scipy.linalg.norm#
- scipy.linalg.norm(a, ord=None, axis=None, keepdims=False, check_finite=True)[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. For tensors with rank different from 1 or 2, only ord=None is supported.- Parameters:
- aarray_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.
- keepdimsbool, 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_finitebool, 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:
- nfloat 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]:
\(||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}\)
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