# nlcpy.linalg.norm

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

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).

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

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))