]>
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 from mjo
.cone
.symmetric_psd
import (factor_psd
,
18 from mjo
.basis_repr
import basis_repr
21 def is_doubly_nonnegative(A
):
23 Determine whether or not the matrix ``A`` is doubly-nonnegative.
27 - ``A`` - The matrix in question
31 Either ``True`` if ``A`` is doubly-nonnegative, or ``False``
36 sage: from mjo.cone.doubly_nonnegative import is_doubly_nonnegative
40 Every completely positive matrix is doubly-nonnegative::
42 sage: v = vector(map(abs, random_vector(ZZ, 10)))
43 sage: A = v.column() * v.row()
44 sage: is_doubly_nonnegative(A)
47 The following matrix is nonnegative but non positive semidefinite::
49 sage: A = matrix(ZZ, [[1, 2], [2, 1]])
50 sage: is_doubly_nonnegative(A)
55 if A
.base_ring() == SR
:
56 msg
= 'The matrix ``A`` cannot be the symbolic.'
57 raise ValueError.new(msg
)
59 # Check that all of the entries of ``A`` are nonnegative.
60 if not all( a
>= 0 for a
in A
.list() ):
63 # It's nonnegative, so all we need to do is check that it's
64 # symmetric positive-semidefinite.
65 return A
.is_positive_semidefinite()
69 def is_admissible_extreme_rank(r
, n
):
71 The extreme matrices of the doubly-nonnegative cone have some
72 restrictions on their ranks. This function checks to see whether the
73 rank ``r`` would be an admissible rank for an ``n``-by-``n`` matrix.
77 - ``r`` - The rank of the matrix.
79 - ``n`` - The dimension of the vector space on which the matrix acts.
83 Either ``True`` if a rank ``r`` matrix could be an extreme vector of
84 the doubly-nonnegative cone in `$\mathbb{R}^{n}$`, or ``False``
89 sage: from mjo.cone.doubly_nonnegative import is_admissible_extreme_rank
93 For dimension 5, only ranks zero, one, and three are admissible::
95 sage: is_admissible_extreme_rank(0,5)
97 sage: is_admissible_extreme_rank(1,5)
99 sage: is_admissible_extreme_rank(2,5)
101 sage: is_admissible_extreme_rank(3,5)
103 sage: is_admissible_extreme_rank(4,5)
105 sage: is_admissible_extreme_rank(5,5)
108 When given an impossible rank, we just return false::
110 sage: is_admissible_extreme_rank(100,5)
115 # Zero is in the doubly-nonnegative cone.
119 # Impossible, just return False
122 # See Theorem 3.1 in the cited reference.
128 return r
<= max(1, n
-3)
131 return r
<= max(1, n
-2)
134 def has_admissible_extreme_rank(A
):
136 The extreme matrices of the doubly-nonnegative cone have some
137 restrictions on their ranks. This function checks to see whether or
138 not ``A`` could be extreme based on its rank.
142 - ``A`` - The matrix in question
146 ``False`` if the rank of ``A`` precludes it from being an extreme
147 matrix of the doubly-nonnegative cone, ``True`` otherwise.
151 Hamilton-Jester, Crista Lee; Li, Chi-Kwong. Extreme Vectors of
152 Doubly Nonnegative Matrices. Rocky Mountain Journal of Mathematics
153 26 (1996), no. 4, 1371--1383. doi:10.1216/rmjm/1181071993.
154 http://projecteuclid.org/euclid.rmjm/1181071993.
158 sage: from mjo.cone.doubly_nonnegative import has_admissible_extreme_rank
162 The zero matrix has rank zero, which is admissible::
164 sage: A = zero_matrix(QQ, 5, 5)
165 sage: has_admissible_extreme_rank(A)
168 Likewise, rank one is admissible for dimension 5::
170 sage: v = vector(QQ, [1,2,3,4,5])
171 sage: A = v.column()*v.row()
172 sage: has_admissible_extreme_rank(A)
175 But rank 2 is never admissible::
177 sage: v1 = vector(QQ, [1,0,0,0,0])
178 sage: v2 = vector(QQ, [0,1,0,0,0])
179 sage: A = v1.column()*v1.row() + v2.column()*v2.row()
180 sage: has_admissible_extreme_rank(A)
183 In dimension 5, three is the only other admissible rank::
185 sage: v1 = vector(QQ, [1,0,0,0,0])
186 sage: v2 = vector(QQ, [0,1,0,0,0])
187 sage: v3 = vector(QQ, [0,0,1,0,0])
188 sage: A = v1.column()*v1.row()
189 sage: A += v2.column()*v2.row()
190 sage: A += v3.column()*v3.row()
191 sage: has_admissible_extreme_rank(A)
195 if not A
.is_symmetric():
196 # This function is more or less internal, so blow up if passed
197 # something unexpected.
198 raise ValueError('The matrix ``A`` must be symmetric.')
201 n
= ZZ(A
.nrows()) # Columns would work, too, since ``A`` is symmetric.
