]>
gitweb.michael.orlitzky.com - sage.d.git/blob - mjo/cone/doubly_nonnegative.py
2 The doubly-nonnegative cone in `S^{n}` is the set of all such matrices
5 a) are positive semidefinite
7 b) have only nonnegative entries
9 It is represented typically by either `\mathcal{D}^{n}` or
14 from sage
.all
import *
16 # Sage doesn't load ~/.sage/init.sage during testing (sage -t), so we
17 # have to explicitly mangle our sitedir here so that our module names
19 from os
.path
import abspath
20 from site
import addsitedir
21 addsitedir(abspath('../../'))
22 from mjo
.cone
.symmetric_psd
import factor_psd
, is_symmetric_psd
, random_psd
23 from mjo
.matrix_vector
import isomorphism
26 def is_doubly_nonnegative(A
):
28 Determine whether or not the matrix ``A`` is doubly-nonnegative.
32 - ``A`` - The matrix in question
36 Either ``True`` if ``A`` is doubly-nonnegative, or ``False``
41 Every completely positive matrix is doubly-nonnegative::
43 sage: v = vector(map(abs, random_vector(ZZ, 10)))
44 sage: A = v.column() * v.row()
45 sage: is_doubly_nonnegative(A)
48 The following matrix is nonnegative but non positive semidefinite::
50 sage: A = matrix(ZZ, [[1, 2], [2, 1]])
51 sage: is_doubly_nonnegative(A)
56 if A
.base_ring() == SR
:
57 msg
= 'The matrix ``A`` cannot be the symbolic.'
58 raise ValueError.new(msg
)
60 # Check that all of the entries of ``A`` are nonnegative.
61 if not all([ a
>= 0 for a
in A
.list() ]):
64 # It's nonnegative, so all we need to do is check that it's
65 # symmetric positive-semidefinite.
66 return is_symmetric_psd(A
)
70 def is_admissible_extreme_rank(r
, n
):
72 The extreme matrices of the doubly-nonnegative cone have some
73 restrictions on their ranks. This function checks to see whether the
74 rank ``r`` would be an admissible rank for an ``n``-by-``n`` matrix.
78 - ``r`` - The rank of the matrix.
80 - ``n`` - The dimension of the vector space on which the matrix acts.
84 Either ``True`` if a rank ``r`` matrix could be an extreme vector of
85 the doubly-nonnegative cone in `$\mathbb{R}^{n}$`, or ``False``
90 For dimension 5, only ranks zero, one, and three are admissible::
92 sage: is_admissible_extreme_rank(0,5)
94 sage: is_admissible_extreme_rank(1,5)
96 sage: is_admissible_extreme_rank(2,5)
98 sage: is_admissible_extreme_rank(3,5)
100 sage: is_admissible_extreme_rank(4,5)
102 sage: is_admissible_extreme_rank(5,5)
105 When given an impossible rank, we just return false::
107 sage: is_admissible_extreme_rank(100,5)
112 # Zero is in the doubly-nonnegative cone.
116 # Impossible, just return False
119 # See Theorem 3.1 in the cited reference.
125 return r
<= max(1, n
-3)
128 return r
<= max(1, n
-2)
131 def has_admissible_extreme_rank(A
):
133 The extreme matrices of the doubly-nonnegative cone have some
134 restrictions on their ranks. This function checks to see whether or
135 not ``A`` could be extreme based on its rank.
139 - ``A`` - The matrix in question
143 ``False`` if the rank of ``A`` precludes it from being an extreme
144 matrix of the doubly-nonnegative cone, ``True`` otherwise.
148 Hamilton-Jester, Crista Lee; Li, Chi-Kwong. Extreme Vectors of
149 Doubly Nonnegative Matrices. Rocky Mountain Journal of Mathematics
150 26 (1996), no. 4, 1371--1383. doi:10.1216/rmjm/1181071993.
151 http://projecteuclid.org/euclid.rmjm/1181071993.
155 The zero matrix has rank zero, which is admissible::
157 sage: A = zero_matrix(QQ, 5, 5)
158 sage: has_admissible_extreme_rank(A)
161 Likewise, rank one is admissible for dimension 5::
163 sage: v = vector(QQ, [1,2,3,4,5])
164 sage: A = v.column()*v.row()
165 sage: has_admissible_extreme_rank(A)
168 But rank 2 is never admissible::
170 sage: v1 = vector(QQ, [1,0,0,0,0])
171 sage: v2 = vector(QQ, [0,1,0,0,0])
172 sage: A = v1.column()*v1.row() + v2.column()*v2.row()
173 sage: has_admissible_extreme_rank(A)
176 In dimension 5, three is the only other admissible rank::
178 sage: v1 = vector(QQ, [1,0,0,0,0])
179 sage: v2 = vector(QQ, [0,1,0,0,0])
180 sage: v3 = vector(QQ, [0,0,1,0,0])
181 sage: A = v1.column()*v1.row()
182 sage: A += v2.column()*v2.row()
183 sage: A += v3.column()*v3.row()
184 sage: has_admissible_extreme_rank(A)
188 if not A
.is_symmetric():
189 # This function is more or less internal, so blow up if passed
190 # something unexpected.
