return True # vacuously
return A.is_hermitian() and all( v >= 0 for v in A.eigenvalues() )
-def ldlt_naive(A):
- r"""
- Perform a pivoted `LDL^{T}` factorization of the Hermitian
- positive-semidefinite matrix `A`.
-
- This is a naive, recursive implementation that is inefficient due
- to Python's lack of tail-call optimization. The pivot strategy is
- to choose the largest diagonal entry of the matrix at each step,
- and to permute it into the top-left position. Ultimately this
- results in a factorization `A = PLDL^{T}P^{T}`, where `P` is a
- permutation matrix, `L` is unit-lower-triangular, and `D` is
- diagonal decreasing from top-left to bottom-right.
-
- ALGORITHM:
-
- The algorithm is based on the discussion in Golub and Van Loan, but with
- some "typos" fixed.
- OUTPUT:
-
- A triple `(P,L,D)` such that `A = PLDL^{T}P^{T}` and where,
-
- * `P` is a permutaiton matrix
- * `L` is unit lower-triangular
- * `D` is a diagonal matrix whose entries are decreasing from top-left
- to bottom-right
+def is_positive_semidefinite(A):
+ r"""
+ A fast positive-semidefinite check based on the block-LDLT
+ factorization.
SETUP::
- sage: from mjo.ldlt import ldlt_naive, is_positive_semidefinite_naive
+ sage: from mjo.ldlt import (is_positive_semidefinite,
+ ....: is_positive_semidefinite_naive)
- EXAMPLES:
+ TESTS:
- All three factors should be the identity when the original matrix is::
+ Check that the naive and fast answers are the same, in general::
- sage: I = matrix.identity(QQ,4)
- sage: P,L,D = ldlt_naive(I)
- sage: P == I and L == I and D == I
+ sage: set_random_seed()
+ sage: F = NumberField(x^2 + 1, 'I')
+ sage: from sage.misc.prandom import choice
+ sage: ring = choice([ZZ,QQ,F])
+ sage: A = matrix.random(ring, 10)
+ sage: is_positive_semidefinite(A) == is_positive_semidefinite_naive(A)
True
- TESTS:
-
- Ensure that a "random" positive-semidefinite matrix is factored correctly::
+ Check that the naive and fast answers are the same for a Hermitian
+ matrix::
sage: set_random_seed()
- sage: n = ZZ.random_element(5)
- sage: A = matrix.random(QQ, n)
- sage: A = A*A.transpose()
- sage: is_positive_semidefinite_naive(A)
- True
- sage: P,L,D = ldlt_naive(A)
- sage: A == P*L*D*L.transpose()*P.transpose()
+ sage: F = NumberField(x^2 + 1, 'I')
+ sage: from sage.misc.prandom import choice
+ sage: ring = choice([ZZ,QQ,F])
+ sage: A = matrix.random(ring, 10); A = A + A.conjugate_transpose()
+ sage: is_positive_semidefinite(A) == is_positive_semidefinite_naive(A)
True
"""
- n = A.nrows()
-
- # Use the fraction field of the given matrix so that division will work
- # when (for example) our matrix consists of integer entries.
- ring = A.base_ring().fraction_field()
-
- if n == 0 or n == 1:
- # We can get n == 0 if someone feeds us a trivial matrix.
- P = matrix.identity(ring, n)
- L = matrix.identity(ring, n)
- D = A
- return (P,L,D)
-
- A1 = A.change_ring(ring)
- diags = A1.diagonal()
- s = diags.index(max(diags))
- P1 = copy(A1.matrix_space().identity_matrix())
- P1.swap_rows(0,s)
- A1 = P1.T * A1 * P1
- alpha1 = A1[0,0]
-
- # Golub and Van Loan mention in passing what to do here. This is
- # only sensible if the matrix is positive-semidefinite, because we
- # are assuming that we can set everything else to zero as soon as
- # we hit the first on-diagonal zero.
- if alpha1 == 0:
- P = A1.matrix_space().identity_matrix()
- L = P
- D = A1.matrix_space().zero()
- return (P,L,D)
-
- v1 = A1[1:n,0]
- A2 = A1[1:,1:]
-
- P2, L2, D2 = ldlt_naive(A2 - (v1*v1.transpose())/alpha1)
-
- P1 = P1*block_matrix(2,2, [[ZZ(1), ZZ(0)],
- [0*v1, P2]])
- L1 = block_matrix(2,2, [[ZZ(1), ZZ(0)],
- [P2.transpose()*v1/alpha1, L2]])
- D1 = block_matrix(2,2, [[alpha1, ZZ(0)],
- [0*v1, D2]])
-
- return (P1,L1,D1)
+ if not A.is_hermitian():
+ return False
+ _,_,d = A._block_ldlt()
+ for d_i in d:
+ if d_i.nrows() == 1:
+ if d_i < 0:
+ return False
+ else:
+ # A 2x2 block indicates that it's indefinite
+ return False
+ return True
projects / sage.d.git / blobdiff