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"""QR decomposition functions."""
import numpy
from numpy import asarray_chkfinite
# Local imports
import special_matrices
from blas import get_blas_funcs
from lapack import get_lapack_funcs, find_best_lapack_type
from misc import _datacopied
# XXX: what is qr_old, should it be kept?
__all__ = ['qr', 'rq', 'qr_old']
def qr(a, overwrite_a=False, lwork=None, mode='full', pivoting=False):
"""Compute QR decomposition of a matrix.
Calculate the decomposition :lm:`A = Q R` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : array, shape (M, N)
Matrix to be decomposed
overwrite_a : bool, optional
Whether data in a is overwritten (may improve performance)
lwork : int, optional
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
is computed.
mode : {'full', 'r', 'economic'}
Determines what information is to be returned: either both Q and R
('full', default), only R ('r') or both Q and R but computed in
economy-size ('economic', see Notes).
pivoting : bool, optional
Whether or not factorization should include pivoting for rank-revealing
qr decomposition. If pivoting, compute the decomposition
:lm:`A P = Q R` as above, but where P is chosen such that the diagonal
of R is non-increasing.
Returns
-------
Q : double or complex ndarray
Of shape (M, M), or (M, K) for ``mode='economic'``. Not returned if
``mode='r'``.
R : double or complex ndarray
Of shape (M, N), or (K, N) for ``mode='economic'``. ``K = min(M, N)``.
P : integer ndarray
Of shape (N,) for ``pivoting=True``. Not returned if ``pivoting=False``.
Raises
------
LinAlgError
Raised if decomposition fails
Notes
-----
This is an interface to the LAPACK routines dgeqrf, zgeqrf,
dorgqr, zungqr, dgeqp3, and zgeqp3.
If ``mode=economic``, the shapes of Q and R are (M, K) and (K, N) instead
of (M,M) and (M,N), with ``K=min(M,N)``.
Examples
--------
>>> from scipy import random, linalg, dot, diag, all, allclose
>>> a = random.randn(9, 6)
>>> q, r = linalg.qr(a)
>>> allclose(a, dot(q, r))
True
>>> q.shape, r.shape
((9, 9), (9, 6))
>>> r2 = linalg.qr(a, mode='r')
>>> allclose(r, r2)
True
>>> q3, r3 = linalg.qr(a, mode='economic')
>>> q3.shape, r3.shape
((9, 6), (6, 6))
>>> q4, r4, p4 = linalg.qr(a, pivoting=True)
>>> d = abs(diag(r4))
>>> all(d[1:] <= d[:-1])
True
>>> allclose(a[:, p4], dot(q4, r4))
True
>>> q4.shape, r4.shape, p4.shape
((9, 9), (9, 6), (6,))
>>> q5, r5, p5 = linalg.qr(a, mode='economic', pivoting=True)
>>> q5.shape, r5.shape, p5.shape
((9, 6), (6, 6), (6,))
"""
if mode == 'qr':
# 'qr' was the old default, equivalent to 'full'. Neither 'full' nor
# 'qr' are used below, but set to 'full' anyway to be sure
mode = 'full'
if not mode in ['full', 'qr', 'r', 'economic']:
raise ValueError(\
"Mode argument should be one of ['full', 'r', 'economic']")
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError("expected 2D array")
M, N = a1.shape
overwrite_a = overwrite_a or (_datacopied(a1, a))
if pivoting:
geqp3, = get_lapack_funcs(('geqp3',), (a1,))
if lwork is None or lwork == -1:
# get optimal work array
qr, jpvt, tau, work, info = geqp3(a1, lwork=-1, overwrite_a=1)
lwork = work[0].real.astype(numpy.int)
qr, jpvt, tau, work, info = geqp3(a1, lwork=lwork,
overwrite_a=overwrite_a)
jpvt -= 1 # geqp3 returns a 1-based index array, so subtract 1
if info < 0:
raise ValueError("illegal value in %d-th argument of internal geqp3"
% -info)
else:
geqrf, = get_lapack_funcs(('geqrf',), (a1,))
if lwork is None or lwork == -1:
# get optimal work array
qr, tau, work, info = geqrf(a1, lwork=-1, overwrite_a=1)
lwork = work[0].real.astype(numpy.int)
qr, tau, work, info = geqrf(a1, lwork=lwork, overwrite_a=overwrite_a)
if info < 0:
raise ValueError("illegal value in %d-th argument of internal geqrf"
% -info)
if not mode == 'economic' or M < N:
R = special_matrices.triu(qr)
else:
R = special_matrices.triu(qr[0:N, 0:N])
if mode == 'r':
if pivoting:
return R, jpvt
else:
return R
if find_best_lapack_type((a1,))[0] in ('s', 'd'):
gor_un_gqr, = get_lapack_funcs(('orgqr',), (qr,))
else:
gor_un_gqr, = get_lapack_funcs(('ungqr',), (qr,))
if M < N:
# get optimal work array
Q, work, info = gor_un_gqr(qr[:,0:M], tau, lwork=-1, overwrite_a=1)
lwork = work[0].real.astype(numpy.int)
Q, work, info = gor_un_gqr(qr[:,0:M], tau, lwork=lwork, overwrite_a=1)
elif mode == 'economic':
# get optimal work array
Q, work, info = gor_un_gqr(qr, tau, lwork=-1, overwrite_a=1)
lwork = work[0].real.astype(numpy.int)
Q, work, info = gor_un_gqr(qr, tau, lwork=lwork, overwrite_a=1)
else:
t = qr.dtype.char
qqr = numpy.empty((M, M), dtype=t)
qqr[:,0:N] = qr
# get optimal work array
Q, work, info = gor_un_gqr(qqr, tau, lwork=-1, overwrite_a=1)
lwork = work[0].real.astype(numpy.int)
Q, work, info = gor_un_gqr(qqr, tau, lwork=lwork, overwrite_a=1)
if info < 0:
raise ValueError("illegal value in %d-th argument of internal gorgqr"
% -info)
if pivoting:
return Q, R, jpvt
return Q, R
def qr_old(a, overwrite_a=False, lwork=None):
"""Compute QR decomposition of a matrix.
