scipy.sparse.linalg.onenormest

scipy.sparse.linalg.onenormest(A, t=2, itmax=5, compute_v=False, compute_w=False)

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.”

New in version 0.13.0.

References

[R390]

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.

[R391]

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