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Python

"""QR decomposition functions."""
from __future__ import division, print_function, absolute_import
import numpy
# Local imports
from .lapack import get_lapack_funcs
from .misc import _datacopied
__all__ = ['qr', 'qr_multiply', 'rq']
def safecall(f, name, *args, **kwargs):
"""Call a LAPACK routine, determining lwork automatically and handling
error return values"""
lwork = kwargs.get("lwork", None)
if lwork in (None, -1):
kwargs['lwork'] = -1
ret = f(*args, **kwargs)
kwargs['lwork'] = ret[-2][0].real.astype(numpy.int)
ret = f(*args, **kwargs)
if ret[-1] < 0:
raise ValueError("illegal value in %d-th argument of internal %s"
% (-ret[-1], name))
return ret[:-2]
def qr(a, overwrite_a=False, lwork=None, mode='full', pivoting=False,
check_finite=True):
"""
Compute QR decomposition of a matrix.
Calculate the decomposition ``A = Q R`` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : (M, N) array_like
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', 'raw'}, optional
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). The final option 'raw'
(added in Scipy 0.11) makes the function return two matrices
(Q, TAU) in the internal format used by LAPACK.
pivoting : bool, optional
Whether or not factorization should include pivoting for rank-revealing
qr decomposition. If pivoting, compute the decomposition
``A P = Q R`` as above, but where P is chosen such that the diagonal
of R is non-increasing.
check_finite : bool, optional
Whether to check that the input matrix contains 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
-------
Q : float or complex ndarray
Of shape (M, M), or (M, K) for ``mode='economic'``. Not returned
if ``mode='r'``.
R : float or complex ndarray
Of shape (M, N), or (K, N) for ``mode='economic'``. ``K = min(M, N)``.
P : int 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, np.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,))
"""
# 'qr' was the old default, equivalent to 'full'. Neither 'full' nor
# 'qr' are used below.
# 'raw' is used internally by qr_multiply
if mode not in ['full', 'qr', 'r', 'economic', 'raw']:
raise ValueError("Mode argument should be one of ['full', 'r',"
"'economic', 'raw']")
if check_finite:
a1 = numpy.asarray_chkfinite(a)
else:
a1 = numpy.asarray(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,))
qr, jpvt, tau = safecall(geqp3, "geqp3", a1, overwrite_a=overwrite_a)
jpvt -= 1 # geqp3 returns a 1-based index array, so subtract 1
else:
geqrf, = get_lapack_funcs(('geqrf',), (a1,))
qr, tau = safecall(geqrf, "geqrf", a1, lwork=lwork,
overwrite_a=overwrite_a)
if mode not in ['economic', 'raw'] or M < N:
R = numpy.triu(qr)
else:
R = numpy.triu(qr[:N, :])
if pivoting:
Rj = R, jpvt
else:
Rj = R,
if mode == 'r':
return Rj
elif mode == 'raw':
return ((qr, tau),) + Rj
gor_un_gqr, = get_lapack_funcs(('orgqr',), (qr,))
if M < N:
Q, = safecall(gor_un_gqr, "gorgqr/gungqr", qr[:, :M], tau,
lwork=lwork, overwrite_a=1)
elif mode == 'economic':
Q, = safecall(gor_un_gqr, "gorgqr/gungqr", qr, tau, lwork=lwork,
overwrite_a=1)
else:
t = qr.dtype.char
qqr = numpy.empty((M, M), dtype=t)
qqr[:, :N] = qr
Q, = safecall(gor_un_gqr, "gorgqr/gungqr", qqr, tau, lwork=lwork,
overwrite_a=1)
return (Q,) + Rj
def qr_multiply(a, c, mode='right', pivoting=False, conjugate=False,
overwrite_a=False, overwrite_c=False):
"""
Calculate the QR decomposition and multiply Q with a matrix.
Calculate the decomposition ``A = Q R`` where Q is unitary/orthogonal
and R upper triangular. Multiply Q with a vector or a matrix c.
Parameters
----------
a : (M, N), array_like
Input array
c : array_like
Input array to be multiplied by ``q``.
mode : {'left', 'right'}, optional
``Q @ c`` is returned if mode is 'left', ``c @ Q`` is returned if
mode is 'right'.
The shape of c must be appropriate for the matrix multiplications,
if mode is 'left', ``min(a.shape) == c.shape[0]``,
if mode is 'right', ``a.shape[0] == c.shape[1]``.
pivoting : bool, optional
Whether or not factorization should include pivoting for rank-revealing
qr decomposition, see the documentation of qr.
conjugate : bool, optional
Whether Q should be complex-conjugated. This might be faster
than explicit conjugation.
overwrite_a : bool, optional
Whether data in a is overwritten (may improve performance)
overwrite_c : bool, optional
Whether data in c is overwritten (may improve performance).
If this is used, c must be big enough to keep the result,
i.e. ``c.shape[0]`` = ``a.shape[0]`` if mode is 'left'.
Returns
-------
CQ : ndarray
The product of ``Q`` and ``c``.
R : (K, N), ndarray
R array of the resulting QR factorization where ``K = min(M, N)``.
P : (N,) ndarray
Integer pivot array. Only returned when ``pivoting=True``.
Raises
------
LinAlgError
Raised if QR decomposition fails.
Notes
-----
This is an interface to the LAPACK routines ``?GEQRF``, ``?ORMQR``,
``?UNMQR``, and ``?GEQP3``.
