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File Content: decomp_cholesky.cpython-35.pyc

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�d�d���Z�d�S)z!Cholesky decomposition functions.�����)�division�print_function�absolute_import)�asarray_chkfinite�asarray����)�LinAlgError�_datacopied)�get_lapack_funcs�cholesky�
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�|�����|�|�f�S)z,Common code for cholesky() and cho_factor().����r���r���zexpected square matrix�potrf�lower�overwrite_a�cleanz)%d-th leading minor not positive definitez1illegal value in %d-th argument of internal potrf)r���)r���r����len�shape�
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    Compute the Cholesky decomposition of a matrix.

Returns the Cholesky decomposition, :math:`A = L L^*` or
    :math:`A = U^* U` of a Hermitian positive-definite matrix A.

Parameters
    ----------
    a : (M, M) array_like
        Matrix to be decomposed
    lower : bool, optional
        Whether to compute the upper or lower triangular Cholesky
        factorization.  Default is upper-triangular.
    overwrite_a : bool, optional
        Whether to overwrite data in `a` (may improve performance).
    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
    -------
    c : (M, M) ndarray
        Upper- or lower-triangular Cholesky factor of `a`.

Raises
    ------
    LinAlgError : if decomposition fails.

Examples
    --------
    >>> from scipy import array, linalg, dot
    >>> a = array([[1,-2j],[2j,5]])
    >>> L = linalg.cholesky(a, lower=True)
    >>> L
    array([[ 1.+0.j,  0.+0.j],
           [ 0.+2.j,  1.+0.j]])
    >>> dot(L, L.T.conj())
    array([[ 1.+0.j,  0.-2.j],
           [ 0.+2.j,  5.+0.j]])

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    Compute the Cholesky decomposition of a matrix, to use in cho_solve

Returns a matrix containing the Cholesky decomposition,
    ``A = L L*`` or ``A = U* U`` of a Hermitian positive-definite matrix `a`.
    The return value can be directly used as the first parameter to cho_solve.

.. warning::
        The returned matrix also contains random data in the entries not
        used by the Cholesky decomposition. If you need to zero these
        entries, use the function `cholesky` instead.

Parameters
    ----------
    a : (M, M) array_like
        Matrix to be decomposed
    lower : bool, optional
        Whether to compute the upper or lower triangular Cholesky factorization
        (Default: upper-triangular)
    overwrite_a : bool, optional
        Whether to overwrite data in a (may improve performance)
    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
    -------
    c : (M, M) ndarray
        Matrix whose upper or lower triangle contains the Cholesky factor
        of `a`. Other parts of the matrix contain random data.
    lower : bool
        Flag indicating whether the factor is in the lower or upper triangle

Raises
    ------
    LinAlgError
        Raised if decomposition fails.

See also
    --------
    cho_solve : Solve a linear set equations using the Cholesky factorization
                of a matrix.

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������C���s��|��\�}�}�|�r-�t��|���}�t��|���}�n�t�|���}�t�|���}�|�j�d�k�sn�|�j�d�|�j�d�k�rz�t�d�����|�j�d�|�j�d�k�r��t�d�����|�p��t�|�|���}�t�d
�|�|�f���\�}�|�|�|�d�|�d�|��\�}�}	�|	�d�k�rt�d	�|	�����|�S)a��Solve the linear equations A x = b, given the Cholesky factorization of A.

Parameters
    ----------
    (c, lower) : tuple, (array, bool)
        Cholesky factorization of a, as given by cho_factor
    b : array
        Right-hand side
    overwrite_b : bool, optional
        Whether to overwrite data in b (may improve performance)
    check_finite : bool, optional
        Whether to check that the input matrices contain 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
    -------
    x : array
        The solution to the system A x = b

See also
    --------
    cho_factor : Cholesky factorization of a matrix

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���)
Zc_and_lower�br ���r���r���r���Zb1r����xr���r���r���r���r
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    Cholesky decompose a banded Hermitian positive-definite matrix

The matrix a is stored in ab either in lower diagonal or upper
    diagonal ordered form::

ab[u + i - j, j] == a[i,j]        (if upper form; i <= j)
        ab[    i - j, j] == a[i,j]        (if lower form; i >= j)

Example of ab (shape of a is (6,6), u=2)::

upper form:
        *   *   a02 a13 a24 a35
        *   a01 a12 a23 a34 a45
        a00 a11 a22 a33 a44 a55

lower form:
        a00 a11 a22 a33 a44 a55
        a10 a21 a32 a43 a54 *
        a20 a31 a42 a53 *   *

Parameters
    ----------
    ab : (u + 1, M) array_like
        Banded matrix
    overwrite_ab : bool, optional
        Discard data in ab (may enhance performance)
    lower : bool, optional
        Is the matrix in the lower form. (Default is upper form)
    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
    -------
    c : (u + 1, M) ndarray
        Cholesky factorization of a, in the same banded format as ab

�pbtrfr����overwrite_abr���z)%d-th leading minor not positive definitez1illegal value in %d-th argument of internal pbtrf)r$���)r���r���r
���r���r���)�abr%���r���r���r$���r���r���r���r���r���r�������s����)c�������	������C���s����|��\�}�}�|�r-�t��|���}�t��|���}�n�t�|���}�t�|���}�|�j�d	�|�j�d�k�rk�t�d�����t�d
�|�|�f���\�}�|�|�|�d�|�d�|��\�}�}�|�d�k�r��t�d�|�����|�d�k��r��t�d�|�����|�S)a���Solve the linear equations A x = b, given the Cholesky factorization of A.

Parameters
    ----------
    (cb, lower) : tuple, (array, bool)
        `cb` is the Cholesky factorization of A, as given by cholesky_banded.
        `lower` must be the same value that was given to cholesky_banded.
    b : array
        Right-hand side
    overwrite_b : bool, optional
        If True, the function will overwrite the values in `b`.
    check_finite : bool, optional
        Whether to check that the input matrices contain 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
    -------
    x : array
        The solution to the system A x = b

See also
    --------
    cholesky_banded : Cholesky factorization of a banded matrix

Notes
    -----

.. versionadded:: 0.8.0

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���r���)	Zcb_and_lowerr"���r ���r����cbr���r'���r#���r���r���r���r���r�������s ���� !N)�__doc__Z
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����__all__r���r���r���r
���r���r���r���r���r���r����<module>���s���		0308

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