scipy.sparse.linalg.onenormest¶

scipy.sparse.linalg.
onenormest
(A, t=2, itmax=5, compute_v=False, compute_w=False)[source]¶ Compute a lower bound of the 1norm 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 normmaximizing linear operator input vector if True.
compute_w : bool, optional
Request a normmaximizing linear operator output vector if True.
Returns: est : float
An underestimate of the 1norm 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 1norm. 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 matrixvector 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
[R396] Nicholas J. Higham and Francoise Tisseur (2000), “A Block Algorithm for Matrix 1Norm Estimation, with an Application to 1Norm Pseudospectra.” SIAM J. Matrix Anal. Appl. Vol. 21, No. 4, pp. 11851201. [R397] Awad H. AlMohy 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. 970989. 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