.. versionadded:: 0.11.0
Examples
--------
>>> from scipy.linalg import qr_multiply, qr
>>> A = np.array([[1, 3, 3], [2, 3, 2], [2, 3, 3], [1, 3, 2]])
>>> qc, r1, piv1 = qr_multiply(A, 2*np.eye(4), pivoting=1)
>>> qc
array([[-1., 1., -1.],
[-1., -1., 1.],
[-1., -1., -1.],
[-1., 1., 1.]])
>>> r1
array([[-6., -3., -5. ],
[ 0., -1., -1.11022302e-16],
[ 0., 0., -1. ]])
>>> piv1
array([1, 0, 2], dtype=int32)
>>> q2, r2, piv2 = qr(A, mode='economic', pivoting=1)
>>> np.allclose(2*q2 - qc, np.zeros((4, 3)))
True
"""
if mode not in ['left', 'right']:
raise ValueError("Mode argument can only be 'left' or 'right' but "
"not '{}'".format(mode))
c = numpy.asarray_chkfinite(c)
if c.ndim < 2:
onedim = True
c = numpy.atleast_2d(c)
if mode == "left":
c = c.T
else:
onedim = False
a = numpy.atleast_2d(numpy.asarray(a)) # chkfinite done in qr
M, N = a.shape
if mode == 'left':
if c.shape[0] != min(M, N + overwrite_c*(M-N)):
raise ValueError('Array shapes are not compatible for Q @ c'
' operation: {} vs {}'.format(a.shape, c.shape))
else:
if M != c.shape[1]:
raise ValueError('Array shapes are not compatible for c @ Q'
' operation: {} vs {}'.format(c.shape, a.shape))
raw = qr(a, overwrite_a, None, "raw", pivoting)
Q, tau = raw[0]
gor_un_mqr, = get_lapack_funcs(('ormqr',), (Q,))
if gor_un_mqr.typecode in ('s', 'd'):
trans = "T"
else:
trans = "C"
Q = Q[:, :min(M, N)]
if M > N and mode == "left" and not overwrite_c:
if conjugate:
cc = numpy.zeros((c.shape[1], M), dtype=c.dtype, order="F")
cc[:, :N] = c.T
else:
cc = numpy.zeros((M, c.shape[1]), dtype=c.dtype, order="F")
cc[:N, :] = c
trans = "N"
if conjugate:
lr = "R"
else:
lr = "L"
overwrite_c = True
elif c.flags["C_CONTIGUOUS"] and trans == "T" or conjugate:
cc = c.T
if mode == "left":
lr = "R"
else:
lr = "L"
else:
trans = "N"
cc = c
if mode == "left":
lr = "L"
else:
lr = "R"
cQ, = safecall(gor_un_mqr, "gormqr/gunmqr", lr, trans, Q, tau, cc,
overwrite_c=overwrite_c)
if trans != "N":
cQ = cQ.T
if mode == "right":
cQ = cQ[:, :min(M, N)]
if onedim:
cQ = cQ.ravel()
return (cQ,) + raw[1:]
def rq(a, overwrite_a=False, lwork=None, mode='full', check_finite=True):
"""
Compute RQ decomposition of a matrix.
Calculate the decomposition ``A = R Q`` where Q is unitary/orthogonal
and R upper triangular.
Parameters
----------
a : (M, N) array_like
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'}, optional
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).
check_finite : bool, optional
Whether to check that the input matrix contains 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
-------
R : float or complex ndarray
Of shape (M, N) or (M, K) for ``mode='economic'``. ``K = min(M, N)``.
Q : float or complex ndarray
Of shape (N, N) or (K, N) for ``mode='economic'``. Not returned
if ``mode='r'``.
Raises
------
LinAlgError
If decomposition fails.
Notes
-----
This is an interface to the LAPACK routines sgerqf, dgerqf, cgerqf, zgerqf,
sorgrq, dorgrq, cungrq and zungrq.
If ``mode=economic``, the shapes of Q and R are (K, N) and (M, K) instead
of (N,N) and (M,N), with ``K=min(M,N)``.
Examples
--------
>>> from scipy import linalg
>>> a = np.random.randn(6, 9)
>>> r, q = linalg.rq(a)
>>> np.allclose(a, r @ q)
True
>>> r.shape, q.shape
((6, 9), (9, 9))
>>> r2 = linalg.rq(a, mode='r')
>>> np.allclose(r, r2)
True
>>> r3, q3 = linalg.rq(a, mode='economic')
>>> r3.shape, q3.shape
((6, 6), (6, 9))
"""
if mode not in ['full', 'r', 'economic']:
raise ValueError(
"Mode argument should be one of ['full', 'r', 'economic']")
if check_finite:
a1 = numpy.asarray_chkfinite(a)
else:
a1 = numpy.asarray(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,))
rq, tau = safecall(gerqf, 'gerqf', a1, lwork=lwork,
overwrite_a=overwrite_a)
if not mode == 'economic' or N < M:
R = numpy.triu(rq, N-M)
else:
R = numpy.triu(rq[-M:, -M:])
if mode == 'r':
return R
gor_un_grq, = get_lapack_funcs(('orgrq',), (rq,))
if N < M:
Q, = safecall(gor_un_grq, "gorgrq/gungrq", rq[-N:], tau, lwork=lwork,
overwrite_a=1)
elif mode == 'economic':
Q, = safecall(gor_un_grq, "gorgrq/gungrq", rq, tau, lwork=lwork,
overwrite_a=1)
else:
rq1 = numpy.empty((N, N), dtype=rq.dtype)
rq1[-M:] = rq
Q, = safecall(gor_un_grq, "gorgrq/gungrq", rq1, tau, lwork=lwork,
overwrite_a=1)
return R, Q