X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fldlt.py;h=682eecf2c44a5f9a6bb7855d7b1ce6e88e631af3;hb=e172e94f5e162397401144b6b2c61760d4b9a726;hp=6db747fee7095900d9c2c45ea2000f4a323de5f5;hpb=0c818df3caa413ff3c70876a9c981e704ce9db73;p=sage.d.git diff --git a/mjo/ldlt.py b/mjo/ldlt.py index 6db747f..682eecf 100644 --- a/mjo/ldlt.py +++ b/mjo/ldlt.py @@ -26,357 +26,29 @@ def is_positive_semidefinite_naive(A): return A.is_hermitian() and all( v >= 0 for v in A.eigenvalues() ) -def ldlt_naive(A): +def _block_ldlt(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 - - SETUP:: - - sage: from mjo.ldlt import ldlt_naive, is_positive_semidefinite_naive - - EXAMPLES: - - All three factors should be the identity when the original matrix is:: - - sage: I = matrix.identity(QQ,4) - sage: P,L,D = ldlt_naive(I) - sage: P == I and L == I and D == I - True - - TESTS: - - Ensure that a "random" positive-semidefinite matrix is factored correctly:: - - 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() - 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. + 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! """ ring = A.base_ring().fraction_field() A = A.change_ring(ring) - - # Keep track of the permutations in a vector rather than in a - # matrix, for efficiency. - n = A.nrows() - 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. - # - # Since "L" is stored in the lower-left "half" of "A", it's a - # good thing that we need to permute "L," too. This is due to - # how P2.T appears in the recursive algorithm applied to the - # "current" column of L There, P2.T is computed recusively, as - # 1 x P3.T, and P3.T = 1 x P4.T, etc, from the bottom up. All - # are eventually applied to "v" in order. Here we're working - # from the top down, and rather than keep track of what - # permutations we need to perform, we just perform them as we - # go along. No recursion needed. - A.swap_columns(k,s) - A.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 largest diagonal is in the top-left corner of the - # block below and to the right of index k,k. 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+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, being sure to set - # the lower-left "half" from the upper-right "half" - A[k+i+1,k] = A[k,k+1+i]/alpha - - MS = A.matrix_space() - P = MS.matrix(lambda i,j: p[j] == i) - D = MS.diagonal_matrix(A.diagonal()) - - for i in range(n): - A[i,i] = 1 - for j in range(i+1,n): - A[i,j] = 0 - - return P,A,D - - -def block_ldlt_naive(A, check_hermitian=False): - r""" - Perform a block-`LDL^{T}` factorization of the Hermitian - matrix `A`. - - This is a naive, recursive implementation akin to - ``ldlt_naive()``, where the pivots (and resulting diagonals) are - either `1 \times 1` or `2 \times 2` blocks. The pivots are chosen - using the Bunch-Kaufmann scheme that is both fast and numerically - stable. - - OUTPUT: - - A triple `(P,L,D)` such that `A = PLDL^{T}P^{T}` and where, - - * `P` is a permutation matrix - * `L` is unit lower-triangular - * `D` is a block-diagonal matrix whose blocks are of size - one or two. - - """ - 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. - # For block-LDLT, n=2 is a base case. - P = matrix.identity(ring, n) - L = matrix.identity(ring, n) - D = A - return (P,L,D) - - alpha = (1 + ZZ(17).sqrt()) * ~ZZ(8) - A1 = A.change_ring(ring) - - # Bunch-Kaufmann step 1, Higham step "zero." We use Higham's - # "omega" notation instead of Bunch-Kaufman's "lamda" because - # lambda means other things in the same context. - column_1_subdiag = [ a_i1.abs() for a_i1 in A1[1:,0].list() ] - omega_1 = max([ a_i1 for a_i1 in column_1_subdiag ]) - - if omega_1 == 0: - # "There's nothing to do at this step of the algorithm," - # which means that our matrix looks like, - # - # [ 1 0 ] - # [ 0 B ] - # - # We could still