mjo/ldlt.py

   1 from sage.all import *
   2
   3 def is_positive_semidefinite_naive(A):
   4     r"""
   5     A naive positive-semidefinite check that tests the eigenvalues for
   6     nonnegativity.  We follow the sage convention that positive
   7     (semi)definite matrices must be symmetric or Hermitian.
   8
   9     SETUP::
  10
  11         sage: from mjo.ldlt import is_positive_semidefinite_naive
  12
  13     TESTS:
  14
  15     The trivial matrix is vaciously positive-semidefinite::
  16
  17         sage: A = matrix(QQ, 0)
  18         sage: A
  19         []
  20         sage: is_positive_semidefinite_naive(A)
  21         True
  22
  23     """
  24     if A.nrows() == 0:
  25         return True # vacuously
  26     return A.is_hermitian() and all( v >= 0 for v in A.eigenvalues() )
  27
  28 def ldlt_naive(A):
  29     r"""
  30     Perform a pivoted `LDL^{T}` factorization of the Hermitian
  31     positive-semidefinite matrix `A`.
  32
  33     This is a naive, recursive implementation that is inefficient due
  34     to Python's lack of tail-call optimization. The pivot strategy is
  35     to choose the largest diagonal entry of the matrix at each step,
  36     and to permute it into the top-left position. Ultimately this
  37     results in a factorization `A = PLDL^{T}P^{T}`, where `P` is a
  38     permutation matrix, `L` is unit-lower-triangular, and `D` is
  39     diagonal decreasing from top-left to bottom-right.
  40
  41     ALGORITHM:
  42
  43     The algorithm is based on the discussion in Golub and Van Loan, but with
  44     some "typos" fixed.
  45
  46     OUTPUT:
  47
  48     A triple `(P,L,D)` such that `A = PLDL^{T}P^{T}` and where,
  49
  50       * `P` is a permutaiton matrix
  51       * `L` is unit lower-triangular
  52       * `D` is a diagonal matrix whose entries are decreasing from top-left
  53         to bottom-right
  54
  55     SETUP::
  56
  57         sage: from mjo.ldlt import ldlt_naive, is_positive_semidefinite_naive
  58
  59     EXAMPLES:
  60
  61     All three factors should be the identity when the original matrix is::
  62
  63         sage: I = matrix.identity(QQ,4)
  64         sage: P,L,D = ldlt_naive(I)
  65         sage: P == I and L == I and D == I
  66         True
  67
  68     TESTS:
  69
  70     Ensure that a "random" positive-semidefinite matrix is factored correctly::
  71
  72         sage: set_random_seed()
  73         sage: n = ZZ.random_element(5)
  74         sage: A = matrix.random(QQ, n)
  75         sage: A = A*A.transpose()
  76         sage: is_positive_semidefinite_naive(A)
  77         True
  78         sage: P,L,D = ldlt_naive(A)
  79         sage: A == P*L*D*L.transpose()*P.transpose()
  80         True
  81
  82     """
  83     n = A.nrows()
  84
  85     # Use the fraction field of the given matrix so that division will work
  86     # when (for example) our matrix consists of integer entries.
  87     ring = A.base_ring().fraction_field()
  88
  89     if n == 0 or n == 1:
  90         # We can get n == 0 if someone feeds us a trivial matrix.
  91         P = matrix.identity(ring, n)
  92         L = matrix.identity(ring, n)
  93         D = A
  94         return (P,L,D)
  95
  96     A1 = A.change_ring(ring)
  97     diags = A1.diagonal()
  98     s = diags.index(max(diags))
  99     P1 = copy(A1.matrix_space().identity_matrix())
 100     A1 = P1.T * A1 * P1
 101     alpha1 = A1[0,0]
 102
 103     # Golub and Van Loan mention in passing what to do here. This is
 104     # only sensible if the matrix is positive-semidefinite, because we
 105     # are assuming that we can set everything else to zero as soon as
 106     # we hit the first on-diagonal zero.
 107     if alpha1 == 0:
 108         P = A1.matrix_space().identity_matrix()
 109         L = P
 110         D = A1.matrix_space().zero()
 111         return (P,L,D)
 112
 113     v1 = A1[1:n,0]
 114     A2 = A1[1:,1:]
 115
 116     P2, L2, D2 = ldlt_naive(A2 - (v1*v1.transpose())/alpha1)
 117
 118     P1 = P1*block_matrix(2,2, [[ZZ(1), ZZ(0)],
 119                                [0*v1,  P2]])
 120     L1 = block_matrix(2,2, [[ZZ(1),                    ZZ(0)],
 121                             [P2.transpose()*v1/alpha1, L2]])
 122     D1 = block_matrix(2,2, [[alpha1, ZZ(0)],
 123                             [0*v1,   D2]])
 124
 125     return (P1,L1,D1)