# Vector Norms

$p$-norms can be computed in two different ways in numpy:

In :
import numpy as np
import numpy.linalg as la

In :
x = np.array([1.,2,3])


First, let's compute the 2-norm by hand:

In :
np.sum(x**2)**(1/2)

Out:
3.7416573867739413


Next, let's use numpy machinery to compute it:

In :
la.norm(x, 2)

Out:
3.7416573867739413


Both of the values above represent the 2-norm: $\|x\|_2$.

Different values of $p$ work similarly:

In :
np.sum(np.abs(x)**5)**(1/5)

Out:
3.0773848853940629

In :
la.norm(x, 5)

Out:
3.0773848853940629


The $\infty$ norm represents a special case, because it's actually (in some sense) the limit of $p$-norms as $p\to\infty$.

Recall that: $\|x\|_\infty = \max(|x_1|, |x_2|, |x_3|)$.

Where does that come from? Let's try with $p=100$:

In :
x**100

Out:
array([  1.00000000e+00,   1.26765060e+30,   5.15377521e+47])

In :
np.sum(x**100)

Out:
5.1537752073201132e+47


Compare to last value in vector: the addition has essentially taken the maximum:

In :
np.sum(x**100)**(1/100)

Out:
3.0


Numpy can compute that, too:

In :
la.norm(x, np.inf)

Out:
3.0