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from __future__ import division, print_function, absolute_import
import numpy as np
from numpy.linalg import LinAlgError
from .blas import get_blas_funcs
from .lapack import get_lapack_funcs
__all__ = ['LinAlgError', 'norm']
def norm(a, ord=None, axis=None, keepdims=False):
"""
Matrix or vector norm.
This function is able to return one of seven different matrix norms,
or one of an infinite number of vector norms (described below), depending
on the value of the ``ord`` parameter.
Parameters
----------
a : (M,) or (M, N) array_like
Input array. If `axis` is None, `a` must be 1-D or 2-D.
ord : {non-zero int, inf, -inf, 'fro'}, 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`.
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 --
inf max(sum(abs(x), axis=1)) max(abs(x))
-inf min(sum(abs(x), axis=1)) min(abs(x))
0 -- sum(x != 0)
1 max(sum(abs(x), axis=0)) as below
-1 min(sum(abs(x), axis=0)) as below
2 2-norm (largest sing. value) as below
-2 smallest singular value as below
other -- sum(abs(x)**ord)**(1./ord)
===== ============================ ==========================
The Frobenius norm is given by [1]_:
:math:`||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}`
The ``axis`` and ``keepdims`` arguments are passed directly to
``numpy.linalg.norm`` and are only usable if they are supported
by the version of numpy in use.
References
----------
.. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*,
Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
Examples
--------
>>> 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
"""
# Differs from numpy only in non-finite handling and the use of blas.
a = np.asarray_chkfinite(a)
# Only use optimized norms if axis and keepdims are not specified.
if a.dtype.char in 'fdFD' and axis is None and not keepdims:
if ord in (None, 2) and (a.ndim == 1):
# use blas for fast and stable euclidean norm
nrm2 = get_blas_funcs('nrm2', dtype=a.dtype)
return nrm2(a)
if a.ndim == 2 and axis is None and not keepdims:
# Use lapack for a couple fast matrix norms.
# For some reason the *lange frobenius norm is slow.
lange_args = None
# Make sure this works if the user uses the axis keywords
# to apply the norm to the transpose.
if ord == 1:
if np.isfortran(a):
lange_args = '1', a
elif np.isfortran(a.T):
lange_args = 'i', a.T
elif ord == np.inf:
if np.isfortran(a):
lange_args = 'i', a
elif np.isfortran(a.T):
lange_args = '1', a.T
if lange_args:
lange = get_lapack_funcs('lange', dtype=a.dtype)
return lange(*lange_args)
# Filter out the axis and keepdims arguments if they aren't used so they
# are never inadvertently passed to a version of numpy that doesn't
# support them.
if axis is not None:
if keepdims:
return np.linalg.norm(a, ord=ord, axis=axis, keepdims=keepdims)
return np.linalg.norm(a, ord=ord, axis=axis)
return np.linalg.norm(a, ord=ord)
def _datacopied(arr, original):
"""
Strict check for `arr` not sharing any data with `original`,
under the assumption that arr = asarray(original)
"""
if arr is original:
return False
if not isinstance(original, np.ndarray) and hasattr(original, '__array__'):
return False
return arr.base is None
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