return A.is_hermitian() and all( v >= 0 for v in A.eigenvalues() )
-def block_ldlt(A):
+def _block_ldlt(A):
r"""
- 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:
-
- 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,
- * `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.
-
- SETUP::
-
- sage: from mjo.ldlt import block_ldlt
-
- 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.transpose()*A*P == L*D*L.transpose()
- 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: A = matrix.random(QQ, 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.conjugate_transpose()*P.conjugate_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.conjugate_transpose()*P.conjugate_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
-
+ Perform a user-unfriendly block-`LDL^{T}` factorization of the
+ Hermitian matrix `A`
+
+ This function is used internally to compute the factorization for
+ the user-friendly ``block_ldlt`` function. Whereas that function
+ returns three nice matrices, this one returns
+
+ * A list ``p`` of the first ``n`` natural numbers, permuted.
+ * A matrix whose lower-triangular portion is ``L``, but whose
+ * (strict) upper-triangular portion is junk.
+ * A list of the block-diagonal entries of ``D``
+
+ This is mainly useful to avoid havinf to "undo" the construction
+ of the matrix ``D`` when we don't need it. For example, it's much
+ easier to compute the inertia of a matrix from the list of blocks
+ than it is from the block-diagonal matrix itself, because given a
+ block-diagonal matrix, you first have to figure out where the
+ blocks are!
"""
-
- # We have to make at least one copy of the input matrix so that we
- # can change the base ring to its fraction field. Both "L" and the
- # intermediate Schur complements will potentially have entries in
- # the fraction field. However, we don't need to make *two* copies.
- # We can't store the entries of "D" and "L" in the same matrix if
- # "D" will contain any 2x2 blocks; but we can still store the
- # 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()
k += 2
- MS = A.matrix_space()
+ for i in range(n):
+ # We skipped this during the main loop, but it's necessary for
+ # correctness.
+ A[i,i] = 1
+
+ return (p,A,d)
+
+def block_ldlt(A):
+ r"""
+ 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:
+
+ 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,
+ * `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.
+
+ SETUP::
+
+ sage: from mjo.ldlt import block_ldlt
+
+ 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.transpose()*A*P == L*D*L.transpose()
+ 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: A = matrix.random(QQ, 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.conjugate_transpose()*P.conjugate_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.conjugate_transpose()*P.conjugate_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
+ # can change the base ring to its fraction field. Both "L" and the
+ # intermediate Schur complements will potentially have entries in
+ # the fraction field. However, we don't need to make *two* copies.
+ # We can't store the entries of "D" and "L" in the same matrix if
+ # "D" will contain any 2x2 blocks; but we can still store the
+ # 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.
+ p,L,d = _block_ldlt(A)
+ MS = L.matrix_space()
P = MS.matrix(lambda i,j: p[j] == i)
# Warning: when n == 0, this works, but returns a matrix
# the base ring of P and L.
D = block_diagonal_matrix(d)
- # Overwrite the diagonal and upper-right half of "A",
- # since we're about to return it as the unit-lower-
- # triangular "L".
+ # Overwrite the (strict) upper-triangular part of "L", since a
+ # priori it contains the same entries as "A" did after _block_ldlt().
+ n = L.nrows()
for i in range(n):
- A[i,i] = 1
for j in range(i+1,n):
- A[i,j] = 0
+ L[i,j] = 0
- return (P,A,D)
+ return (P,L,D)
projects / sage.d.git / commitdiff