Calculate the decomposition :lm:`A = Q R` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : array, shape (M, N)
Matrix to be decomposed
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
lwork : integer
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
is computed.
Returns
-------
Q : double or complex array, shape (M, M)
R : double or complex array, shape (M, N)
Size K = min(M, N)
Raises LinAlgError if decomposition fails
"""
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
M,N = a1.shape
overwrite_a = overwrite_a or (_datacopied(a1, a))
geqrf, = get_lapack_funcs(('geqrf',), (a1,))
if lwork is None or lwork == -1:
# get optimal work array
qr, tau, work, info = geqrf(a1, lwork=-1, overwrite_a=1)
lwork = work[0]
qr, tau, work, info = geqrf(a1, lwork=lwork, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal geqrf'
% -info)
gemm, = get_blas_funcs(('gemm',), (qr,))
t = qr.dtype.char
R = special_matrices.triu(qr)
Q = numpy.identity(M, dtype=t)
ident = numpy.identity(M, dtype=t)
zeros = numpy.zeros
for i in range(min(M, N)):
v = zeros((M,), t)
v[i] = 1
v[i+1:M] = qr[i+1:M, i]
H = gemm(-tau[i], v, v, 1+0j, ident, trans_b=2)
Q = gemm(1, Q, H)
return Q, R
def rq(a, overwrite_a=False, lwork=None, mode='full'):
"""Compute RQ decomposition of a square real matrix.
Calculate the decomposition :lm:`A = R Q` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : array, shape (M, M)
Matrix to be decomposed
overwrite_a : boolean
Whether data in a is overwritten (may improve performance)
lwork : integer
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
is computed.
mode : {'full', 'r', 'economic'}
Determines what information is to be returned: either both Q and R
('full', default), only R ('r') or both Q and R but computed in
economy-size ('economic', see Notes).
Returns
-------
R : double array, shape (M, N)
Q : double or complex array, shape (M, M)
Raises LinAlgError if decomposition fails
Examples
--------
>>> from scipy import linalg
>>> from numpy import random, dot, allclose
>>> a = random.randn(6, 9)
>>> r, q = linalg.rq(a)
>>> allclose(a, dot(r, q))
True
>>> r.shape, q.shape
((6, 9), (9, 9))
>>> r2 = linalg.rq(a, mode='r')
>>> allclose(r, r2)
True
>>> r3, q3 = linalg.rq(a, mode='economic')
>>> r3.shape, q3.shape
((6, 6), (6, 9))
"""
if not mode in ['full', 'r', 'economic']:
raise ValueError(\
"Mode argument should be one of ['full', 'r', 'economic']")
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError('expected matrix')
M, N = a1.shape
overwrite_a = overwrite_a or (_datacopied(a1, a))
gerqf, = get_lapack_funcs(('gerqf',), (a1,))
if lwork is None or lwork == -1:
# get optimal work array
rq, tau, work, info = gerqf(a1, lwork=-1, overwrite_a=1)
lwork = work[0].real.astype(numpy.int)
rq, tau, work, info = gerqf(a1, lwork=lwork, overwrite_a=overwrite_a)
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gerqf'
% -info)
if not mode == 'economic' or N < M:
R = special_matrices.triu(rq, N-M)
else:
R = special_matrices.triu(rq[-M:, -M:])
if mode == 'r':
return R
if find_best_lapack_type((a1,))[0] in ('s', 'd'):
gor_un_grq, = get_lapack_funcs(('orgrq',), (rq,))
else:
gor_un_grq, = get_lapack_funcs(('ungrq',), (rq,))
if N < M:
# get optimal work array
Q, work, info = gor_un_grq(rq[-N:], tau, lwork=-1, overwrite_a=1)
lwork = work[0].real.astype(numpy.int)
Q, work, info = gor_un_grq(rq[-N:], tau, lwork=lwork, overwrite_a=1)
elif mode == 'economic':
# get optimal work array
Q, work, info = gor_un_grq(rq, tau, lwork=-1, overwrite_a=1)
lwork = work[0].real.astype(numpy.int)
Q, work, info = gor_un_grq(rq, tau, lwork=lwork, overwrite_a=1)
else:
rq1 = numpy.empty((N, N), dtype=rq.dtype)
rq1[-M:] = rq
# get optimal work array
Q, work, info = gor_un_grq(rq1, tau, lwork=-1, overwrite_a=1)
lwork = work[0].real.astype(numpy.int)
Q, work, info = gor_un_grq(rq1, tau, lwork=lwork, overwrite_a=1)
if info < 0:
raise ValueError("illegal value in %d-th argument of internal orgrq"
% -info)
return R, Q
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