192 lines
6.1 KiB
Python
192 lines
6.1 KiB
Python
|
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
|