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425 lines
13 KiB
425 lines
13 KiB
4 years ago
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"""QR decomposition functions."""
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import numpy
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# Local imports
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from .lapack import get_lapack_funcs
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from .misc import _datacopied
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__all__ = ['qr', 'qr_multiply', 'rq']
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def safecall(f, name, *args, **kwargs):
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"""Call a LAPACK routine, determining lwork automatically and handling
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error return values"""
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lwork = kwargs.get("lwork", None)
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if lwork in (None, -1):
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kwargs['lwork'] = -1
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ret = f(*args, **kwargs)
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kwargs['lwork'] = ret[-2][0].real.astype(numpy.int_)
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ret = f(*args, **kwargs)
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if ret[-1] < 0:
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raise ValueError("illegal value in %dth argument of internal %s"
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% (-ret[-1], name))
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return ret[:-2]
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def qr(a, overwrite_a=False, lwork=None, mode='full', pivoting=False,
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check_finite=True):
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"""
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Compute QR decomposition of a matrix.
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Calculate the decomposition ``A = Q R`` where Q is unitary/orthogonal
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and R upper triangular.
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Parameters
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----------
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a : (M, N) array_like
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Matrix to be decomposed
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overwrite_a : bool, optional
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Whether data in `a` is overwritten (may improve performance if
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`overwrite_a` is set to True by reusing the existing input data
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structure rather than creating a new one.)
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lwork : int, optional
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Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
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is computed.
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mode : {'full', 'r', 'economic', 'raw'}, optional
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Determines what information is to be returned: either both Q and R
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('full', default), only R ('r') or both Q and R but computed in
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economy-size ('economic', see Notes). The final option 'raw'
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(added in SciPy 0.11) makes the function return two matrices
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(Q, TAU) in the internal format used by LAPACK.
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pivoting : bool, optional
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Whether or not factorization should include pivoting for rank-revealing
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qr decomposition. If pivoting, compute the decomposition
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``A P = Q R`` as above, but where P is chosen such that the diagonal
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of R is non-increasing.
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check_finite : bool, optional
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Whether to check that the input matrix contains 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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Q : float or complex ndarray
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Of shape (M, M), or (M, K) for ``mode='economic'``. Not returned
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if ``mode='r'``.
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R : float or complex ndarray
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Of shape (M, N), or (K, N) for ``mode='economic'``. ``K = min(M, N)``.
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P : int ndarray
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Of shape (N,) for ``pivoting=True``. Not returned if
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``pivoting=False``.
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Raises
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------
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LinAlgError
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Raised if decomposition fails
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Notes
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-----
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This is an interface to the LAPACK routines dgeqrf, zgeqrf,
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dorgqr, zungqr, dgeqp3, and zgeqp3.
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If ``mode=economic``, the shapes of Q and R are (M, K) and (K, N) instead
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of (M,M) and (M,N), with ``K=min(M,N)``.
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Examples
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--------
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>>> from scipy import linalg
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>>> a = np.random.randn(9, 6)
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>>> q, r = linalg.qr(a)
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>>> np.allclose(a, np.dot(q, r))
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True
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>>> q.shape, r.shape
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((9, 9), (9, 6))
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>>> r2 = linalg.qr(a, mode='r')
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>>> np.allclose(r, r2)
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True
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>>> q3, r3 = linalg.qr(a, mode='economic')
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>>> q3.shape, r3.shape
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((9, 6), (6, 6))
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>>> q4, r4, p4 = linalg.qr(a, pivoting=True)
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>>> d = np.abs(np.diag(r4))
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>>> np.all(d[1:] <= d[:-1])
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True
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>>> np.allclose(a[:, p4], np.dot(q4, r4))
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True
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>>> q4.shape, r4.shape, p4.shape
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((9, 9), (9, 6), (6,))
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>>> q5, r5, p5 = linalg.qr(a, mode='economic', pivoting=True)
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>>> q5.shape, r5.shape, p5.shape
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((9, 6), (6, 6), (6,))
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"""
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# 'qr' was the old default, equivalent to 'full'. Neither 'full' nor
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# 'qr' are used below.
