# coding: utf-8
# # Vector Norms
# $p$-norms can be computed in two different ways in numpy:
# In[1]:
import numpy as np
import numpy.linalg as la
# In[2]:
x = np.array([1.,2,3])
# First, let's compute the 2-norm by hand:
# In[20]:
np.sum(x**2)**(1/2)
# Next, let's use `numpy` machinery to compute it:
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la.norm(x, 2)
# Both of the values above represent the 2-norm: $\|x\|_2$.
# --------------
# Different values of $p$ work similarly:
# In[3]:
np.sum(np.abs(x)**5)**(1/5)
# In[23]:
la.norm(x, 5)
# ---------------------
#
# 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 $p=100$:
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x**100
# In[25]:
np.sum(x**100)
# Compare to last value in vector: the addition has essentially taken the maximum:
# In[26]:
np.sum(x**100)**(1/100)
# Numpy can compute that, too:
# In[27]:
la.norm(x, np.inf)