do a pivot_one_by_one() here, but it would - # pointlessly subract a bunch of zeros and multiply by one. - B = A1[1:,1:] - one = matrix(ring, 1, 1, [1]) - P2, L2, D2 = block_ldlt_naive(B) - P1 = block_diagonal_matrix(one, P2) - L1 = block_diagonal_matrix(one, L2) - D1 = block_diagonal_matrix(one, D2) - return (P1,L1,D1) - - def pivot_one_by_one(M, c=None): - # Perform a one-by-one pivot on "M," swapping row/columns "c". - # If "c" is None, no swap is performed. - if c is not None: - P1 = copy(M.matrix_space().identity_matrix()) - P1.swap_rows(0,c) - M = P1.T * M * P1 - - # The top-left entry is now our 1x1 pivot. - C = M[1:n,0] - B = M[1:,1:] - - P2, L2, D2 = block_ldlt_naive(B - (C*C.transpose())/M[0,0]) - - if c is None: - P1 = block_matrix(2,2, [[ZZ(1), ZZ(0)], - [0*C, P2]]) - else: - P1 = P1*block_matrix(2,2, [[ZZ(1), ZZ(0)], - [0*C, P2]]) - - L1 = block_matrix(2,2, [[ZZ(1), ZZ(0)], - [P2.transpose()*C/M[0,0], L2]]) - D1 = block_matrix(2,2, [[M[0,0], ZZ(0)], - [0*C, D2]]) - - return (P1,L1,D1) - - - if A1[0,0].abs() > alpha*omega_1: - return pivot_one_by_one(A1) - - r = 1 + column_1_subdiag.index(omega_1) - - # If the matrix is Hermitian, we need only look at the above- - # diagonal entries to find the off-diagonal of maximal magnitude. - omega_r = max( a_rj.abs() for a_rj in A1[:r,r].list() ) - - if A1[0,0].abs()*omega_r >= alpha*(omega_1**2): - return pivot_one_by_one(A1) - - if A1[r,r].abs() > alpha*omega_r: - # Higham step (3) - # Another 1x1 pivot, but this time swapping indices 0,r. - return pivot_one_by_one(A1,r) - - # Higham step (4) - # If we made it here, we have to do a 2x2 pivot. - P1 = copy(A1.matrix_space().identity_matrix()) - P1.swap_rows(1,r) - A1 = P1.T * A1 * P1 - - # The top-left 2x2 submatrix is now our pivot. - E = A1[:2,:2] - C = A1[2:n,0:2] - B = A1[2:,2:] - - if B.nrows() == 0: - # We have a two-by-two matrix that we can do nothing - # useful with. - P = matrix.identity(ring, n) - L = matrix.identity(ring, n) - D = A1 - return (P,L,D) - - P2, L2, D2 = block_ldlt_naive(B - (C*E.inverse()*C.transpose())) - - P1 = P1*block_matrix(2,2, [[ZZ(1), ZZ(0)], - [0*C, P2]]) - - L1 = block_matrix(2,2, [[ZZ(1), ZZ(0)], - [P2.transpose()*C*E.inverse(), L2]]) - D1 = block_diagonal_matrix(E,D2) - - return (P1,L1,D1) - - -def block_ldlt(A): - r""" - Perform a block-`LDL^{T}` factorization of the Hermitian - matrix `A`. - - OUTPUT: - - A triple `(P,L,D)` such that `A = PLDL^{T}P^{T}` and where, - - * `P` is a permutation matrix - * `L` is unit lower-triangular - * `D` is a block-diagonal matrix whose blocks are of size - one or two. - """ - - # 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() # The magic constant used by Bunch-Kaufman @@ -434,16 +106,14 @@ def block_ldlt(A): # 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. @@ -460,7 +130,9 @@ def block_ldlt(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 @@ -469,10 +141,8 @@ def block_ldlt(A): # 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: @@ -482,7 +152,9 @@ def block_ldlt(A): # [ 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 @@ -508,9 +180,8 @@ def block_ldlt(A): # 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. @@ -537,19 +208,25 @@ def block_ldlt(A): C = A[k+2:n,k:k+2] B = A[k+2:,k+2:] - # TODO: don't invert, there are better ways to get the C*E^(-1) - # that we need. - E_inverse = E.inverse() + # 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 - (C*E_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 @@ -559,15 +236,181 @@ def block_ldlt(A): 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. - A[k+i+2,k+j] = (C*E_inverse)[i,j] + # the lower-left-hand corner of "A". + A[k+i+2,k+j] = CE_inverse[i,j] 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 @@ -575,12 +418,11 @@ def block_ldlt(A): # 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)