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Module « scipy.sparse.linalg »

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

help(scipy.sparse.linalg.onenormest)

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

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 array.
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
--------
>>> import numpy as np
>>> from scipy.sparse import csc_array
>>> from scipy.sparse.linalg import onenormest
>>> A = csc_array([[1., 0., 0.], [5., 8., 2.], [0., -1., 0.]], dtype=float)
>>> A.toarray()
array([[ 1.,  0.,  0.],
       [ 5.,  8.,  2.],
       [ 0., -1.,  0.]])
>>> onenormest(A)
9.0
>>> np.linalg.norm(A.toarray(), ord=1)
9.0

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