nlcpy.linalg.norm

nlcpy.linalg.norm(x, ord=None, axis=None, keepdims=False)[source]

Returns 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

xarray_like: Input array. If axis is None, x must be 1-D or 2-D.
ord{non-zero int, inf, -inf, ‘fro’, ‘nuc’}, optional: Order of the norm (see table under Note). inf means nlcpy’s inf object.
axis{None, int, 2-tuple of ints}, optional: If axis is an integer, it specifies the axis of x 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 x is 1-D) or a matrix norm (when x 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 x.

Returns

nndarray: Norm of the matrix or vector(s).

Note

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.

Examples

>>> import nlcpy as vp
>>> a = vp.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]])
>>> vp.linalg.norm(a)    
array(7.74596669)
>>> vp.linalg.norm(b)    
array(7.74596669)
>>> vp.linalg.norm(b, 'fro')   
array(7.74596669)
>>> vp.linalg.norm(a, vp.inf)  
array(4.)
>>> vp.linalg.norm(b, vp.inf)  
array(9.)
>>> vp.linalg.norm(a, -vp.inf)  
array(0.)
>>> vp.linalg.norm(b, -vp.inf)  
array(2.)
>>> vp.linalg.norm(a, 1)   
array(20.)
>>> vp.linalg.norm(b, 1)   
array(7.)
>>> vp.linalg.norm(a, -1)  
array(0.)
>>> vp.linalg.norm(b, -1)  
array(6.)
>>> vp.linalg.norm(a, 2)   
array(7.74596669)
>>> vp.linalg.norm(b, 2)  
array(7.34846923)
>>> vp.linalg.norm(a, -2) 
array(0.)
>>> vp.linalg.norm(b, -2) 
array(3.75757704e-16)
>>> vp.linalg.norm(a, 3)  
array(5.84803548)
>>> vp.linalg.norm(a, -3) 
array(0.)

Using the axis argument to compute vector norms:

>>> c = vp.array([[ 1, 2, 3],
...               [-1, 1, 4]])
>>> vp.linalg.norm(c, axis=0)   
array([1.41421356, 2.23606798, 5.        ])
>>> vp.linalg.norm(c, axis=1)   
array([3.74165739, 4.24264069])
>>> vp.linalg.norm(c, ord=1, axis=1)  
array([6., 6.])

Using the axis argument to compute matrix norms:

>>> m = vp.arange(8).reshape(2,2,2)
>>> vp.linalg.norm(m, axis=(1,2))   
array([ 3.74165739, 11.22497216])
>>> vp.linalg.norm(m[0, :, :]), vp.linalg.norm(m[1, :, :])  
(array(3.74165739), array(11.22497216))