191 raise ValueError('The matrix ``A`` must be symmetric.')
194 n
= ZZ(A
.nrows()) # Columns would work, too, since ``A`` is symmetric.
196 return is_admissible_extreme_rank(r
,n
)
199 def E(matrix_space
, i
,j
):
201 Return the ``i``,``j``th element of the standard basis in
206 - ``matrix_space`` - The underlying matrix space of whose basis
207 the returned matrix is an element
209 - ``i`` - The row index of the single nonzero entry
211 - ``j`` - The column index of the single nonzero entry
215 A basis element of ``matrix_space``. It has a single \"1\" in the
216 ``i``,``j`` row,column and zeros elsewhere.
220 sage: M = MatrixSpace(ZZ, 2, 2)
234 Traceback (most recent call last):
236 IndexError: Index `i` is out of bounds.
238 Traceback (most recent call last):
240 IndexError: Index `j` is out of bounds.
243 # We need to check these ourselves, see below.
244 if i
>= matrix_space
.nrows():
245 raise IndexError('Index `i` is out of bounds.')
246 if j
>= matrix_space
.ncols():
247 raise IndexError('Index `j` is out of bounds.')
249 # The basis here is returned as a one-dimensional list, so we need
250 # to compute the offset into it based on ``i`` and ``j``. Since we
251 # compute the index ourselves, we need to do bounds-checking
252 # manually. Otherwise for e.g. a 2x2 matrix space, the index (0,2)
253 # would be computed as offset 3 into a four-element list and we
254 # would succeed incorrectly.
255 idx
= matrix_space
.ncols()*i
+ j
256 return matrix_space
.basis()[idx
]
260 def is_extreme_doubly_nonnegative(A
):
262 Returns ``True`` if the given matrix is an extreme matrix of the
263 doubly-nonnegative cone, and ``False`` otherwise.
267 1. Hamilton-Jester, Crista Lee; Li, Chi-Kwong. Extreme Vectors of
268 Doubly Nonnegative Matrices. Rocky Mountain Journal of Mathematics
269 26 (1996), no. 4, 1371--1383. doi:10.1216/rmjm/1181071993.
270 http://projecteuclid.org/euclid.rmjm/1181071993.
272 2. Berman, Abraham and Shaked-Monderer, Naomi. Completely Positive
273 Matrices. World Scientific, 2003.
277 The zero matrix is an extreme matrix::
279 sage: A = zero_matrix(QQ, 5, 5)
280 sage: is_extreme_doubly_nonnegative(A)
283 Any extreme vector of the completely positive cone is an extreme
284 vector of the doubly-nonnegative cone::
286 sage: v = vector([1,2,3,4,5,6])
287 sage: A = v.column() * v.row()
288 sage: A = A.change_ring(QQ)
289 sage: is_extreme_doubly_nonnegative(A)
292 We should be able to generate the extreme completely positive
295 sage: v = vector(map(abs, random_vector(ZZ, 4)))
296 sage: A = v.column() * v.row()
297 sage: A = A.change_ring(QQ)
298 sage: is_extreme_doubly_nonnegative(A)
300 sage: v = vector(map(abs, random_vector(ZZ, 10)))
301 sage: A = v.column() * v.row()
302 sage: A = A.change_ring(QQ)
303 sage: is_extreme_doubly_nonnegative(A)
306 The following matrix is completely positive but has rank 3, so by a
307 remark in reference #1 it is not extreme::
309 sage: A = matrix(QQ, [[1,2,1],[2,6,3],[1,3,5]])
310 sage: is_extreme_doubly_nonnegative(A)
313 The following matrix is completely positive (diagonal) with rank 2,
314 so it is also not extreme::
316 sage: A = matrix(QQ, [[1,0,0],[2,0,0],[0,0,0]])
317 sage: is_extreme_doubly_nonnegative(A)
322 if not A
.base_ring().is_exact() and not A
.base_ring() is SR
:
323 msg
= 'The base ring of ``A`` must be either exact or symbolic.'
324 raise ValueError(msg
)
326 if not A
.base_ring().is_field():
327 raise ValueError('The base ring of ``A`` must be a field.')
329 if not A
.base_ring() is SR
:
330 # Change the base field of ``A`` so that we are sure we can take
331 # roots. The symbolic ring has no algebraic_closure method.
332 A
= A
.change_ring(A
.base_ring().algebraic_closure())
334 # Step 1 (see reference #1)
338 # Short circuit, we know the zero matrix is extreme.
