Vector Norms

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

In [18]: 
import numpy as np
import numpy.linalg as la
In [19]: 
x = np.array([1.,2,3])
In [20]: 
np.sum(x**2)**(1/2)
Out[20]:
3.7416573867739413
In [21]: 
la.norm(x, 2)
Out[21]:
3.7416573867739413

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

Different values of $p$ work similarly:

In [22]: 
np.sum(np.abs(x)**5)**(1/5)
Out[22]:
3.0773848853940629
In [23]: 
la.norm(x, 5)
Out[23]:
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 [24]: 
x**100
Out[24]:
array([  1.00000000e+00,   1.26765060e+30,   5.15377521e+47])
In [25]: 
np.sum(x**100)
Out[25]:
5.1537752073201132e+47

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

In [26]: 
np.sum(x**100)**(1/100)
Out[26]:
3.0

Numpy can compute that, too:

In [27]: 
la.norm(x, np.inf)
Out[27]:
3.0