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Fixed database typo and removed unnecessary class identifier.

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Batuhan Berk Başoğlu 2020-10-14 10:10:37 -04:00
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from warnings import warn
import numpy as np
from numpy import (atleast_2d, ComplexWarning, arange, zeros_like, imag, diag,
iscomplexobj, tril, triu, argsort, empty_like)
from .decomp import _asarray_validated
from .lapack import get_lapack_funcs, _compute_lwork
__all__ = ['ldl']
def ldl(A, lower=True, hermitian=True, overwrite_a=False, check_finite=True):
""" Computes the LDLt or Bunch-Kaufman factorization of a symmetric/
hermitian matrix.
This function returns a block diagonal matrix D consisting blocks of size
at most 2x2 and also a possibly permuted unit lower triangular matrix
``L`` such that the factorization ``A = L D L^H`` or ``A = L D L^T``
holds. If ``lower`` is False then (again possibly permuted) upper
triangular matrices are returned as outer factors.
The permutation array can be used to triangularize the outer factors
simply by a row shuffle, i.e., ``lu[perm, :]`` is an upper/lower
triangular matrix. This is also equivalent to multiplication with a
permutation matrix ``P.dot(lu)``, where ``P`` is a column-permuted
identity matrix ``I[:, perm]``.
Depending on the value of the boolean ``lower``, only upper or lower
triangular part of the input array is referenced. Hence, a triangular
matrix on entry would give the same result as if the full matrix is
supplied.
Parameters
----------
a : array_like
Square input array
lower : bool, optional
This switches between the lower and upper triangular outer factors of
the factorization. Lower triangular (``lower=True``) is the default.
hermitian : bool, optional
For complex-valued arrays, this defines whether ``a = a.conj().T`` or
``a = a.T`` is assumed. For real-valued arrays, this switch has no
effect.
overwrite_a : bool, optional
Allow overwriting data in ``a`` (may enhance performance). The default
is False.
check_finite : bool, optional
Whether to check that the input matrices contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
lu : ndarray
The (possibly) permuted upper/lower triangular outer factor of the
factorization.
d : ndarray
The block diagonal multiplier of the factorization.
perm : ndarray
The row-permutation index array that brings lu into triangular form.
Raises
------
ValueError
If input array is not square.
ComplexWarning
If a complex-valued array with nonzero imaginary parts on the
diagonal is given and hermitian is set to True.
Examples
--------
Given an upper triangular array `a` that represents the full symmetric
array with its entries, obtain `l`, 'd' and the permutation vector `perm`:
>>> import numpy as np
>>> from scipy.linalg import ldl
>>> a = np.array([[2, -1, 3], [0, 2, 0], [0, 0, 1]])
>>> lu, d, perm = ldl(a, lower=0) # Use the upper part
>>> lu
array([[ 0. , 0. , 1. ],
[ 0. , 1. , -0.5],
[ 1. , 1. , 1.5]])
>>> d
array([[-5. , 0. , 0. ],
[ 0. , 1.5, 0. ],
[ 0. , 0. , 2. ]])
>>> perm
array([2, 1, 0])
>>> lu[perm, :]
array([[ 1. , 1. , 1.5],
[ 0. , 1. , -0.5],
[ 0. , 0. , 1. ]])
>>> lu.dot(d).dot(lu.T)
array([[ 2., -1., 3.],
[-1., 2., 0.],
[ 3., 0., 1.]])
Notes
-----
This function uses ``?SYTRF`` routines for symmetric matrices and
``?HETRF`` routines for Hermitian matrices from LAPACK. See [1]_ for
the algorithm details.
Depending on the ``lower`` keyword value, only lower or upper triangular
part of the input array is referenced. Moreover, this keyword also defines
the structure of the outer factors of the factorization.
