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Fonction onenormest - module scipy.sparse.linalg

Signature de la fonction onenormest

def onenormest(A, t=2, itmax=5, compute_v=False, compute_w=False) 

Description

onenormest.__doc__

    Compute a lower bound of the 1-norm of a sparse matrix.

    Parameters
    ----------
    A : ndarray or other linear operator
        A linear operator that can be transposed and that can
        produce matrix products.
    t : int, optional
        A positive parameter controlling the tradeoff between
        accuracy versus time and memory usage.
        Larger values take longer and use more memory
        but give more accurate output.
    itmax : int, optional
        Use at most this many iterations.
    compute_v : bool, optional
        Request a norm-maximizing linear operator input vector if True.
    compute_w : bool, optional
        Request a norm-maximizing linear operator output vector if True.

    Returns
    -------
    est : float
        An underestimate of the 1-norm of the sparse matrix.
    v : ndarray, optional
        The vector such that ||Av||_1 == est*||v||_1.
        It can be thought of as an input to the linear operator
        that gives an output with particularly large norm.
    w : ndarray, optional
        The vector Av which has relatively large 1-norm.
        It can be thought of as an output of the linear operator
        that is relatively large in norm compared to the input.

    Notes
    -----
    This is algorithm 2.4 of [1].

    In [2] it is described as follows.
    "This algorithm typically requires the evaluation of
    about 4t matrix-vector products and almost invariably
    produces a norm estimate (which is, in fact, a lower
    bound on the norm) correct to within a factor 3."

    .. versionadded:: 0.13.0

    References
    ----------
    .. [1] Nicholas J. Higham and Francoise Tisseur (2000),
           "A Block Algorithm for Matrix 1-Norm Estimation,
           with an Application to 1-Norm Pseudospectra."
           SIAM J. Matrix Anal. Appl. Vol. 21, No. 4, pp. 1185-1201.

    .. [2] Awad H. Al-Mohy and Nicholas J. Higham (2009),
           "A new scaling and squaring algorithm for the matrix exponential."
           SIAM J. Matrix Anal. Appl. Vol. 31, No. 3, pp. 970-989.

    Examples
    --------
    >>> from scipy.sparse import csc_matrix
    >>> from scipy.sparse.linalg import onenormest
    >>> A = csc_matrix([[1., 0., 0.], [5., 8., 2.], [0., -1., 0.]], dtype=float)
    >>> A.todense()
    matrix([[ 1.,  0.,  0.],
            [ 5.,  8.,  2.],
            [ 0., -1.,  0.]])
    >>> onenormest(A)
    9.0
    >>> np.linalg.norm(A.todense(), ord=1)
    9.0