# The top-left 2x2 submatrix is now our pivot.
E = A1[:2,:2]
- C = A1[2:n,0]
+ C = A1[2:n,0:2]
B = A1[2:,2:]
if B.nrows() == 0:
Perform a block-`LDL^{T}` factorization of the Hermitian
matrix `A`.
+ The standard `LDL^{T}` factorization of a positive-definite matrix
+ `A` factors it as `A = LDL^{T}` where `L` is unit-lower-triangular
+ and `D` is diagonal. If one allows row/column swaps via a
+ permutation matrix `P`, then this factorization can be extended to
+ some positive-semidefinite matrices `A` via the factorization
+ `P^{T}AP = LDL^{T}` that places the zeros at the bottom of `D` to
+ avoid division by zero. These factorizations extend easily to
+ complex Hermitian matrices when one replaces the transpose by the
+ conjugate-transpose.
+
+ However, we can go one step further. If, in addition, we allow `D`
+ to potentially contain `2 \times 2` blocks on its diagonal, then
+ every real or complex Hermitian matrix `A` can be factored as `A =
+ PLDL^{*}P^{T}`. When the row/column swaps are made intelligently,
+ this process is numerically stable over inexact rings like ``RDF``.
+ Bunch and Kaufman describe such a "pivot" scheme that is suitable
+ for the solution of Hermitian systems, and that is how we choose
+ our row and column swaps.
+
OUTPUT:
- A triple `(P,L,D)` such that `A = PLDL^{T}P^{T}` and where,
+ If the input matrix is Hermitian, we return a triple `(P,L,D)`
+ such that `A = PLDL^{*}P^{T}` and
- * `P` is a permutation matrix
- * `L` is unit lower-triangular
+ * `P` is a permutation matrix,
+ * `L` is unit lower-triangular,
* `D` is a block-diagonal matrix whose blocks are of size
one or two.
+
+ If the input matrix is not Hermitian, the output from this function
+ is undefined.
+
+ EXAMPLES:
+
+ This three-by-three real symmetric matrix has one positive, one
+ negative, and one zero eigenvalue -- so it is not any flavor of
+ (semi)definite, yet we can still factor it::
+
+ sage: A = matrix(QQ, [[0, 1, 0],
+ ....: [1, 1, 2],
+ ....: [0, 2, 0]])
+ sage: P,L,D = block_ldlt(A)
+ sage: P
+ [0 0 1]
+ [1 0 0]
+ [0 1 0]
+ sage: L
+ [ 1 0 0]
+ [ 2 1 0]
+ [ 1 1/2 1]
+ sage: D
+ [ 1| 0| 0]
+ [--+--+--]
+ [ 0|-4| 0]
+ [--+--+--]
+ [ 0| 0| 0]
+ sage: P.T*A*P == L*D*L.T
+ True
+
+ This two-by-two matrix has no standard factorization, but it
+ constitutes its own block-factorization::
+
+ sage: A = matrix(QQ, [ [0,1],
+ ....: [1,0] ])
+ sage: block_ldlt(A)
+ (
+ [1 0] [1 0] [0 1]
+ [0 1], [0 1], [1 0]
+ )
+
+ The same is true of the following complex Hermitian matrix::
+
+ sage: A = matrix(QQbar, [ [ 0,I],
+ ....: [-I,0] ])
+ sage: block_ldlt(A)
+ (
+ [1 0] [1 0] [ 0 I]
+ [0 1], [0 1], [-I 0]
+ )
+
+ TESTS:
+
+ All three factors should be the identity when the original matrix is::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(6)
+ sage: I = matrix.identity(QQ,n)
+ sage: P,L,D = block_ldlt(I)
+ sage: P == I and L == I and D == I
+ True
+
+ Ensure that a "random" real symmetric matrix is factored correctly::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(6)
+ sage: F = NumberField(x^2 +1, 'I')
+ sage: A = matrix.random(F, n)
+ sage: A = A + A.transpose()
+ sage: P,L,D = block_ldlt(A)
+ sage: A == P*L*D*L.transpose()*P.transpose()
+ True
+
+ Ensure that a "random" complex Hermitian matrix is factored correctly::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(6)
+ sage: F = NumberField(x^2 +1, 'I')
+ sage: A = matrix.random(F, n)
+ sage: A = A + A.conjugate_transpose()
+ sage: P,L,D = block_ldlt(A)
+ sage: A == P*L*D*L.transpose()*P.transpose()
+ True
+
+ Ensure that a "random" complex positive-semidefinite matrix is
+ factored correctly and that the resulting block-diagonal matrix is
+ in fact diagonal::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(6)
+ sage: F = NumberField(x^2 +1, 'I')
+ sage: A = matrix.random(F, n)
+ sage: A = A*A.conjugate_transpose()
+ sage: P,L,D = block_ldlt(A)
+ sage: A == P*L*D*L.transpose()*P.transpose()
+ True
+ sage: diagonal_matrix(D.diagonal()) == D
+ True
+
+ The factorization should be a no-op on diagonal matrices::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(6)
+ sage: A = matrix.diagonal(random_vector(QQ, n))
+ sage: I = matrix.identity(QQ,n)
+ sage: P,L,D = block_ldlt(A)
+ sage: P == I and L == I and A == D
+ True
+
"""
