X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fldlt.py;h=682eecf2c44a5f9a6bb7855d7b1ce6e88e631af3;hb=e172e94f5e162397401144b6b2c61760d4b9a726;hp=696d78d2053793164a49cd1ac7f25b452b1e2d2c;hpb=f471b4e3c164cbc6ed79594a30540f4ef07911bc;p=sage.d.git diff --git a/mjo/ldlt.py b/mjo/ldlt.py index 696d78d..682eecf 100644 --- a/mjo/ldlt.py +++ b/mjo/ldlt.py @@ -26,185 +26,403 @@ 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`. + 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) + MS = A.matrix_space() + + # The magic constant used by Bunch-Kaufman + alpha = (1 + ZZ(17).sqrt()) * ~ZZ(8) + + # Keep track of the permutations and diagonal blocks in a vector + # rather than in a matrix, for efficiency. + n = A.nrows() + p = list(range(n)) + d = [] + + def swap_rows_columns(M, k, s): + r""" + 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. 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) + + p_k = p[k] + p[k] = p[s] + p[s] = p_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 + + + 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! + + 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: + # At each step, we're considering the k-by-k submatrix + # contained in the lower-right half of "A", because that's + # where we're storing the next iterate. So our indices are + # always "k" greater than those of Higham or B&K. Note that + # ``n == 0`` is handled by skipping this loop entirely. + + if k == (n-1): + # Handle this trivial case manually, since otherwise the + # algorithm's references to the e.g. "subdiagonal" are + # 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 + + # Find the largest subdiagonal entry (in magnitude) in the + # 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. + 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: + # In this case, our matrix looks like + # + # [ a 0 ] + # [ 0 B ] + # + # and we can simply skip to the next step after recording + # 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 + + if A[k,k].abs() > alpha*omega_1: + # This is the first case in Higham's Step (1), and B&K's + # Step (2). Note that we have skipped the part of B&K's + # Step (1) where we determine "r", since "r" is not yet + # needed and we may waste some time computing it + # otherwise. We are performing a 1x1 pivot, but the + # rows/columns are already where we want them, so nothing + # needs to be permuted. + pivot1x1(A,k,k) + k += 1 + continue + + # Now back to Step (1) of Higham, where we find the index "r" + # that corresponds to omega_1. This is the "else" branch of + # Higham's Step (1). + r = k + 1 + column_1_subdiag.index(omega_1) + + # Continuing the "else" branch of Higham's Step (1), and onto + # 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. + 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. + pivot1x1(A,k,k) + k += 1 + continue + + 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(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. + 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 - 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. + for i in range(n): + # We skipped this during the main loop, but it's necessary for + # correctness. + A[i,i] = 1 - ALGORITHM: + return (p,A,d) - The algorithm is based on the discussion in Golub and Van Loan, but with - some "typos" fixed. +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: - 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, + * `D` is a block-diagonal matrix whose blocks are of size + one or two. - * `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 + If the input matrix is not Hermitian, the output from this function + is undefined. SETUP:: - sage: from mjo.ldlt import ldlt_naive, is_positive_semidefinite_naive + sage: from mjo.ldlt import block_ldlt 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 + 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 - TESTS: + This two-by-two matrix has no standard factorization, but it + constitutes its own block-factorization:: - Ensure that a "random" positive-semidefinite matrix is factored correctly:: + sage: A = matrix(QQ, [ [0,1], + ....: [1,0] ]) + sage: block_ldlt(A) + ( + [1 0] [1 0] [0 1] + [0 1], [0 1], [1 0] + ) - 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 + The same is true of the following complex Hermitian matrix:: - """ - n = A.nrows() + sage: A = matrix(QQbar, [ [ 0,I], + ....: [-I,0] ]) + sage: block_ldlt(A) + ( + [1 0] [1 0] [ 0 I] + [0 1], [0 1], [-I 0] + ) - # 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() + TESTS: - 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) + All three factors should be the identity when the original matrix is:: - 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] + 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 - # 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) + Ensure that a "random" real symmetric matrix is factored correctly:: - v1 = A1[1:n,0] - A2 = A1[1:,1:] + 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 - P2, L2, D2 = ldlt_naive(A2 - (v1*v1.transpose())/alpha1) + Ensure that a "random" complex Hermitian matrix is factored correctly:: - 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]]) + 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 - return (P1,L1,D1) + 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:: -def ldlt_fast(A): - r""" - Perform a fast, pivoted `LDL^{T}` factorization of the Hermitian - positive-semidefinite matrix `A`. + 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 - This function is much faster than ``ldlt_naive`` because the - tail-recursion has been unrolled into a loop. """ - 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 permuts "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() + # 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) - D = MS.diagonal_matrix(A.diagonal()) + # Warning: when n == 0, this works, but returns a matrix + # whose (nonexistent) entries are in ZZ rather than in + # the base ring of P and L. + D = block_diagonal_matrix(d) + + # 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)