353 lines
12 KiB
Python
353 lines
12 KiB
Python
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from warnings import warn
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import numpy as np
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from numpy import (atleast_2d, ComplexWarning, arange, zeros_like, imag, diag,
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iscomplexobj, tril, triu, argsort, empty_like)
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from ._decomp import _asarray_validated
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from .lapack import get_lapack_funcs, _compute_lwork
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__all__ = ['ldl']
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def ldl(A, lower=True, hermitian=True, overwrite_a=False, check_finite=True):
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""" Computes the LDLt or Bunch-Kaufman factorization of a symmetric/
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hermitian matrix.
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This function returns a block diagonal matrix D consisting blocks of size
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at most 2x2 and also a possibly permuted unit lower triangular matrix
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``L`` such that the factorization ``A = L D L^H`` or ``A = L D L^T``
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holds. If `lower` is False then (again possibly permuted) upper
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triangular matrices are returned as outer factors.
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The permutation array can be used to triangularize the outer factors
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simply by a row shuffle, i.e., ``lu[perm, :]`` is an upper/lower
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triangular matrix. This is also equivalent to multiplication with a
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permutation matrix ``P.dot(lu)``, where ``P`` is a column-permuted
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identity matrix ``I[:, perm]``.
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Depending on the value of the boolean `lower`, only upper or lower
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triangular part of the input array is referenced. Hence, a triangular
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matrix on entry would give the same result as if the full matrix is
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supplied.
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Parameters
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----------
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A : array_like
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Square input array
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lower : bool, optional
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This switches between the lower and upper triangular outer factors of
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the factorization. Lower triangular (``lower=True``) is the default.
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hermitian : bool, optional
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For complex-valued arrays, this defines whether ``A = A.conj().T`` or
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``A = A.T`` is assumed. For real-valued arrays, this switch has no
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effect.
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overwrite_a : bool, optional
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Allow overwriting data in `A` (may enhance performance). The default
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is False.
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check_finite : bool, optional
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Whether to check that the input matrices contain only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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Returns
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-------
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lu : ndarray
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The (possibly) permuted upper/lower triangular outer factor of the
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factorization.
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d : ndarray
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The block diagonal multiplier of the factorization.
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perm : ndarray
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The row-permutation index array that brings lu into triangular form.
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Raises
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------
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ValueError
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If input array is not square.
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ComplexWarning
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If a complex-valued array with nonzero imaginary parts on the
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diagonal is given and hermitian is set to True.
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See Also
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--------
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cholesky, lu
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Notes
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-----
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This function uses ``?SYTRF`` routines for symmetric matrices and
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``?HETRF`` routines for Hermitian matrices from LAPACK. See [1]_ for
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the algorithm details.
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Depending on the `lower` keyword value, only lower or upper triangular
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part of the input array is referenced. Moreover, this keyword also defines
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the structure of the outer factors of the factorization.
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.. versionadded:: 1.1.0
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References
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----------
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.. [1] J.R. Bunch, L. Kaufman, Some stable methods for calculating
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inertia and solving symmetric linear systems, Math. Comput. Vol.31,
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1977. :doi:`10.2307/2005787`
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Examples
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--------
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Given an upper triangular array ``a`` that represents the full symmetric
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array with its entries, obtain ``l``, 'd' and the permutation vector `perm`:
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>>> import numpy as np
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>>> from scipy.linalg import ldl
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>>> a = np.array([[2, -1, 3], [0, 2, 0], [0, 0, 1]])
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>>> lu, d, perm = ldl(a, lower=0) # Use the upper part
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>>> lu
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array([[ 0. , 0. , 1. ],
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[ 0. , 1. , -0.5],
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[ 1. , 1. , 1.5]])
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>>> d
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array([[-5. , 0. , 0. ],
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[ 0. , 1.5, 0. ],
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[ 0. , 0. , 2. ]])
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>>> perm
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array([2, 1, 0])
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>>> lu[perm, :]
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array([[ 1. , 1. , 1.5],
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[ 0. , 1. , -0.5],
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[ 0. , 0. , 1. ]])
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>>> lu.dot(d).dot(lu.T)
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array([[ 2., -1., 3.],
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[-1., 2., 0.],
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[ 3., 0., 1.]])