# We have to make at least one copy of the input matrix so that we
# entries of "L" in the copy of "A" that we're going to make.
# Contrast this with the non-block LDL^T factorization where the
# entries of both "L" and "D" overwrite the lower-left half of "A".
+ #
+ # This grants us an additional speedup, since we don't have to
+ # permute the rows/columns of "L" *and* "A" at each iteration.
ring = A.base_ring().fraction_field()
A = A.change_ring(ring)
MS = A.matrix_space()
p = list(range(n))
d = []
- def pivot1x1(M, k, s):
+ def swap_rows_columns(M, k, s):
r"""
- Perform a 1x1 pivot swapping rows/columns `k` and `s >= k`.
- Relies on the fact that matrices are passed by reference,
- since for performance reasons this routine should overwrite
- its argument. Updates the local variables ``p`` and ``d`` as
- well.
-
- Note that ``A`` is passed in by reference here, so it doesn't
- matter if we shadow the name ``A`` with itself.
+ Swap rows/columns ``k`` and ``s`` of the matrix ``M``, and update
+ the list ``p`` accordingly.
"""
if s > k:
# s == k would swap row/column k with itself, and we don't
- # actually want to perform the identity permutation.
- # We don't have to permute "L" separately so long as "L"
- # is stored within "A".
- A.swap_columns(k,s)
- A.swap_rows(k,s)
+ # actually want to perform the identity permutation. If
+ # you work out the recursive factorization by hand, you'll
+ # notice that the rows/columns of "L" need to be permuted
+ # as well. A nice side effect of storing "L" within "A"
+ # itself is that we can skip that step. The first column
+ # of "L" is hit by all of the transpositions in
+ # succession, and the second column is hit by all but the
+ # first transposition, and so on.
+ M.swap_columns(k,s)
+ M.swap_rows(k,s)
- # Update the permutation "matrix" with the swap we just did.
p_k = p[k]
p[k] = p[s]
p[s] = p_k
- # Now the pivot is in the (k,k)th position.
- d.append( matrix(ring, 1, [[A[k,k]]]) )
+ # No return value, we're only interested in the "side effects"
+ # of modifing the matrix M (by reference) and the permutation
+ # list p (which is in scope when this function is defined).
+ return
- # Compute the Schur complement that we'll work on during
- # the following iteration, and store it back in the lower-
- # right-hand corner of "A".
- for i in range(n-k-1):
- for j in range(i+1):
- A[k+1+j,k+1+i] = ( A[k+1+j,k+1+i] -
- A[k,k+1+j]*A[k,k+1+i]/alpha )
- A[k+1+i,k+1+j] = A[k+1+j,k+1+i] # keep it symmetric!
- for i in range(n-k-1):
- # Store the new (kth) column of "L" within the lower-
- # left-hand corner of "A", being sure to set the lower-
- # left entries from the upper-right ones to avoid
- #collisions.
- A[k+i+1,k] = A[k,k+1+i]/alpha
+ def pivot1x1(M, k, s):
+ r"""
+ Perform a 1x1 pivot swapping rows/columns `k` and `s >= k`.
+ Relies on the fact that matrices are passed by reference,
+ since for performance reasons this routine should overwrite
+ its argument. Updates the local variables ``p`` and ``d`` as
+ well.
+ """
+ swap_rows_columns(M,k,s)
+
+ # Now the pivot is in the (k,k)th position.
+ d.append( matrix(ring, 1, [[A[k,k]]]) )
+
+ # Compute the Schur complement that we'll work on during
+ # the following iteration, and store it back in the lower-
+ # right-hand corner of "A".
+ 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+1+i,k]*A[k,k+1+j]/A[k,k] )
+ A[k+1+j,k+1+i] = A[k+1+i,k+1+j].conjugate() # stay hermitian!
- # No return value, only the desired side effects of updating
- # p, d, and A.