203 return is_admissible_extreme_rank(r
,n
)
206 def stdE(matrix_space
, i
,j
):
208 Return the ``i``,``j``th element of the standard basis in
213 - ``matrix_space`` - The underlying matrix space of whose basis
214 the returned matrix is an element
216 - ``i`` - The row index of the single nonzero entry
218 - ``j`` - The column index of the single nonzero entry
222 A basis element of ``matrix_space``. It has a single \"1\" in the
223 ``i``,``j`` row,column and zeros elsewhere.
227 sage: from mjo.cone.doubly_nonnegative import stdE
231 sage: M = MatrixSpace(ZZ, 2, 2)
245 Traceback (most recent call last):
247 IndexError: Index `i` is out of bounds.
249 Traceback (most recent call last):
251 IndexError: Index `j` is out of bounds.
254 # We need to check these ourselves, see below.
255 if i
>= matrix_space
.nrows():
256 raise IndexError('Index `i` is out of bounds.')
257 if j
>= matrix_space
.ncols():
258 raise IndexError('Index `j` is out of bounds.')
260 # The basis here is returned as a one-dimensional list, so we need
261 # to compute the offset into it based on ``i`` and ``j``. Since we
262 # compute the index ourselves, we need to do bounds-checking
263 # manually. Otherwise for e.g. a 2x2 matrix space, the index (0,2)
264 # would be computed as offset 3 into a four-element list and we
265 # would succeed incorrectly.
266 idx
= matrix_space
.ncols()*i
+ j
267 return list(matrix_space
.basis())[idx
]
271 def is_extreme_doubly_nonnegative(A
):
273 Returns ``True`` if the given matrix is an extreme matrix of the
274 doubly-nonnegative cone, and ``False`` otherwise.
278 1. Hamilton-Jester, Crista Lee; Li, Chi-Kwong. Extreme Vectors of
279 Doubly Nonnegative Matrices. Rocky Mountain Journal of Mathematics
280 26 (1996), no. 4, 1371--1383. doi:10.1216/rmjm/1181071993.
281 http://projecteuclid.org/euclid.rmjm/1181071993.
283 2. Berman, Abraham and Shaked-Monderer, Naomi. Completely Positive
284 Matrices. World Scientific, 2003.
288 sage: from mjo.cone.doubly_nonnegative import is_extreme_doubly_nonnegative
292 The zero matrix is an extreme matrix::
294 sage: A = zero_matrix(QQ, 5, 5)
295 sage: is_extreme_doubly_nonnegative(A)
298 Any extreme vector of the completely positive cone is an extreme
299 vector of the doubly-nonnegative cone::
301 sage: v = vector([1,2,3,4,5,6])
302 sage: A = v.column() * v.row()
303 sage: A = A.change_ring(QQ)
304 sage: is_extreme_doubly_nonnegative(A)
307 We should be able to generate the extreme completely positive
310 sage: v = vector(map(abs, random_vector(ZZ, 4)))
311 sage: A = v.column() * v.row()
312 sage: A = A.change_ring(QQ)
313 sage: is_extreme_doubly_nonnegative(A)
315 sage: v = vector(map(abs, random_vector(ZZ, 10)))
316 sage: A = v.column() * v.row()
317 sage: A = A.change_ring(QQ)
318 sage: is_extreme_doubly_nonnegative(A)
321 The following matrix is completely positive but has rank 3, so by a
322 remark in reference #1 it is not extreme::
324 sage: A = matrix(QQ, [[1,2,1],[2,6,3],[1,3,5]])
325 sage: is_extreme_doubly_nonnegative(A)
328 The following matrix is completely positive (diagonal) with rank 2,
329 so it is also not extreme::
331 sage: A = matrix(QQ, [[1,0,0],[2,0,0],[0,0,0]])
332 sage: is_extreme_doubly_nonnegative(A)
337 if not A
.base_ring().is_exact() and not A
.base_ring() is SR
:
338 msg
= 'The base ring of ``A`` must be either exact or symbolic.'
339 raise ValueError(msg
)
341 if not A
.base_ring().is_field():
342 raise ValueError('The base ring of ``A`` must be a field.')
344 if not A
.base_ring() is SR
:
345 # Change the base field of ``A`` so that we are sure we can take
346 # roots. The symbolic ring has no algebraic_closure method.
347 A
= A
.change_ring(A
.base_ring().algebraic_closure())
349 # Step 1 (see reference #1)
353 # Short circuit, we know the zero matrix is extreme.
356 if not A
.is_positive_semidefinite():
359 # Step 1.5, appeal to Theorem 3.1 in reference #1 to short
361 if not has_admissible_extreme_rank(A
):
369 # Begin with an empty spanning set, and add a new matrix to it
370 # whenever we come across an index pair `$(i,j)$` with
373 for j
in range(A
.ncols()):
377 S
= X
.transpose() * (stdE(M
,i
,j
) + stdE(M
,j
,i
)) * X
378 spanning_set
.append(S
)
380 # The spanning set that we have at this point is of matrices. We
381 # only care about the dimension of the spanned space, and Sage
382 # can't compute the dimension of a set of matrices anyway, so we
383 # convert them all to vectors and just ask for the dimension of the
384 # resulting vector space.
