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[sage.d.git] / mjo / cone / rearrangement.py
1 from sage.all import *
2
3 def rearrangement_cone(p,n,lattice=None):
4 r"""
5 Return the rearrangement cone of order ``p`` in ``n`` dimensions.
6
7 The rearrangement cone in ``n`` dimensions has as its elements
8 vectors of length ``n``. For inclusion in the cone, the smallest
9 ``p`` components of a vector must sum to a nonnegative number.
10
11 For example, the rearrangement cone of order ``p == 1`` has its
12 single smallest component nonnegative. This implies that all
13 components are nonnegative, and that therefore the rearrangement
14 cone of order one is the nonnegative orthant.
15
16 When ``p == n``, the sum of all components of a vector must be
17 nonnegative for inclusion in the cone. That is, the cone is a
18 half-space in ``n`` dimensions.
19
20 INPUT:
21
22 - ``p`` -- The number of components to "rearrange."
23
24 - ``n`` -- The dimension of the ambient space for the resulting cone.
25
26 - ``lattice`` -- (default: ``None``) an ambient lattice of rank ``n``
27 to be passed to the :func:`Cone` constructor.
28
29 OUTPUT:
30
31 A polyhedral closed convex cone object representing a rearrangement
32 cone of order ``p`` in ``n`` dimensions. Each generating ray will
33 have the integer ring as its base ring.
34
35 If a ``lattice`` was specified, then the resulting cone will live in
36 that lattice unless its rank is incompatible with the dimension
37 ``n`` (in which case a ``ValueError`` is raised).
38
39 ALGORITHM:
40
41 The generators for the rearrangement cone are given by [Jeong]_
42 Theorem 5.2.3.
43
44 REFERENCES:
45
46 .. [GowdaJeong] Muddappa Seetharama Gowda and Juyoung Jeong.
47 Spectral cones in Euclidean Jordan algebras.
48 Linear Algebra and its Applications, 509, 286-305.
49 doi:10.1016/j.laa.2016.08.004
50
51 .. [HenrionSeeger] Rene Henrion and Alberto Seeger.
52 Inradius and Circumradius of Various Convex Cones Arising in
53 Applications. Set-Valued and Variational Analysis, 18(3-4),
54 483-511, 2010. doi:10.1007/s11228-010-0150-z
55
56 .. [Jeong] Juyoung Jeong.
57 Spectral sets and functions on Euclidean Jordan algebras.
58 University of Maryland, Baltimore County, Ph.D. thesis, 2017.
59
60 SETUP::
61
62 sage: from mjo.cone.rearrangement import rearrangement_cone
63
64 EXAMPLES:
65
66 The rearrangement cones of order one are nonnegative orthants::
67
68 sage: rearrangement_cone(1,1) == Cone([(1,)])
69 True
70 sage: rearrangement_cone(1,2) == Cone([(0,1),(1,0)])
71 True
72 sage: rearrangement_cone(1,3) == Cone([(0,0,1),(0,1,0),(1,0,0)])
73 True
74
75 When ``p == n``, the resulting cone will be a half-space, so we
76 expect its lineality to be one less than ``n`` because it will
77 contain a hyperplane but is not the entire space::
78
79 sage: rearrangement_cone(5,5).lineality()
80 4
81
82 All rearrangement cones are proper when ``p`` is less than ``n`` by
83 [Jeong]_ Proposition 5.2.1::
84
85 sage: all( rearrangement_cone(p,n).is_proper()
86 ....: for n in range(10)
87 ....: for p in range(1, n) )
88 True
89
90 The Lyapunov rank of the rearrangement cone of order ``p`` in ``n``
91 dimensions is ``n`` for ``p == 1`` or ``p == n`` and one otherwise,
92 by [Jeong]_ Corollary 5.2.4::
93
