norm(x, ord=None, axis=None, keepdims=False)
This docstring was copied from numpy.linalg.norm.
Some inconsistencies with the Dask version may exist.
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 values of ord < 1
, 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(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 :
$||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}$
The nuclear norm is the sum of the singular values.
Both the Frobenius and nuclear norm orders are only defined for matrices and raise a ValueError when x.ndim != 2
.
Input array. If :None:None:`axis` is None, x must be 1-D or 2-D, unless :None:None:`ord` is None. If both :None:None:`axis` and :None:None:`ord` are None, the 2-norm of x.ravel
will be returned.
Order of the norm (see table under Notes
). inf means numpy's :None:None:`inf` object. The default is None.
If :None:None:`axis` is an integer, it specifies the axis of x along which to compute the vector norms. If :None:None:`axis` is a 2-tuple, it specifies the axes that hold 2-D matrices, and the matrix norms of these matrices are computed. If :None:None:`axis` is None then either a vector norm (when x is 1-D) or a matrix norm (when x is 2-D) is returned. The default is None.
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 x.
Norm of the matrix or vector(s).
Matrix or vector norm.
scipy.linalg.norm
Similar function in SciPy.
>>> from numpy import linalg as LA # doctest: +SKIPThis example is valid syntax, but we were not able to check execution
... a = np.arange(9) - 4 # doctest: +SKIP
... a # doctest: +SKIP array([-4, -3, -2, ..., 2, 3, 4])
>>> b = a.reshape((3, 3)) # doctest: +SKIPThis example is valid syntax, but we were not able to check execution
... b # doctest: +SKIP array([[-4, -3, -2], [-1, 0, 1], [ 2, 3, 4]])
>>> LA.norm(a) # doctest: +SKIP 7.745966692414834This example is valid syntax, but we were not able to check execution
>>> LA.norm(b) # doctest: +SKIP 7.745966692414834This example is valid syntax, but we were not able to check execution
>>> LA.norm(b, 'fro') # doctest: +SKIP 7.745966692414834This example is valid syntax, but we were not able to check execution
>>> LA.norm(a, np.inf) # doctest: +SKIP 4.0This example is valid syntax, but we were not able to check execution
>>> LA.norm(b, np.inf) # doctest: +SKIP 9.0This example is valid syntax, but we were not able to check execution
>>> LA.norm(a, -np.inf) # doctest: +SKIP 0.0This example is valid syntax, but we were not able to check execution
>>> LA.norm(b, -np.inf) # doctest: +SKIP 2.0This example is valid syntax, but we were not able to check execution
>>> LA.norm(a, 1) # doctest: +SKIP 20.0This example is valid syntax, but we were not able to check execution
>>> LA.norm(b, 1) # doctest: +SKIP 7.0This example is valid syntax, but we were not able to check execution
>>> LA.norm(a, -1) # doctest: +SKIP -4.6566128774142013e-010This example is valid syntax, but we were not able to check execution
>>> LA.norm(b, -1) # doctest: +SKIP 6.0This example is valid syntax, but we were not able to check execution
>>> LA.norm(a, 2) # doctest: +SKIP 7.745966692414834This example is valid syntax, but we were not able to check execution
>>> LA.norm(b, 2) # doctest: +SKIP 7.3484692283495345This example is valid syntax, but we were not able to check execution
>>> LA.norm(a, -2) # doctest: +SKIP 0.0This example is valid syntax, but we were not able to check execution
>>> LA.norm(b, -2) # doctest: +SKIP 1.8570331885190563e-016 # may varyThis example is valid syntax, but we were not able to check execution
>>> LA.norm(a, 3) # doctest: +SKIP 5.8480354764257312 # may varyThis example is valid syntax, but we were not able to check execution
>>> LA.norm(a, -3) # doctest: +SKIP 0.0
Using the :None:None:`axis` argument to compute vector norms:
>>> c = np.array([[ 1, 2, 3], # doctest: +SKIPThis example is valid syntax, but we were not able to check execution
... [-1, 1, 4]])
... LA.norm(c, axis=0) # doctest: +SKIP array([ 1.41421356, 2.23606798, 5. ])
>>> LA.norm(c, axis=1) # doctest: +SKIP array([ 3.74165739, 4.24264069])This example is valid syntax, but we were not able to check execution
>>> LA.norm(c, ord=1, axis=1) # doctest: +SKIP array([ 6., 6.])
Using the :None:None:`axis` argument to compute matrix norms:
>>> m = np.arange(8).reshape(2,2,2) # doctest: +SKIPThis example is valid syntax, but we were not able to check execution
... LA.norm(m, axis=(1,2)) # doctest: +SKIP array([ 3.74165739, 11.22497216])
>>> LA.norm(m[0, :, :]), LA.norm(m[1, :, :]) # doctest: +SKIP (3.7416573867739413, 11.224972160321824)See :
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