ó ˽÷Xc@`sådZddlmZmZmZddlZddlmZmZmZddl m Z m Z ddl m Z mZddlmZdd lmZd d d d gZeeeedd„Zeed„Zd„Zd„ZdS(sSVD decomposition functions.i(tdivisiontprint_functiontabsolute_importN(tzerostr_tdiagi(t LinAlgErrort _datacopied(tget_lapack_funcst_compute_lwork(t_asarray_validated(t string_typestsvdtsvdvalstdiagsvdtorthtgesddc C`stt|d|ƒ}t|jƒdkr6tdƒ‚n|j\}}|pWt||ƒ}t|tƒsxtdƒ‚n|dkrštd|fƒ‚n||df} t| |fƒ\} } t | |jd |jd d |d |ƒ} | |d |d | d |d|ƒ\} }}}|d kr9t dƒ‚n|d krYtd| ƒ‚n|rl| ||fS|SdS(s6 Singular Value Decomposition. Factorizes the matrix a into two unitary matrices U and Vh, and a 1-D array s of singular values (real, non-negative) such that ``a == U*S*Vh``, where S is a suitably shaped matrix of zeros with main diagonal s. Parameters ---------- a : (M, N) array_like Matrix to decompose. full_matrices : bool, optional If True, `U` and `Vh` are of shape ``(M,M)``, ``(N,N)``. If False, the shapes are ``(M,K)`` and ``(K,N)``, where ``K = min(M,N)``. compute_uv : bool, optional Whether to compute also `U` and `Vh` in addition to `s`. Default is True. overwrite_a : bool, optional Whether to overwrite `a`; may improve performance. Default is False. 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. lapack_driver : {'gesdd', 'gesvd'}, optional Whether to use the more efficient divide-and-conquer approach (``'gesdd'``) or general rectangular approach (``'gesvd'``) to compute the SVD. MATLAB and Octave use the ``'gesvd'`` approach. Default is ``'gesdd'``. .. versionadded:: 0.18 Returns ------- U : ndarray Unitary matrix having left singular vectors as columns. Of shape ``(M,M)`` or ``(M,K)``, depending on `full_matrices`. s : ndarray The singular values, sorted in non-increasing order. Of shape (K,), with ``K = min(M, N)``. Vh : ndarray Unitary matrix having right singular vectors as rows. Of shape ``(N,N)`` or ``(K,N)`` depending on `full_matrices`. For ``compute_uv=False``, only `s` is returned. Raises ------ LinAlgError If SVD computation does not converge. See also -------- svdvals : Compute singular values of a matrix. diagsvd : Construct the Sigma matrix, given the vector s. Examples -------- >>> from scipy import linalg >>> a = np.random.randn(9, 6) + 1.j*np.random.randn(9, 6) >>> U, s, Vh = linalg.svd(a) >>> U.shape, Vh.shape, s.shape ((9, 9), (6, 6), (6,)) >>> U, s, Vh = linalg.svd(a, full_matrices=False) >>> U.shape, Vh.shape, s.shape ((9, 6), (6, 6), (6,)) >>> S = linalg.diagsvd(s, 6, 6) >>> np.allclose(a, np.dot(U, np.dot(S, Vh))) True >>> s2 = linalg.svd(a, compute_uv=False) >>> np.allclose(s, s2) True t check_finiteisexpected matrixslapack_driver must be a stringRtgesvds2lapack_driver must be "gesdd" or "gesvd", not "%s"t_lworkiit compute_uvt full_matricestlworkt overwrite_asSVD did not converges1illegal value in %d-th argument of internal gesddN(RR( R tlentshapet ValueErrorRt isinstanceR t TypeErrorRR R(taRRRRt lapack_driverta1tmtntfuncstgesXdt gesXd_lworkRtutstvtinfo((s6/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp_svd.pyR s0P    cC`slt|d|ƒ}|jr7t|ddd|dtƒSt|jƒdkr[tdƒ‚n tjdƒSdS(s‘ Compute singular values of a matrix. Parameters ---------- a : (M, N) array_like Matrix to decompose. overwrite_a : bool, optional Whether to overwrite `a`; may improve performance. Default is False. 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 ------- s : (min(M, N),) ndarray The singular values, sorted in decreasing order. Raises ------ LinAlgError If SVD computation does not converge. Notes ----- ``svdvals(a)`` only differs from ``svd(a, compute_uv=False)`` by its handling of the edge case of empty ``a``, where it returns an empty sequence: >>> a = np.empty((0, 2)) >>> from scipy.linalg import svdvals >>> svdvals(a) array([], dtype=float64) See also -------- svd : Compute the full singular value decomposition of a matrix. diagsvd : Construct the Sigma matrix, given the vector s. RRiRisexpected matrixN( R tsizeR tFalseRRRtnumpytempty(RRR((s6/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp_svd.pyR s+ cC`s‘t|ƒ}|jj}t|ƒ}||krTtd|t|||f|ƒfS||krt|t|||f|ƒfStdƒ‚dS(sœ Construct the sigma matrix in SVD from singular values and size M, N. Parameters ---------- s : (M,) or (N,) array_like Singular values M : int Size of the matrix whose singular values are `s`. N : int Size of the matrix whose singular values are `s`. Returns ------- S : (M, N) ndarray The S-matrix in the singular value decomposition s-1sLength of s must be M or N.N(RtdtypetcharRRRR(R&tMtNtpartttyptMorN((s6/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp_svd.pyR¶s    $ !c C`s—t|dtƒ\}}}|j\}}tjtƒj}t||ƒtj|ƒ|}tj ||kdt ƒ}|dd…d|…f} | S(s‚ Construct an orthonormal basis for the range of A using SVD Parameters ---------- A : (M, N) array_like Input array Returns ------- Q : (M, K) ndarray Orthonormal basis for the range of A. K = effective rank of A, as determined by automatic cutoff See also -------- svd : Singular value decomposition of a matrix RR-N( R R*RR+tfinfotfloattepstmaxtamaxtsumtint( tAR%R&tvhR/R0R6ttoltnumtQ((s6/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp_svd.pyRÖs (t__doc__t __future__RRRR+RRRtmiscRRtlapackRR tdecompR tscipy._lib.sixR t__all__tTrueR*R R RR(((s6/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp_svd.pyts  p5