- return
+ for i in range(n-k-1):
+ # Store the new (kth) column of "L" within the lower-
+ # left-hand corner of "A".
+ A[k+i+1,k] /= A[k,k]
+
+ # No return value, only the desired side effects of updating
+ # p, d, and A.
+ return
k = 0
while k < n:
if k == (n-1):
# Handle this trivial case manually, since otherwise the
# algorithm's references to the e.g. "subdiagonal" are
- # meaningless.
+ # meaningless. The corresponding entry of "L" will be
+ # fixed later (since it's an on-diagonal element, it gets
+ # set to one eventually).
d.append( matrix(ring, 1, [[A[k,k]]]) )
k += 1
continue
# kth column. This occurs prior to Step (1) in Higham,
# but is part of Step (1) in Bunch and Kaufman. We adopt
# Higham's "omega" notation instead of B&K's "lambda"
- # because "lambda" can lead to some confusion. Beware:
- # the subdiagonals of our matrix are being overwritten!
- # So we actually use the corresponding row entries instead.
- column_1_subdiag = [ a_ki.abs() for a_ki in A[k,1:].list() ]
+ # because "lambda" can lead to some confusion.
+ column_1_subdiag = [ a_ki.abs() for a_ki in A[k+1:,k].list() ]
omega_1 = max([ a_ki for a_ki in column_1_subdiag ])
if omega_1 == 0:
# [ 0 B ]
#
# and we can simply skip to the next step after recording
- # the 1x1 pivot "1" in the top-left position.
+ # the 1x1 pivot "a" in the top-left position. The entry "a"
+ # will be adjusted to "1" later on to ensure that "L" is
+ # (block) unit-lower-triangular.
d.append( matrix(ring, 1, [[A[k,k]]]) )
k += 1
continue
# B&K's Step (3) where we find the largest off-diagonal entry
# (in magniture) in column "r". Since the matrix is Hermitian,
# we need only look at the above-diagonal entries to find the
- # off-diagonal of maximal magnitude. (Beware: the subdiagonal
- # entries are being overwritten.)
- omega_r = max( a_rj.abs() for a_rj in A[:r,r].list() )
+ # off-diagonal of maximal magnitude.
+ omega_r = max( a_rj.abs() for a_rj in A[r,k:r].list() )
if A[k,k].abs()*omega_r >= alpha*(omega_1**2):
# Step (2) in Higham or Step (4) in B&K.
if A[r,r].abs() > alpha*omega_r:
# This is Step (3) in Higham or Step (5) in B&K. Still a 1x1
# pivot, but this time we need to swap rows/columns k and r.
- pivot1x1(A1,k,r)
+ pivot1x1(A,k,r)
k += 1
continue
# If we've made it this far, we're at Step (4) in Higham or
# Step (6) in B&K, where we perform a 2x2 pivot.
- k += 2
+ swap_rows_columns(A,k+1,r)
+
+ # The top-left 2x2 submatrix (starting at position k,k) is now
+ # our pivot.
+ E = A[k:k+2,k:k+2]
+ d.append(E)
+ C = A[k+2:n,k:k+2]
+ B = A[k+2:,k+2:]
+
+ # We don't actually need the inverse of E, what we really need
+ # is C*E.inverse(), and that can be found by setting
+ #
+ # X = C*E.inverse() <====> XE = C.
+ #
+ # Then "X" can be found easily by solving a system. Note: I
+ # do not actually know that sage solves the system more
+ # intelligently, but this is still The Right Thing To Do.
+ CE_inverse = E.solve_left(C)
+
+ schur_complement = B - (CE_inverse*C.conjugate_transpose())
+
+ # Compute the Schur complement that we'll work on during
+ # the following iteration, and store it back in the lower-
+ # right-hand corner of "A".
+ for i in range(n-k-2):
+ for j in range(i+1):
+ A[k+2+i,k+2+j] = schur_complement[i,j]
+ A[k+2+j,k+2+i] = schur_complement[j,i]
+
+ # The on- and above-diagonal entries of "L" will be fixed
+ # later, so we only need to worry about the lower-left entry
+ # of the 2x2 identity matrix that belongs at the top of the
+ # new column of "L".
+ A[k+1,k] = 0
+ for i in range(n-k-2):
+ for j in range(2):
+ # Store the new (k and (k+1)st) columns of "L" within
+ # the lower-left-hand corner of "A".
+ A[k+i+2,k+j] = CE_inverse[i,j]
+
+
+ k += 2
MS = A.matrix_space()
P = MS.matrix(lambda i,j: p[j] == i)
projects / sage.d.git / blobdiff