flopscope.numpy.linalg.norm

fnp.linalg.norm(x, ord=None, axis=None, keepdims=False)[flopscope source][numpy source]

Matrix or vector norm.

Adapted from NumPy docs np.linalg.norm

Arealinalg

Typecustom

NumPy Refnp.linalg.norm

Cost

depends on ord

Flopscope Context

Norm. Cost depends on ord: numel for L1/inf, 2*numel for Frobenius, m*n*min(m,n) for ord=2.

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 axis is None, x must be 1-D or 2-D, unless ord is None. If both axis and ord are None, the 2-norm of x.ravel will be returned.
ord:{int, float, inf, -inf, 'fro', 'nuc'}, optional: Order of the norm (see table under Notes for what values are supported for matrices and vectors respectively). inf means numpy's inf object. The default is None.
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. The default is None.
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 x.

Returns

n:float or ndarray: Norm of the matrix or vector(s).

Notes

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:

table







ord

norm for matrices

norm for vectors

None

Frobenius norm

2-norm

'fro'

Frobenius norm

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Treating it as ordinary text because it's so short.

--

'nuc'

nuclear norm

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--

inf

max(sum(abs(x), axis=1))

max(abs(x))

-inf

min(sum(abs(x), axis=1))

min(abs(x))

0

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--

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

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--

sum(abs(x)**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 and raise a ValueError when x.ndim != 2.

References

footnote

1

G. H. Golub and C. F. Van Loan, Matrix Computations,
Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15

Examples

>>> import flopscope.numpy as fnp
>>> from numpy import linalg as LA
>>> a = flops.arange(9) - 4
>>> a
array([-4, -3, -2, ...,  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, flops.inf)
4.0
>>> LA.norm(b, flops.inf)
9.0
>>> LA.norm(a, -flops.inf)
0.0
>>> LA.norm(b, -flops.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)
0.0
>>> LA.norm(b, -2)
1.8570331885190563e-016 # may vary
>>> LA.norm(a, 3)
5.8480354764257312 # may vary
>>> LA.norm(a, -3)
0.0

Using the axis argument to compute vector norms:

>>> c = flops.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 = flops.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)