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# 'raw' is used internally by qr_multiply
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if mode not in ['full', 'qr', 'r', 'economic', 'raw']:
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raise ValueError("Mode argument should be one of ['full', 'r',"
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"'economic', 'raw']")
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if check_finite:
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a1 = numpy.asarray_chkfinite(a)
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else:
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a1 = numpy.asarray(a)
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if len(a1.shape) != 2:
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raise ValueError("expected a 2-D array")
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M, N = a1.shape
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overwrite_a = overwrite_a or (_datacopied(a1, a))
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if pivoting:
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geqp3, = get_lapack_funcs(('geqp3',), (a1,))
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qr, jpvt, tau = safecall(geqp3, "geqp3", a1, overwrite_a=overwrite_a)
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jpvt -= 1 # geqp3 returns a 1-based index array, so subtract 1
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else:
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geqrf, = get_lapack_funcs(('geqrf',), (a1,))
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qr, tau = safecall(geqrf, "geqrf", a1, lwork=lwork,
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overwrite_a=overwrite_a)
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if mode not in ['economic', 'raw'] or M < N:
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R = numpy.triu(qr)
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else:
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R = numpy.triu(qr[:N, :])
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if pivoting:
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Rj = R, jpvt
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else:
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Rj = R,
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if mode == 'r':
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return Rj
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elif mode == 'raw':
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return ((qr, tau),) + Rj
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gor_un_gqr, = get_lapack_funcs(('orgqr',), (qr,))
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if M < N:
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Q, = safecall(gor_un_gqr, "gorgqr/gungqr", qr[:, :M], tau,
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lwork=lwork, overwrite_a=1)
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elif mode == 'economic':
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Q, = safecall(gor_un_gqr, "gorgqr/gungqr", qr, tau, lwork=lwork,
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overwrite_a=1)
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else:
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t = qr.dtype.char
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qqr = numpy.empty((M, M), dtype=t)
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qqr[:, :N] = qr
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Q, = safecall(gor_un_gqr, "gorgqr/gungqr", qqr, tau, lwork=lwork,
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overwrite_a=1)
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return (Q,) + Rj
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def qr_multiply(a, c, mode='right', pivoting=False, conjugate=False,
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overwrite_a=False, overwrite_c=False):
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"""
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Calculate the QR decomposition and multiply Q with a matrix.
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Calculate the decomposition ``A = Q R`` where Q is unitary/orthogonal
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and R upper triangular. Multiply Q with a vector or a matrix c.
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Parameters
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----------
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a : (M, N), array_like
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Input array
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c : array_like
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Input array to be multiplied by ``q``.
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mode : {'left', 'right'}, optional
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``Q @ c`` is returned if mode is 'left', ``c @ Q`` is returned if
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mode is 'right'.
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The shape of c must be appropriate for the matrix multiplications,
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if mode is 'left', ``min(a.shape) == c.shape[0]``,
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if mode is 'right', ``a.shape[0] == c.shape[1]``.
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pivoting : bool, optional
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Whether or not factorization should include pivoting for rank-revealing
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qr decomposition, see the documentation of qr.
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conjugate : bool, optional
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Whether Q should be complex-conjugated. This might be faster
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than explicit conjugation.
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overwrite_a : bool, optional
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Whether data in a is overwritten (may improve performance)
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overwrite_c : bool, optional
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Whether data in c is overwritten (may improve performance).
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If this is used, c must be big enough to keep the result,
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i.e. ``c.shape[0]`` = ``a.shape[0]`` if mode is 'left'.
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Returns
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-------
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CQ : ndarray
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The product of ``Q`` and ``c``.
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R : (K, N), ndarray
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R array of the resulting QR factorization where ``K = min(M, N)``.
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P : (N,) ndarray
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Integer pivot array. Only returned when ``pivoting=True``.
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Raises
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------
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LinAlgError
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Raised if QR decomposition fails.
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Notes
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-----
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This is an interface to the LAPACK routines ``?GEQRF``, ``?ORMQR``,
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``?UNMQR``, and ``?GEQP3``.