.. versionadded:: 1.1.0
See also
--------
cholesky, lu
References
----------
.. [1] J.R. Bunch, L. Kaufman, Some stable methods for calculating
inertia and solving symmetric linear systems, Math. Comput. Vol.31,
1977. DOI: 10.2307/2005787
"""
a = atleast_2d(_asarray_validated(A, check_finite=check_finite))
if a.shape[0] != a.shape[1]:
raise ValueError('The input array "a" should be square.')
# Return empty arrays for empty square input
if a.size == 0:
return empty_like(a), empty_like(a), np.array([], dtype=int)
n = a.shape[0]
r_or_c = complex if iscomplexobj(a) else float
# Get the LAPACK routine
if r_or_c is complex and hermitian:
s, sl = 'hetrf', 'hetrf_lwork'
if np.any(imag(diag(a))):
warn('scipy.linalg.ldl():\nThe imaginary parts of the diagonal'
'are ignored. Use "hermitian=False" for factorization of'
'complex symmetric arrays.', ComplexWarning, stacklevel=2)
else:
s, sl = 'sytrf', 'sytrf_lwork'
solver, solver_lwork = get_lapack_funcs((s, sl), (a,))
lwork = _compute_lwork(solver_lwork, n, lower=lower)
ldu, piv, info = solver(a, lwork=lwork, lower=lower,
overwrite_a=overwrite_a)
if info < 0:
raise ValueError('{} exited with the internal error "illegal value '
'in argument number {}". See LAPACK documentation '
'for the error codes.'.format(s.upper(), -info))
swap_arr, pivot_arr = _ldl_sanitize_ipiv(piv, lower=lower)
d, lu = _ldl_get_d_and_l(ldu, pivot_arr, lower=lower, hermitian=hermitian)
lu, perm = _ldl_construct_tri_factor(lu, swap_arr, pivot_arr, lower=lower)
return lu, d, perm
def _ldl_sanitize_ipiv(a, lower=True):
"""
This helper function takes the rather strangely encoded permutation array
returned by the LAPACK routines ?(HE/SY)TRF and converts it into
regularized permutation and diagonal pivot size format.
Since FORTRAN uses 1-indexing and LAPACK uses different start points for
upper and lower formats there are certain offsets in the indices used
below.
Let's assume a result where the matrix is 6x6 and there are two 2x2
and two 1x1 blocks reported by the routine. To ease the coding efforts,
we still populate a 6-sized array and fill zeros as the following ::
pivots = [2, 0, 2, 0, 1, 1]
This denotes a diagonal matrix of the form ::
[x x ]
[x x ]
[ x x ]
[ x x ]
[ x ]
[ x]
In other words, we write 2 when the 2x2 block is first encountered and
automatically write 0 to the next entry and skip the next spin of the
loop. Thus, a separate counter or array appends to keep track of block
sizes are avoided. If needed, zeros can be filtered out later without
losing the block structure.
Parameters
----------
a : ndarray
The permutation array ipiv returned by LAPACK
lower : bool, optional
The switch to select whether upper or lower triangle is chosen in
the LAPACK call.
Returns
-------
swap_ : ndarray
The array that defines the row/column swap operations. For example,
if row two is swapped with row four, the result is [0, 3, 2, 3].
pivots : ndarray
The array that defines the block diagonal structure as given above.
"""
n = a.size
swap_ = arange(n)
pivots = zeros_like(swap_, dtype=int)
skip_2x2 = False
# Some upper/lower dependent offset values
# range (s)tart, r(e)nd, r(i)ncrement
x, y, rs, re, ri = (1, 0, 0, n, 1) if lower else (-1, -1, n-1, -1, -1)
for ind in range(rs, re, ri):
# If previous spin belonged already to a 2x2 block
if skip_2x2:
skip_2x2 = False
continue
cur_val = a[ind]
# do we have a 1x1 block or not?
if cur_val > 0:
if cur_val != ind+1:
# Index value != array value --> permutation required
swap_[ind] = swap_[cur_val-1]
pivots[ind] = 1
# Not.
elif cur_val < 0 and cur_val == a[ind+x]:
# first neg entry of 2x2 block identifier
if -cur_val != ind+2:
# Index value != array value --> permutation required
swap_[ind+x] = swap_[-cur_val-1]
pivots[ind+y] = 2
skip_2x2 = True
else: # Doesn't make sense, give up
raise ValueError('While parsing the permutation array '
'in "scipy.linalg.ldl", invalid entries '
'found. The array syntax is invalid.')
return swap_, pivots
def _ldl_get_d_and_l(ldu, pivs, lower=True, hermitian=True):
"""
Helper function to extract the diagonal and triangular matrices for
LDL.T factorization.
