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File Content: decomp_svd.cpython-35.pyc

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����Z�d�S)zSVD decomposition functions.�����)�division�print_function�absolute_importN)�zeros�r_�diag����)�LinAlgError�_datacopied)�get_lapack_funcs�_compute_lwork)�_asarray_validated)�string_types�svd�svdvals�diagsvd�orthTF�gesddc����������
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    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

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c�������������C���sl���t��|��d�|��}��|��j�r7�t�|��d�d�d�|�d�d��St�|��j���d�k�r[�t�d�����n
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    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.

r���r���r���r���Fr���zexpected matrixN)r
����sizer���r���r���r����numpy�empty)r ���r���r���r'���r'���r(���r�������s����+	c�������������C���s����t��|����}�|�j�j�}�t�|����}�|�|�k�rT�t�d�|�t�|�|�|�f�|���f�S|�|�k�r��t�|�t�|�|�|�f�|���f�St�d�����d�S)a���
    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

z-1zLength of s must be M or N.N)r����dtype�charr���r���r���r���)r$����M�N�part�typZMorNr'���r'���r(���r�������s����$!c�������
������C���s����t��|��d�d��\�}�}�}�|��j�\�}�}�t�j�t���j�}�t�|�|���t�j�|���|�}�t�j�|�|�k�d�t	��}�|�d�d���d�|���f�}	�|	�S)a���
    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

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