Module « numpy.linalg »

Fonction cond - module numpy.linalg

Signature de la fonction cond

def cond(x, p=None) 

Description

cond.__doc__

    Compute the condition number of a matrix.

    This function is capable of returning the condition number using
    one of seven different norms, depending on the value of `p` (see
    Parameters below).

    Parameters
    ----------
    x : (..., M, N) array_like
        The matrix whose condition number is sought.
    p : {None, 1, -1, 2, -2, inf, -inf, 'fro'}, optional
        Order of the norm:

        =====  ============================
        p      norm for matrices
        =====  ============================
        None   2-norm, computed directly using the ``SVD``
        'fro'  Frobenius norm
        inf    max(sum(abs(x), axis=1))
        -inf   min(sum(abs(x), axis=1))
        1      max(sum(abs(x), axis=0))
        -1     min(sum(abs(x), axis=0))
        2      2-norm (largest sing. value)
        -2     smallest singular value
        =====  ============================

        inf means the numpy.inf object, and the Frobenius norm is
        the root-of-sum-of-squares norm.

    Returns
    -------
    c : {float, inf}
        The condition number of the matrix. May be infinite.

    See Also
    --------
    numpy.linalg.norm

    Notes
    -----
    The condition number of `x` is defined as the norm of `x` times the
    norm of the inverse of `x` [1]_; the norm can be the usual L2-norm
    (root-of-sum-of-squares) or one of a number of other matrix norms.

    References
    ----------
    .. [1] G. Strang, *Linear Algebra and Its Applications*, Orlando, FL,
           Academic Press, Inc., 1980, pg. 285.

    Examples
    --------
    >>> from numpy import linalg as LA
    >>> a = np.array([[1, 0, -1], [0, 1, 0], [1, 0, 1]])
    >>> a
    array([[ 1,  0, -1],
           [ 0,  1,  0],
           [ 1,  0,  1]])
    >>> LA.cond(a)
    1.4142135623730951
    >>> LA.cond(a, 'fro')
    3.1622776601683795
    >>> LA.cond(a, np.inf)
    2.0
    >>> LA.cond(a, -np.inf)
    1.0
    >>> LA.cond(a, 1)
    2.0
    >>> LA.cond(a, -1)
    1.0
    >>> LA.cond(a, 2)
    1.4142135623730951
    >>> LA.cond(a, -2)
    0.70710678118654746 # may vary
    >>> min(LA.svd(a, compute_uv=False))*min(LA.svd(LA.inv(a), compute_uv=False))
    0.70710678118654746 # may vary