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"""
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a = atleast_2d(_asarray_validated(A, check_finite=check_finite))
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if a.shape[0] != a.shape[1]:
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raise ValueError('The input array "a" should be square.')
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# Return empty arrays for empty square input
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if a.size == 0:
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return empty_like(a), empty_like(a), np.array([], dtype=int)
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n = a.shape[0]
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r_or_c = complex if iscomplexobj(a) else float
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# Get the LAPACK routine
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if r_or_c is complex and hermitian:
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s, sl = 'hetrf', 'hetrf_lwork'
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if np.any(imag(diag(a))):
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warn('scipy.linalg.ldl():\nThe imaginary parts of the diagonal'
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'are ignored. Use "hermitian=False" for factorization of'
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'complex symmetric arrays.', ComplexWarning, stacklevel=2)
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else:
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s, sl = 'sytrf', 'sytrf_lwork'
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solver, solver_lwork = get_lapack_funcs((s, sl), (a,))
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lwork = _compute_lwork(solver_lwork, n, lower=lower)
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ldu, piv, info = solver(a, lwork=lwork, lower=lower,
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overwrite_a=overwrite_a)
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if info < 0:
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raise ValueError('{} exited with the internal error "illegal value '
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'in argument number {}". See LAPACK documentation '
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'for the error codes.'.format(s.upper(), -info))
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swap_arr, pivot_arr = _ldl_sanitize_ipiv(piv, lower=lower)
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d, lu = _ldl_get_d_and_l(ldu, pivot_arr, lower=lower, hermitian=hermitian)
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lu, perm = _ldl_construct_tri_factor(lu, swap_arr, pivot_arr, lower=lower)
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return lu, d, perm
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def _ldl_sanitize_ipiv(a, lower=True):
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"""
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This helper function takes the rather strangely encoded permutation array
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returned by the LAPACK routines ?(HE/SY)TRF and converts it into
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regularized permutation and diagonal pivot size format.
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Since FORTRAN uses 1-indexing and LAPACK uses different start points for
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upper and lower formats there are certain offsets in the indices used
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below.
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Let's assume a result where the matrix is 6x6 and there are two 2x2
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and two 1x1 blocks reported by the routine. To ease the coding efforts,
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we still populate a 6-sized array and fill zeros as the following ::
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pivots = [2, 0, 2, 0, 1, 1]
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This denotes a diagonal matrix of the form ::
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[x x ]
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[x x ]
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[ x x ]
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[ x x ]
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[ x ]
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[ x]
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In other words, we write 2 when the 2x2 block is first encountered and
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automatically write 0 to the next entry and skip the next spin of the
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loop. Thus, a separate counter or array appends to keep track of block
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sizes are avoided. If needed, zeros can be filtered out later without
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losing the block structure.
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Parameters
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----------
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a : ndarray
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The permutation array ipiv returned by LAPACK
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lower : bool, optional
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The switch to select whether upper or lower triangle is chosen in
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the LAPACK call.
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Returns
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-------
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swap_ : ndarray
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The array that defines the row/column swap operations. For example,
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if row two is swapped with row four, the result is [0, 3, 2, 3].
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pivots : ndarray
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The array that defines the block diagonal structure as given above.
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"""
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n = a.size
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swap_ = arange(n)
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pivots = zeros_like(swap_, dtype=int)
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skip_2x2 = False
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# Some upper/lower dependent offset values
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# range (s)tart, r(e)nd, r(i)ncrement
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x, y, rs, re, ri = (1, 0, 0, n, 1) if lower else (-1, -1, n-1, -1, -1)
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for ind in range(rs, re, ri):
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# If previous spin belonged already to a 2x2 block
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if skip_2x2:
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skip_2x2 = False
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continue
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cur_val = a[ind]
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# do we have a 1x1 block or not?
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if cur_val > 0:
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if cur_val != ind+1:
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# Index value != array value --> permutation required
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swap_[ind] = swap_[cur_val-1]
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pivots[ind] = 1
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# Not.
