numpy.linalg.norm(x, ord=None, axis=None, keepdims=False)
[source]
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.
Parameters: |
x : array_like Input array. If ord : {non-zero int, inf, -inf, ‘fro’, ‘nuc’}, optional Order of the norm (see table under axis : {int, 2-tuple of ints, None}, optional If 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 New in version 1.10.0. |
---|---|
Returns: |
n : float or ndarray Norm of the matrix or vector(s). |
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(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 [R8585]:
The nuclear norm is the sum of the singular values.
[R8585] | (1, 2) G. H. Golub and C. F. Van Loan, Matrix Computations, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15 |
>>> from numpy import linalg as LA >>> a = np.arange(9) - 4 >>> 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]])
>>> LA.norm(a) 7.745966692414834 >>> LA.norm(b) 7.745966692414834 >>> LA.norm(b, 'fro') 7.745966692414834 >>> LA.norm(a, np.inf) 4.0 >>> LA.norm(b, np.inf) 9.0 >>> LA.norm(a, -np.inf) 0.0 >>> LA.norm(b, -np.inf) 2.0
>>> LA.norm(a, 1) 20.0 >>> LA.norm(b, 1) 7.0 >>> LA.norm(a, -1) -4.6566128774142013e-010 >>> LA.norm(b, -1) 6.0 >>> LA.norm(a, 2) 7.745966692414834 >>> LA.norm(b, 2) 7.3484692283495345
>>> LA.norm(a, -2) nan >>> LA.norm(b, -2) 1.8570331885190563e-016 >>> LA.norm(a, 3) 5.8480354764257312 >>> LA.norm(a, -3) nan
Using the axis
argument to compute vector norms:
>>> c = np.array([[ 1, 2, 3], ... [-1, 1, 4]]) >>> LA.norm(c, axis=0) array([ 1.41421356, 2.23606798, 5. ]) >>> LA.norm(c, axis=1) array([ 3.74165739, 4.24264069]) >>> LA.norm(c, ord=1, axis=1) array([ 6., 6.])
Using the axis
argument to compute matrix norms:
>>> m = np.arange(8).reshape(2,2,2) >>> LA.norm(m, axis=(1,2)) array([ 3.74165739, 11.22497216]) >>> LA.norm(m[0, :, :]), LA.norm(m[1, :, :]) (3.7416573867739413, 11.224972160321824)
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https://docs.scipy.org/doc/numpy-1.14.2/reference/generated/numpy.linalg.norm.html