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).
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)``.
Raises LinAlgError if decomposition fails
Notes
-----
This is an interface to the LAPACK routines dgeqrf, zgeqrf,
dorgqr, and zungqr.
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
>>> 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)
>>> q3, r3 = linalg.qr(a, mode='economic')
>>> q3.shape, r3.shape
((9, 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 (_datanotshared(a1, a))
if pivoting:
qr = cvxopt.matrix(0, a1.shape, tc = 'd')
qr[:, :] = a1
tau = cvxopt.matrix(0, (min(M, N), 1), tc = 'd')
jpvt = cvxopt.matrix(0, (N, 1), tc = 'i')
lapack.geqp3(qr, jpvt, tau)
qr = numpy.asarray(qr)
tau = numpy.asarray(tau)
jpvt = (numpy.asarray(jpvt) - 1).ravel()
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':
return R
if find_best_lapack_type((a1,))[0] == 's' or \
find_best_lapack_type((a1,))[0] == '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(a, overwrite_a=0, lwork=None, mode='qr', 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 : 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 : {'qr', 'r', 'economic'}
Determines what information is to be returned: either both Q and R
or only R, or Q and R and P, a permutation matrix. Any of these can
be combined with 'economic' using the '+' sign as a separator.
Economic mode means the following:
Compute the economy-size QR decomposition, making shapes
of Q and R (M, K) and (K, N) instead of (M,M) and (M,N).
K=min(M,N).
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. P represents the new column order of A.
Returns
-------
(if mode == 'qr')
Q : double or complex array, shape (M, M) or (M, K) for econ==True
(for any mode)
R : double or complex array, shape (M, N) or (K, N) for econ==True
Size K = min(M, N)
P : integer ndarray
Of shape (N,) for ``pivoting=True``.
Not returned if ``pivoting=False``.
Raises LinAlgError if decomposition fails
Notes
-----
This is an interface to the LAPACK routines dgeqrf, zgeqrf,
dorgqr, and zungqr.
Examples
--------
>>> from scipy import random, linalg, dot
>>> 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)
>>> q3, r3 = linalg.qr(a, mode='economic')
>>> q3.shape, r3.shape
((9, 6), (6, 6))
"""
mode = mode.split("+")
if "economic" in mode:
econ = True
else:
econ = False
a1 = asarray_chkfinite(a)
if len(a1.shape) != 2:
raise ValueError("expected 2D array")
M, N = a1.shape
overwrite_a = overwrite_a or (_datanotshared(a1,a))
if pivoting:
qr = cvxopt.matrix(0, a1.shape, tc = 'd')
qr[:, :] = a1
tau = cvxopt.matrix(0, (min(M, N), 1), tc = 'd')
jpvt = cvxopt.matrix(0, (N, 1), tc = 'i')
lapack.geqp3(qr, jpvt, tau)
qr = numpy.asarray(qr)
tau = numpy.asarray(tau)
jpvt = (numpy.asarray(jpvt) - 1).ravel()
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]
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 econ or M<N:
R = basic.triu(qr)
else:
R = basic.triu(qr[0:N,0:N])
if 'r' in mode:
return R
if find_best_lapack_type((a1,))[0]=='s' or find_best_lapack_type((a1,))[0]=='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]
Q,work,info = gor_un_gqr(qr[:,0:M],tau,lwork=lwork,overwrite_a=1)
elif econ:
# get optimal work array
Q,work,info = gor_un_gqr(qr,tau,lwork=-1,overwrite_a=1)
lwork = work[0]
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]
Q,work,info = gor_un_gqr(qqr,tau,lwork=lwork,overwrite_a=1)
if info < 0:
raise ValueError("illegal value in %-th argument of internal gorgqr"
% -info)
if pivoting:
return Q, R, jpvt
return Q, R