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
# 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]/A[k,k] )
- A[k+1+i,k+1+j] = A[k+1+j,k+1+i] # keep it symmetric!
+ 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!
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]/A[k,k]
+ # 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.
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,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.
# We don't actually need the inverse of E, what we really need
# is C*E.inverse(), and that can be found by setting
#
- # C*E.inverse() == X <====> XE == C.
+ # X = C*E.inverse() <====> XE = C.
#
- # The latter can be found much more easily by solving a system.
- # Note: I do not actually know that sage solves the system more
+ # 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.transpose())
+ 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+j,k+2+i] = A[k+2+j,k+2+i] - schur_complement[j,i]
- A[k+2+i,k+2+j] = A[k+2+j,k+2+i] # keep it symmetric!
+ 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
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", being sure to set
- # the lower-left entries from the upper-right ones to
- # avoid collisions.
+ # the lower-left-hand corner of "A".
A[k+i+2,k+j] = CE_inverse[i,j]
projects / sage.d.git / blobdiff