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elif cur_val < 0 and cur_val == a[ind+x]:
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# first neg entry of 2x2 block identifier
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if -cur_val != ind+2:
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# Index value != array value --> permutation required
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swap_[ind+x] = swap_[-cur_val-1]
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pivots[ind+y] = 2
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skip_2x2 = True
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else: # Doesn't make sense, give up
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raise ValueError('While parsing the permutation array '
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'in "scipy.linalg.ldl", invalid entries '
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'found. The array syntax is invalid.')
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return swap_, pivots
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def _ldl_get_d_and_l(ldu, pivs, lower=True, hermitian=True):
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"""
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Helper function to extract the diagonal and triangular matrices for
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LDL.T factorization.
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Parameters
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----------
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ldu : ndarray
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The compact output returned by the LAPACK routing
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pivs : ndarray
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The sanitized array of {0, 1, 2} denoting the sizes of the pivots. For
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every 2 there is a succeeding 0.
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lower : bool, optional
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If set to False, upper triangular part is considered.
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hermitian : bool, optional
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If set to False a symmetric complex array is assumed.
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Returns
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-------
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d : ndarray
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The block diagonal matrix.
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lu : ndarray
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The upper/lower triangular matrix
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"""
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is_c = iscomplexobj(ldu)
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d = diag(diag(ldu))
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n = d.shape[0]
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blk_i = 0 # block index
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# row/column offsets for selecting sub-, super-diagonal
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x, y = (1, 0) if lower else (0, 1)
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lu = tril(ldu, -1) if lower else triu(ldu, 1)
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diag_inds = arange(n)
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lu[diag_inds, diag_inds] = 1
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for blk in pivs[pivs != 0]:
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# increment the block index and check for 2s
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# if 2 then copy the off diagonals depending on uplo
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inc = blk_i + blk
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if blk == 2:
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d[blk_i+x, blk_i+y] = ldu[blk_i+x, blk_i+y]
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# If Hermitian matrix is factorized, the cross-offdiagonal element
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# should be conjugated.
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if is_c and hermitian:
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d[blk_i+y, blk_i+x] = ldu[blk_i+x, blk_i+y].conj()
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else:
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d[blk_i+y, blk_i+x] = ldu[blk_i+x, blk_i+y]
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lu[blk_i+x, blk_i+y] = 0.
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blk_i = inc
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return d, lu
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def _ldl_construct_tri_factor(lu, swap_vec, pivs, lower=True):
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"""
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Helper function to construct explicit outer factors of LDL factorization.
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If lower is True the permuted factors are multiplied as L(1)*L(2)*...*L(k).
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Otherwise, the permuted factors are multiplied as L(k)*...*L(2)*L(1). See
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LAPACK documentation for more details.
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Parameters
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----------
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lu : ndarray
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The triangular array that is extracted from LAPACK routine call with
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ones on the diagonals.
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swap_vec : ndarray
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The array that defines the row swapping indices. If the kth entry is m
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then rows k,m are swapped. Notice that the mth entry is not necessarily
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k to avoid undoing the swapping.
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pivs : ndarray
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The array that defines the block diagonal structure returned by
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_ldl_sanitize_ipiv().
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lower : bool, optional
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The boolean to switch between lower and upper triangular structure.
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Returns
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-------
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lu : ndarray
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The square outer factor which satisfies the L * D * L.T = A
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perm : ndarray
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The permutation vector that brings the lu to the triangular form
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Notes
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-----
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Note that the original argument "lu" is overwritten.
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"""
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n = lu.shape[0]
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perm = arange(n)
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# Setup the reading order of the permutation matrix for upper/lower
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rs, re, ri = (n-1, -1, -1) if lower else (0, n, 1)
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for ind in range(rs, re, ri):
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s_ind = swap_vec[ind]
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if s_ind != ind:
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# Column start and end positions
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col_s = ind if lower else 0
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col_e = n if lower else ind+1
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# If we stumble upon a 2x2 block include both cols in the perm.
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if pivs[ind] == (0 if lower else 2):
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col_s += -1 if lower else 0
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col_e += 0 if lower else 1
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lu[[s_ind, ind], col_s:col_e] = lu[[ind, s_ind], col_s:col_e]
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perm[[s_ind, ind]] = perm[[ind, s_ind]]
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return lu, argsort(perm)
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