# scipy/scipy

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 """QR decomposition functions.""" from __future__ import division, print_function, absolute_import import numpy # Local imports from .blas import get_blas_funcs 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.pop("lwork", None) if lwork is None: 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 : boolean, 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. .. versionadded:: 0.11.0 Parameters ---------- a : ndarray, shape (M, N) Matrix to be decomposed c : ndarray, one- or two-dimensional calculate the product of c and q, depending on the mode: mode : {'left', 'right'}, optional ``dot(Q, c)`` is returned if mode is 'left', ``dot(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 : float or complex ndarray the product of Q and c, as defined in mode R : float or complex ndarray Of shape (K, N), ``K = min(M, N)``. P : ndarray of ints 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, dormqr, zunmqr, dgeqp3, and zgeqp3. """ if not mode in ['left', 'right']: raise ValueError("Mode argument should be one of ['left', 'right']") c = numpy.asarray_chkfinite(c) onedim = c.ndim == 1 if onedim: c = c.reshape(1, len(c)) if mode == "left": c = c.T a = numpy.asarray(a) # chkfinite done in qr M, N = a.shape if not (mode == "left" and (not overwrite_c and min(M, N) == c.shape[0] or overwrite_c and M == c.shape[0]) or mode == "right" and M == c.shape[1]): raise ValueError("objects are not aligned") 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 square real matrix. Calculate the decomposition ``A = R Q`` where ``Q`` is unitary/orthogonal and ``R`` upper triangular. Parameters ---------- a : array, shape (M, M) 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 array, shape (M, N) Upper triangular Q : float or complex array, shape (M, M) Unitary/orthogonal 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']") 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,)) 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 = 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: # 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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