/
_misc.py
191 lines (163 loc) · 6.14 KB
/
_misc.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
import numpy as np
from numpy.linalg import LinAlgError
from .blas import get_blas_funcs
from .lapack import get_lapack_funcs
__all__ = ['LinAlgError', 'LinAlgWarning', 'norm']
class LinAlgWarning(RuntimeWarning):
"""
The warning emitted when a linear algebra related operation is close
to fail conditions of the algorithm or loss of accuracy is expected.
"""
pass
def norm(a, ord=None, axis=None, keepdims=False, check_finite=True):
"""
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
"""
# Differs from numpy only in non-finite handling and the use of blas.
if check_finite:
a = np.asarray_chkfinite(a)
else:
a = np.asarray(a)
if a.size and 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, ilp64='preferred')
return nrm2(a)
if a.ndim == 2:
# 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, ilp64='preferred')
return lange(*lange_args)
# fall back to numpy in every other case
return np.linalg.norm(a, ord=ord, axis=axis, keepdims=keepdims)
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