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)
+ 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
- sage: P,L,D = ldlt_naive(A)
- sage: A == P*L*D*L.transpose()*P.transpose()
- 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)
-
-
-
-def ldlt_fast(A):
- r"""
- Perform a fast, pivoted `LDL^{T}` factorization of the Hermitian
- positive-semidefinite matrix `A`.
- This function is much faster than ``ldlt_naive`` because the
- tail-recursion has been unrolled into a loop.
"""
- n = A.nrows()
- ring = A.base_ring().fraction_field()
-
- A = A.change_ring(ring)
-
- # Don't try to store the results in the lower-left-hand corner of
- # "A" itself; there lies madness.
- L = copy(A.matrix_space().identity_matrix())
- D = copy(A.matrix_space().zero())
-
- # Keep track of the permutations in a vector rather than in a
- # matrix, for efficiency.
- p = list(range(n))
-
- for k in range(n):
- # We need to loop once for every diagonal entry in the
- # matrix. So, as many times as it has rows/columns. At each
- # step, we obtain the permutation needed to put things in the
- # right place, then the "next" entry (alpha) of D, and finally
- # another column of L.
- diags = A.diagonal()[k:n]
- alpha = max(diags)
-
- # We're working *within* the matrix ``A``, so every index is
- # offset by k. For example: after the second step, we should
- # only be looking at the lower 3-by-3 block of a 5-by-5 matrix.
- s = k + diags.index(alpha)
-
- # Move the largest diagonal element up into the top-left corner
- # of the block we're working on (the one starting from index k,k).
- # Presumably this is faster than hitting the thing with a
- # permutation matrix.
- A.swap_columns(k,s)
- A.swap_rows(k,s)
-
- # Have to do L, too, to keep track of the "P2.T" (which is 1 x
- # P3.T which is 1 x P4 T)... in the recursive
- # algorithm. There, we compute P2^T from the bottom up. Here,
- # we apply the permutations one at a time, essentially
- # building them top-down (but just applying them instead of
- # building them.
- L.swap_columns(k,s)
- L.swap_rows(k,s)
-
- # Update the permutation "matrix" with the next swap.
- p_k = p[k]
- p[k] = p[s]
- p[s] = p_k
-
- # Now the largest diagonal is in the top-left corner of
- # the block below and to the right of index k,k....
- # Note: same as ``pivot``.
- D[k,k] = alpha
-
- # When alpha is zero, we can just leave the rest of the D/L entries
- # zero... which is exactly how they start out.
- if alpha != 0:
- # Update the "next" block of A that we'll work on during
- # the following iteration. I think it's faster to get the
- # entries of a row than a column here?
- for i in range(n-k-1):
- for j in range(i+1):
- A[k+1+i,k+1+j] = A[k+1+i,k+1+j] - A[k,k+1+i]*A[k,k+1+j]/alpha
- A[k+1+j,k+1+i] = A[k+1+i,k+1+j] # keep it symmetric!
-
- # Store the "new" (kth) column of L.
- for i in range(n-k-1):
- # Set the lower-left "half" from the upper-right "half"...
- L[k+i+1,k] = A[k,k+1+i]/alpha
-
- I = A.matrix_space().identity_matrix()
- P = matrix.column( I.row(p[j]) for j in range(n) )
-
- return P,L,D
+ 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