Parameters
----------
ldu : ndarray
The compact output returned by the LAPACK routing
pivs : ndarray
The sanitized array of {0, 1, 2} denoting the sizes of the pivots. For
every 2 there is a succeeding 0.
lower : bool, optional
If set to False, upper triangular part is considered.
hermitian : bool, optional
If set to False a symmetric complex array is assumed.
Returns
-------
d : ndarray
The block diagonal matrix.
lu : ndarray
The upper/lower triangular matrix
"""
is_c = iscomplexobj(ldu)
d = diag(diag(ldu))
n = d.shape[0]
blk_i = 0 # block index
# row/column offsets for selecting sub-, super-diagonal
x, y = (1, 0) if lower else (0, 1)
lu = tril(ldu, -1) if lower else triu(ldu, 1)
diag_inds = arange(n)
lu[diag_inds, diag_inds] = 1
for blk in pivs[pivs != 0]:
# increment the block index and check for 2s
# if 2 then copy the off diagonals depending on uplo
inc = blk_i + blk
if blk == 2:
d[blk_i+x, blk_i+y] = ldu[blk_i+x, blk_i+y]
# If Hermitian matrix is factorized, the cross-offdiagonal element
# should be conjugated.
if is_c and hermitian:
d[blk_i+y, blk_i+x] = ldu[blk_i+x, blk_i+y].conj()
else:
d[blk_i+y, blk_i+x] = ldu[blk_i+x, blk_i+y]
lu[blk_i+x, blk_i+y] = 0.
blk_i = inc
return d, lu
def _ldl_construct_tri_factor(lu, swap_vec, pivs, lower=True):
"""
Helper function to construct explicit outer factors of LDL factorization.
If lower is True the permuted factors are multiplied as L(1)*L(2)*...*L(k).
Otherwise, the permuted factors are multiplied as L(k)*...*L(2)*L(1). See
LAPACK documentation for more details.
Parameters
----------
lu : ndarray
The triangular array that is extracted from LAPACK routine call with
ones on the diagonals.
swap_vec : ndarray
The array that defines the row swapping indices. If the kth entry is m
then rows k,m are swapped. Notice that the mth entry is not necessarily
k to avoid undoing the swapping.
pivs : ndarray
The array that defines the block diagonal structure returned by
_ldl_sanitize_ipiv().
lower : bool, optional
The boolean to switch between lower and upper triangular structure.
Returns
-------
lu : ndarray
The square outer factor which satisfies the L * D * L.T = A
perm : ndarray
The permutation vector that brings the lu to the triangular form
Notes
-----
Note that the original argument "lu" is overwritten.
"""
n = lu.shape[0]
perm = arange(n)
# Setup the reading order of the permutation matrix for upper/lower
rs, re, ri = (n-1, -1, -1) if lower else (0, n, 1)
for ind in range(rs, re, ri):
s_ind = swap_vec[ind]
if s_ind != ind:
# Column start and end positions
col_s = ind if lower else 0
col_e = n if lower else ind+1
# If we stumble upon a 2x2 block include both cols in the perm.
if pivs[ind] == (0 if lower else 2):
col_s += -1 if lower else 0
col_e += 0 if lower else 1
lu[[s_ind, ind], col_s:col_e] = lu[[ind, s_ind], col_s:col_e]
perm[[s_ind, ind]] = perm[[ind, s_ind]]
return lu, argsort(perm)