341 if not is_symmetric_psd(A
):
344 # Step 1.5, appeal to Theorem 3.1 in reference #1 to short
346 if not has_admissible_extreme_rank(A
):
354 # Begin with an empty spanning set, and add a new matrix to it
355 # whenever we come across an index pair `$(i,j)$` with
358 for j
in range(0, A
.ncols()):
362 S
= X
.transpose() * (E(M
,i
,j
) + E(M
,j
,i
)) * X
363 spanning_set
.append(S
)
365 # The spanning set that we have at this point is of matrices. We
366 # only care about the dimension of the spanned space, and Sage
367 # can't compute the dimension of a set of matrices anyway, so we
368 # convert them all to vectors and just ask for the dimension of the
369 # resulting vector space.
370 (phi
, phi_inverse
) = isomorphism(A
.matrix_space())
371 vectors
= map(phi
,spanning_set
)
373 V
= span(vectors
, A
.base_ring())
376 # Needed to safely divide by two here (we don't want integer
377 # division). We ensured that the base ring of ``A`` is a field
379 two
= A
.base_ring()(2)
380 return d
== (k
*(k
+ 1)/two
- 1)
383 def random_doubly_nonnegative(V
, accept_zero
=True, rank
=None):
385 Generate a random doubly nonnegative matrix over the vector
386 space ``V``. That is, the returned matrix will be a linear
387 transformation on ``V``, with the same base ring as ``V``.
389 We take a very loose interpretation of "random," here. Otherwise we
390 would never (for example) choose a matrix on the boundary of the
395 - ``V`` - The vector space on which the returned matrix will act.
397 - ``accept_zero`` - Do you want to accept the zero matrix (which
398 is doubly nonnegative)? Default to ``True``.
400 - ``rank`` - Require the returned matrix to have the given rank
405 A random doubly nonnegative matrix, i.e. a linear transformation
406 from ``V`` to itself.
410 Well, it doesn't crash at least::
412 sage: V = VectorSpace(QQ, 2)
413 sage: A = random_doubly_nonnegative(V)
414 sage: A.matrix_space()
415 Full MatrixSpace of 2 by 2 dense matrices over Rational Field
416 sage: is_doubly_nonnegative(A)
419 A matrix with the desired rank is returned::
421 sage: V = VectorSpace(QQ, 5)
422 sage: A = random_doubly_nonnegative(V,False,1)
425 sage: A = random_doubly_nonnegative(V,False,2)
428 sage: A = random_doubly_nonnegative(V,False,3)
431 sage: A = random_doubly_nonnegative(V,False,4)
434 sage: A = random_doubly_nonnegative(V,False,5)
440 # Generate random symmetric positive-semidefinite matrices until
441 # one of them is nonnegative, then return that.
442 A
= random_psd(V
, accept_zero
, rank
)
444 while not all([ x
>= 0 for x
in A
.list() ]):
445 A
= random_psd(V
, accept_zero
, rank
)
451 def random_extreme_doubly_nonnegative(V
, accept_zero
=True, rank
=None):
453 Generate a random extreme doubly nonnegative matrix over the
454 vector space ``V``. That is, the returned matrix will be a linear
455 transformation on ``V``, with the same base ring as ``V``.
457 We take a very loose interpretation of "random," here. Otherwise we
458 would never (for example) choose a matrix on the boundary of the
463 - ``V`` - The vector space on which the returned matrix will act.
465 - ``accept_zero`` - Do you want to accept the zero matrix
466 (which is extreme)? Defaults to ``True``.
468 - ``rank`` - Require the returned matrix to have the given rank
469 (optional). WARNING: certain ranks are not possible
470 in any given dimension! If an impossible rank is
471 requested, a ValueError will be raised.
475 A random extreme doubly nonnegative matrix, i.e. a linear
476 transformation from ``V`` to itself.
480 Well, it doesn't crash at least::
482 sage: V = VectorSpace(QQ, 2)
483 sage: A = random_extreme_doubly_nonnegative(V)
484 sage: A.matrix_space()
485 Full MatrixSpace of 2 by 2 dense matrices over Rational Field
486 sage: is_extreme_doubly_nonnegative(A)
489 Rank 2 is never allowed, so we expect an error::
491 sage: V = VectorSpace(QQ, 5)
492 sage: A = random_extreme_doubly_nonnegative(V, False, 2)
493 Traceback (most recent call last):
495 ValueError: Rank 2 not possible in dimension 5.
497 Rank 4 is not allowed in dimension 5::
499 sage: V = VectorSpace(QQ, 5)
500 sage: A = random_extreme_doubly_nonnegative(V, False, 4)
501 Traceback (most recent call last):
503 ValueError: Rank 4 not possible in dimension 5.
507 if not is_admissible_extreme_rank(rank
, V
.dimension()):
508 msg
= 'Rank %d not possible in dimension %d.'
509 raise ValueError(msg
% (rank
, V
.dimension()))
511 # Generate random doubly-nonnegative matrices until
512 # one of them is extreme, then return that.
513 A
= random_doubly_nonnegative(V
, accept_zero
, rank
)
515 while not is_extreme_doubly_nonnegative(A
):
516 A
= random_doubly_nonnegative(V
, accept_zero
, rank
)