94 sage: all( rearrangement_cone(p,n).lyapunov_rank() == n
95 ....: for n in range(2, 10)
96 ....: for p in [1, n-1] )
97 True
98 sage: all( rearrangement_cone(p,n).lyapunov_rank() == 1
99 ....: for n in range(3, 10)
100 ....: for p in range(2, n-1) )
101 True
102
103 TESTS:
104
105 All rearrangement cones are permutation-invariant by [Jeong]_
106 Proposition 5.2.1::
107
108 sage: n = ZZ.random_element(2,10).abs()
109 sage: p = ZZ.random_element(1,n)
110 sage: K = rearrangement_cone(p,n)
111 sage: P = SymmetricGroup(n).random_element().matrix()
112 sage: all( K.contains(P*r) for r in K )
113 True
114
115 The smallest ``p`` components of every element of the rearrangement
116 cone should sum to a nonnegative number (this tests that the
117 generators really are what we think they are)::
118
119 sage: set_random_seed()
120 sage: def _has_rearrangement_property(v,p):
121 ....: return sum( sorted(v)[0:p] ) >= 0
122 sage: all( _has_rearrangement_property(
123 ....: rearrangement_cone(p,n).random_element(),
124 ....: p
125 ....: )
126 ....: for n in range(2, 10)
127 ....: for p in range(1, n-1)
128 ....: )
129 True
130
131 The rearrangenent cone of order ``p`` is contained in the rearrangement
132 cone of order ``p + 1`` by [Jeong]_ Proposition 5.2.1::
133
134 sage: set_random_seed()
135 sage: n = ZZ.random_element(2,10)
136 sage: p = ZZ.random_element(1,n)
137 sage: K1 = rearrangement_cone(p,n)
138 sage: K2 = rearrangement_cone(p+1,n)
139 sage: all( x in K2 for x in K1 )
140 True
141
142 The rearrangement cone of order ``p`` is linearly isomorphic to the
143 rearrangement cone of order ``n - p`` when ``p`` is less than ``n``,
144 by [Jeong]_ Proposition 5.2.1::
145
146 sage: set_random_seed()
147 sage: n = ZZ.random_element(2,10)
148 sage: p = ZZ.random_element(1,n)
149 sage: K1 = rearrangement_cone(p,n)
150 sage: K2 = rearrangement_cone(n-p, n)
151 sage: Mp = (1/p)*matrix.ones(QQ,n) - identity_matrix(QQ,n)
152 sage: Cone( (Mp*K2.rays()).columns() ).is_equivalent(K1)
153 True
154
155 The order ``p`` should be between ``1`` and ``n``, inclusive::
156
157 sage: rearrangement_cone(0,3)
158 Traceback (most recent call last):
159 ...
160 ValueError: order p=0 should be between 1 and n=3, inclusive
161 sage: rearrangement_cone(5,3)
162 Traceback (most recent call last):
163 ...
164 ValueError: order p=5 should be between 1 and n=3, inclusive
165
166 If a ``lattice`` was given, it is actually used::
167
168 sage: L = ToricLattice(3, 'M')
169 sage: rearrangement_cone(2, 3, lattice=L)
170 3-d cone in 3-d lattice M
171
172 Unless the rank of the lattice disagrees with ``n``::
173
174 sage: L = ToricLattice(1, 'M')
175 sage: rearrangement_cone(2, 3, lattice=L)
176 Traceback (most recent call last):
177 ...
178 ValueError: lattice rank=1 and dimension n=3 are incompatible
179
180 """
181 if p < 1 or p > n:
182 raise ValueError('order p=%d should be between 1 and n=%d, inclusive'
183 %
184 (p,n))
185
186 if lattice is None:
187 lattice = ToricLattice(n)
188
189 if lattice.rank() != n:
190 raise ValueError('lattice rank=%d and dimension n=%d are incompatible'
191 %
192 (lattice.rank(), n))
193
194 I = identity_matrix(ZZ,n)
195 M = matrix.ones(ZZ,n) - p*I
196 G = identity_matrix(ZZ,n).rows() + M.rows()
197 return Cone(G, lattice=lattice)