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The warning emitted when a linear algebra related operation is close
to fail conditions of the algorithm or loss of accuracy is expected.
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Matrix or vector norm.
This function is able to return one of eight different matrix norms,
or one of an infinite number of vector norms (described below), depending
on the value of the ``ord`` parameter. For tensors with rank different from
1 or 2, only `ord=None` is supported.
Parameters
----------
a : array_like
Input array. If `axis` is None, `a` must be 1-D or 2-D, unless `ord`
is None. If both `axis` and `ord` are None, the 2-norm of
``a.ravel`` will be returned.
ord : {int, inf, -inf, 'fro', 'nuc', None}, optional
Order of the norm (see table under ``Notes``). inf means NumPy's
`inf` object.
axis : {int, 2-tuple of ints, None}, optional
If `axis` is an integer, it specifies the axis of `a` along which to
compute the vector norms. If `axis` is a 2-tuple, it specifies the
axes that hold 2-D matrices, and the matrix norms of these matrices
are computed. If `axis` is None then either a vector norm (when `a`
is 1-D) or a matrix norm (when `a` is 2-D) is returned.
keepdims : bool, optional
If this is set to True, the axes which are normed over are left in the
result as dimensions with size one. With this option the result will
broadcast correctly against the original `a`.
check_finite : bool, optional
Whether to check that the input matrix contains only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Returns
-------
n : float or ndarray
Norm of the matrix or vector(s).
Notes
-----
For values of ``ord <= 0``, the result is, strictly speaking, not a
mathematical 'norm', but it may still be useful for various numerical
purposes.
The following norms can be calculated:
===== ============================ ==========================
ord norm for matrices norm for vectors
===== ============================ ==========================
None Frobenius norm 2-norm
'fro' Frobenius norm --
'nuc' nuclear norm --
inf max(sum(abs(a), axis=1)) max(abs(a))
-inf min(sum(abs(a), axis=1)) min(abs(a))
0 -- sum(a != 0)
1 max(sum(abs(a), axis=0)) as below
-1 min(sum(abs(a), axis=0)) as below
2 2-norm (largest sing. value) as below
-2 smallest singular value as below
other -- sum(abs(a)**ord)**(1./ord)
===== ============================ ==========================
The Frobenius norm is given by [1]_:
:math:`||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}`
The nuclear norm is the sum of the singular values.
Both the Frobenius and nuclear norm orders are only defined for
matrices.
References
----------
.. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*,
Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
Examples
--------
>>> import numpy as np
>>> from scipy.linalg import norm
>>> a = np.arange(9) - 4.0
>>> a
array([-4., -3., -2., -1., 0., 1., 2., 3., 4.])
>>> b = a.reshape((3, 3))
>>> b
array([[-4., -3., -2.],
[-1., 0., 1.],
[ 2., 3., 4.]])
>>> norm(a)
7.745966692414834
>>> norm(b)
7.745966692414834
>>> norm(b, 'fro')
7.745966692414834
>>> norm(a, np.inf)
4
>>> norm(b, np.inf)
9
>>> norm(a, -np.inf)
0
>>> norm(b, -np.inf)
2
>>> norm(a, 1)
20
>>> norm(b, 1)
7
>>> norm(a, -1)
-4.6566128774142013e-010
>>> norm(b, -1)
6
>>> norm(a, 2)
7.745966692414834
>>> norm(b, 2)
7.3484692283495345
>>> norm(a, -2)
0
>>> norm(b, -2)
1.8570331885190563e-016
>>> norm(a, 3)
5.8480354764257312
>>> norm(a, -3)
0
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