norm — SciPy v1.15.2 Manual (original) (raw)
scipy.linalg.
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 ofa.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) 4.0 norm(b, np.inf) 9.0 norm(a, -np.inf) 0.0 norm(b, -np.inf) 2.0
norm(a, 1) 20.0 norm(b, 1) 7.0 norm(a, -1) -4.6566128774142013e-010 norm(b, -1) 6.0 norm(a, 2) 7.745966692414834 norm(b, 2) 7.3484692283495345
norm(a, -2) 0.0 norm(b, -2) 1.8570331885190563e-016 norm(a, 3) 5.8480354764257312 norm(a, -3) 0.0