# scipy/scipy

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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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