385 (phi
, phi_inverse
) = basis_repr(A
.matrix_space())
386 vectors
= map(phi
,spanning_set
)
388 V
= span(vectors
, A
.base_ring())
391 # Needed to safely divide by two here (we don't want integer
392 # division). We ensured that the base ring of ``A`` is a field
394 two
= A
.base_ring()(2)
395 return d
== (k
*(k
+ 1)/two
- 1)
398 def random_doubly_nonnegative(V
, accept_zero
=True, rank
=None):
400 Generate a random doubly nonnegative matrix over the vector
401 space ``V``. That is, the returned matrix will be a linear
402 transformation on ``V``, with the same base ring as ``V``.
404 We take a very loose interpretation of "random," here. Otherwise we
405 would never (for example) choose a matrix on the boundary of the
410 - ``V`` - The vector space on which the returned matrix will act.
412 - ``accept_zero`` - Do you want to accept the zero matrix (which
413 is doubly nonnegative)? Default to ``True``.
415 - ``rank`` - Require the returned matrix to have the given rank
420 A random doubly nonnegative matrix, i.e. a linear transformation
421 from ``V`` to itself.
425 sage: from mjo.cone.doubly_nonnegative import (is_doubly_nonnegative,
426 ....: random_doubly_nonnegative)
430 Well, it doesn't crash at least::
432 sage: V = VectorSpace(QQ, 2)
433 sage: A = random_doubly_nonnegative(V)
434 sage: A.matrix_space()
435 Full MatrixSpace of 2 by 2 dense matrices over Rational Field
436 sage: is_doubly_nonnegative(A)
439 A matrix with the desired rank is returned::
441 sage: V = VectorSpace(QQ, 5)
442 sage: A = random_doubly_nonnegative(V,False,1)
445 sage: A = random_doubly_nonnegative(V,False,2)
448 sage: A = random_doubly_nonnegative(V,False,3)
451 sage: A = random_doubly_nonnegative(V,False,4)
454 sage: A = random_doubly_nonnegative(V,False,5)
460 # Generate random symmetric positive-semidefinite matrices until
461 # one of them is nonnegative, then return that.
462 A
= random_symmetric_psd(V
, accept_zero
, rank
)
464 while not all( x
>= 0 for x
in A
.list() ):
465 A
= random_symmetric_psd(V
, accept_zero
, rank
)
471 def random_extreme_doubly_nonnegative(V
, accept_zero
=True, rank
=None):
473 Generate a random extreme doubly nonnegative matrix over the
474 vector space ``V``. That is, the returned matrix will be a linear
475 transformation on ``V``, with the same base ring as ``V``.
477 We take a very loose interpretation of "random," here. Otherwise we
478 would never (for example) choose a matrix on the boundary of the
483 - ``V`` - The vector space on which the returned matrix will act.
485 - ``accept_zero`` - Do you want to accept the zero matrix
486 (which is extreme)? Defaults to ``True``.
488 - ``rank`` - Require the returned matrix to have the given rank
489 (optional). WARNING: certain ranks are not possible
490 in any given dimension! If an impossible rank is
491 requested, a ValueError will be raised.
495 A random extreme doubly nonnegative matrix, i.e. a linear
496 transformation from ``V`` to itself.
500 sage: from mjo.cone.doubly_nonnegative import (is_extreme_doubly_nonnegative,
501 ....: random_extreme_doubly_nonnegative)
505 Well, it doesn't crash at least::
507 sage: V = VectorSpace(QQ, 2)
508 sage: A = random_extreme_doubly_nonnegative(V)
509 sage: A.matrix_space()
510 Full MatrixSpace of 2 by 2 dense matrices over Rational Field
511 sage: is_extreme_doubly_nonnegative(A)
514 Rank 2 is never allowed, so we expect an error::
516 sage: V = VectorSpace(QQ, 5)
517 sage: A = random_extreme_doubly_nonnegative(V, False, 2)
518 Traceback (most recent call last):
520 ValueError: Rank 2 not possible in dimension 5.
522 Rank 4 is not allowed in dimension 5::
524 sage: V = VectorSpace(QQ, 5)
525 sage: A = random_extreme_doubly_nonnegative(V, False, 4)
526 Traceback (most recent call last):
528 ValueError: Rank 4 not possible in dimension 5.
532 if rank
is not None and not is_admissible_extreme_rank(rank
, V
.dimension()):
533 msg
= 'Rank %d not possible in dimension %d.'
534 raise ValueError(msg
% (rank
, V
.dimension()))
536 # Generate random doubly-nonnegative matrices until
537 # one of them is extreme, then return that.
538 A
= random_doubly_nonnegative(V
, accept_zero
, rank
)
540 while not is_extreme_doubly_nonnegative(A
):
541 A
= random_doubly_nonnegative(V
, accept_zero
, rank
)