ó î&]\c@`s¯dZddlmZmZmZddlZddlmZddlm Z ddd gZ d „Z e dd e ed „Zd e e e e d„Ze dd ed„ZdS(sQR decomposition functions.i(tdivisiontprint_functiontabsolute_importNi(tget_lapack_funcs(t _datacopiedtqrt qr_multiplytrqcO`s |jddƒ}|dkr[d|d<|||Ž}|ddjjtjƒ|d= 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,)) RRtrteconomictraws>Mode argument should be one of ['full', 'r','economic', 'raw']isexpected 2D arraytgeqp3t overwrite_aitgeqrfRNtorgqrs gorgqr/gungqrtdtype(RRRRR(R(R(RR(R( RR tasarray_chkfinitetasarraytlentshapeRRRttriuRtchartempty(taRRtmodetpivotingt check_finiteta1tMtNRRtjpvtttauRtRtRjt gor_un_gqrtQtttqqr((s5lib/python2.7/site-packages/scipy/linalg/decomp_qr.pyRsN] ! %    (   trightc C`sŒ|dkr$tdj|ƒƒ‚ntj|ƒ}|jdkrrt}tj|ƒ}|dkrx|j}qxnt}tjtj |ƒƒ}|j \}} |dkr÷|j dt || ||| ƒkr+tdj|j |j ƒƒ‚q+n4||j dkr+tdj|j |j ƒƒ‚nt ||dd |ƒ} | d\} } td| fƒ\} | jdkr€d }nd}| dd…dt || ƒ…f} || krˆ|dkrˆ| rˆ|rtj|j d|fd|jddƒ}|j|dd…d| …f>> 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 tleftR5s8Mode argument can only be 'left' or 'right' but not '{}'iis=Array shapes are not compatible for Q @ c operation: {} vs {}is=Array shapes are not compatible for c @ Q operation: {} vs {}RtormqrtstdtTtCNRtordertFR,R/tLt C_CONTIGUOUSs gormqr/gunmqrt overwrite_c(R6R5(R7(R8R9(RtformatR RtndimtTruet atleast_2dR:tFalseR R"tminRR RttypecodetzerosRtflagsRtravel(R&tcR'R(t conjugateRR@tonedimR+R,RR2R.t gor_un_mqrttranstcctlrtcQ((s5lib/python2.7/site-packages/scipy/linalg/decomp_qr.pyR°spK    (   %+"+            (c C`s|dkrtdƒ‚n|r3tj|ƒ}ntj|ƒ}t|jƒdkrftdƒ‚n|j\}}|p‡t||ƒ}td|fƒ\}t|d|d|d |ƒ\} } |dk sÜ||krõtj | ||ƒ} n'tj | | d …| d …fƒ} |dkr,| Std| fƒ\} ||kryt| d | | | d|d d ƒ\} n€|dkr¬t| d | | d|d d ƒ\} nMtj ||fd| j ƒ}| || )t| d || d|d d ƒ\} | | fS(sš 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)) RRRs8Mode argument should be one of ['full', 'r', 'economic']isexpected matrixtgerqfRRNtorgrqs gorgrq/gungrqiR(RRR(RS(RT( RR RR R!R"RRRR#R%R(R&RRR'R)R*R+R,RSRR.R/t gor_un_grqR2trq1((s5lib/python2.7/site-packages/scipy/linalg/decomp_qr.pyRCs<?  '     (t__doc__t __future__RRRR tlapackRtmiscRt__all__RRER RCRRR(((s5lib/python2.7/site-packages/scipy/linalg/decomp_qr.pyts    “ ’