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.. versionadded:: 0.11.0
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Examples
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--------
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>>> from scipy.linalg import qr_multiply, qr
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>>> A = np.array([[1, 3, 3], [2, 3, 2], [2, 3, 3], [1, 3, 2]])
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>>> qc, r1, piv1 = qr_multiply(A, 2*np.eye(4), pivoting=1)
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>>> qc
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array([[-1., 1., -1.],
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[-1., -1., 1.],
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[-1., -1., -1.],
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[-1., 1., 1.]])
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>>> r1
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array([[-6., -3., -5. ],
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[ 0., -1., -1.11022302e-16],
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[ 0., 0., -1. ]])
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>>> piv1
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array([1, 0, 2], dtype=int32)
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>>> q2, r2, piv2 = qr(A, mode='economic', pivoting=1)
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>>> np.allclose(2*q2 - qc, np.zeros((4, 3)))
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True
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"""
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if mode not in ['left', 'right']:
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raise ValueError("Mode argument can only be 'left' or 'right' but "
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"not '{}'".format(mode))
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c = numpy.asarray_chkfinite(c)
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if c.ndim < 2:
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onedim = True
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c = numpy.atleast_2d(c)
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if mode == "left":
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c = c.T
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else:
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onedim = False
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a = numpy.atleast_2d(numpy.asarray(a)) # chkfinite done in qr
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M, N = a.shape
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if mode == 'left':
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if c.shape[0] != min(M, N + overwrite_c*(M-N)):
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raise ValueError('Array shapes are not compatible for Q @ c'
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' operation: {} vs {}'.format(a.shape, c.shape))
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else:
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if M != c.shape[1]:
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raise ValueError('Array shapes are not compatible for c @ Q'
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' operation: {} vs {}'.format(c.shape, a.shape))
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raw = qr(a, overwrite_a, None, "raw", pivoting)
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Q, tau = raw[0]
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gor_un_mqr, = get_lapack_funcs(('ormqr',), (Q,))
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if gor_un_mqr.typecode in ('s', 'd'):
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trans = "T"
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else:
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trans = "C"
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Q = Q[:, :min(M, N)]
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if M > N and mode == "left" and not overwrite_c:
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if conjugate:
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cc = numpy.zeros((c.shape[1], M), dtype=c.dtype, order="F")
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cc[:, :N] = c.T
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else:
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cc = numpy.zeros((M, c.shape[1]), dtype=c.dtype, order="F")
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cc[:N, :] = c
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trans = "N"
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if conjugate:
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lr = "R"
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else:
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lr = "L"
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overwrite_c = True
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elif c.flags["C_CONTIGUOUS"] and trans == "T" or conjugate:
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cc = c.T
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if mode == "left":
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lr = "R"
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else:
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lr = "L"
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else:
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trans = "N"
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cc = c
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if mode == "left":
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lr = "L"
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else:
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lr = "R"
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cQ, = safecall(gor_un_mqr, "gormqr/gunmqr", lr, trans, Q, tau, cc,
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overwrite_c=overwrite_c)
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if trans != "N":
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cQ = cQ.T
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if mode == "right":
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cQ = cQ[:, :min(M, N)]
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if onedim:
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cQ = cQ.ravel()
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return (cQ,) + raw[1:]
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def rq(a, overwrite_a=False, lwork=None, mode='full', check_finite=True):
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"""
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Compute RQ decomposition of a matrix.
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Calculate the decomposition ``A = R Q`` where Q is unitary/orthogonal
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and R upper triangular.
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Parameters
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----------
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a : (M, N) array_like
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Matrix to be decomposed
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overwrite_a : bool, optional
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Whether data in a is overwritten (may improve performance)
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lwork : int, optional
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Work array size, lwork >= a.shape[1]. If None or -1, an optimal size
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is computed.
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mode : {'full', 'r', 'economic'}, optional
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Determines what information is to be returned: either both Q and R
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('full', default), only R ('r') or both Q and R but computed in
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economy-size ('economic', see Notes).
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check_finite : bool, optional
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Whether to check that the input matrix contains 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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R : float or complex ndarray
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Of shape (M, N) or (M, K) for ``mode='economic'``. ``K = min(M, N)``.
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Q : float or complex ndarray
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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
|