-    direct sum of ``P`` and ``S``.
-
-    REFERENCES:
-
-    .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and
-       Optimization in Finite Dimensions I. Springer-Verlag, New
-       York, 1970.
-
-    EXAMPLES:
-
-    The nonnegative orthant is strictly convex, so it is its own
-    strictly convex component and its subspace component is trivial::
-
-        sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
-        sage: (P,S) = motzkin_decomposition(K)
-        sage: K.is_equivalent(P)
-        True
-        sage: S.is_trivial()
-        True
-
-    Likewise, full spaces are their own subspace components::
-
-        sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
-        sage: K.is_full_space()
-        True
-        sage: (P,S) = motzkin_decomposition(K)
-        sage: K.is_equivalent(S)
-        True
-        sage: P.is_trivial()
-        True
-
-    TESTS:
-
-    A random point in the cone should belong to either the strictly
-    convex component or the subspace component. If the point is nonzero,
-    it cannot be in both::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8)
-        sage: (P,S) = motzkin_decomposition(K)
-        sage: x = K.random_element(ring=QQ)
-        sage: P.contains(x) or S.contains(x)
-        True
-        sage: x.is_zero() or (P.contains(x) != S.contains(x))
-        True
-
-    The strictly convex component should always be strictly convex, and
-    the subspace component should always be a subspace::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8)
-        sage: (P,S) = motzkin_decomposition(K)
-        sage: P.is_strictly_convex()
-        True
-        sage: S.lineality() == S.dim()
-        True
-
-    A strictly convex cone should be equal to its strictly convex component::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8, strictly_convex=True)
-        sage: (P,_) = motzkin_decomposition(K)
-        sage: K.is_equivalent(P)
-        True
-
-    The generators of the components are obtained from orthogonal
-    projections of the original generators [Stoer-Witzgall]_::
-
-        sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8)
-        sage: (P,S) = motzkin_decomposition(K)
-        sage: A = S.linear_subspace().complement().matrix()
-        sage: proj_S_perp = A.transpose() * (A*A.transpose()).inverse() * A
-        sage: expected_P = Cone([ proj_S_perp*g for g in K ], K.lattice())
-        sage: P.is_equivalent(expected_P)
-        True
-        sage: A = S.linear_subspace().matrix()
-        sage: proj_S = A.transpose() * (A*A.transpose()).inverse() * A
-        sage: expected_S = Cone([ proj_S*g for g in K ], K.lattice())
-        sage: S.is_equivalent(expected_S)
-        True
-    """
-    # The lines() method only returns one generator per line. For a true
-    # line, we also need a generator pointing in the opposite direction.
-    S_gens = [ direction*gen for direction in [1,-1] for gen in K.lines() ]
-    S = Cone(S_gens, K.lattice(), check=False)
-
-    # Since ``S`` is a subspace, the rays of its dual generate its
-    # orthogonal complement.
-    S_perp = Cone(S.dual(), K.lattice(), check=False)
-    P = K.intersection(S_perp)
-
-    return (P,S)
-
-
 def positive_operator_gens(K):
     r"""
     Compute generators of the cone of positive operators on this cone.