Module « numpy.dual »

Fonction cholesky - module numpy.dual

Signature de la fonction cholesky

def cholesky(a) 

Description

cholesky.__doc__

    Cholesky decomposition.

    Return the Cholesky decomposition, `L * L.H`, of the square matrix `a`,
    where `L` is lower-triangular and .H is the conjugate transpose operator
    (which is the ordinary transpose if `a` is real-valued).  `a` must be
    Hermitian (symmetric if real-valued) and positive-definite. No
    checking is performed to verify whether `a` is Hermitian or not.
    In addition, only the lower-triangular and diagonal elements of `a`
    are used. Only `L` is actually returned.

    Parameters
    ----------
    a : (..., M, M) array_like
        Hermitian (symmetric if all elements are real), positive-definite
        input matrix.

    Returns
    -------
    L : (..., M, M) array_like
        Upper or lower-triangular Cholesky factor of `a`.  Returns a
        matrix object if `a` is a matrix object.

    Raises
    ------
    LinAlgError
       If the decomposition fails, for example, if `a` is not
       positive-definite.

    See Also
    --------
    scipy.linalg.cholesky : Similar function in SciPy.
    scipy.linalg.cholesky_banded : Cholesky decompose a banded Hermitian
                                   positive-definite matrix.
    scipy.linalg.cho_factor : Cholesky decomposition of a matrix, to use in
                              `scipy.linalg.cho_solve`.

    Notes
    -----

    .. versionadded:: 1.8.0

    Broadcasting rules apply, see the `numpy.linalg` documentation for
    details.

    The Cholesky decomposition is often used as a fast way of solving

    .. math:: A \mathbf{x} = \mathbf{b}

    (when `A` is both Hermitian/symmetric and positive-definite).

    First, we solve for :math:`\mathbf{y}` in

    .. math:: L \mathbf{y} = \mathbf{b},

    and then for :math:`\mathbf{x}` in

    .. math:: L.H \mathbf{x} = \mathbf{y}.

    Examples
    --------
    >>> A = np.array([[1,-2j],[2j,5]])
    >>> A
    array([[ 1.+0.j, -0.-2.j],
           [ 0.+2.j,  5.+0.j]])
    >>> L = np.linalg.cholesky(A)
    >>> L
    array([[1.+0.j, 0.+0.j],
           [0.+2.j, 1.+0.j]])
    >>> np.dot(L, L.T.conj()) # verify that L * L.H = A
    array([[1.+0.j, 0.-2.j],
           [0.+2.j, 5.+0.j]])
    >>> A = [[1,-2j],[2j,5]] # what happens if A is only array_like?
    >>> np.linalg.cholesky(A) # an ndarray object is returned
    array([[1.+0.j, 0.+0.j],
           [0.+2.j, 1.+0.j]])
    >>> # But a matrix object is returned if A is a matrix object
    >>> np.linalg.cholesky(np.matrix(A))
    matrix([[ 1.+0.j,  0.+0.j],
            [ 0.+2.j,  1.+0.j]])