#!/usr/bin/env python
# coding: utf-8

# # Matrix norms
# 
# Copyright (C) 2020 Andreas Kloeckner
# 
# <details>
# <summary>MIT License</summary>
# Permission is hereby granted, free of charge, to any person obtaining a copy
# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
# copies of the Software, and to permit persons to whom the Software is
# furnished to do so, subject to the following conditions:
# 
# The above copyright notice and this permission notice shall be included in
# all copies or substantial portions of the Software.
# 
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
# THE SOFTWARE.
# </details>

# In[1]:


import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt


# Here's a matrix of which we're trying to compute the norm:

# In[14]:


n = 2
A = np.random.randn(n, n)
A


# Recall:
# 
# $$||A||=\max_{\|x\|=1} \|Ax\|,$$
# 
# where the vector norm must be specified, and the value of the matrix norm $\|A\|$ depends on the choice of vector norm.
# 
# For instance, for the $p$-norms, we often write:
# 
# $$||A||_2=\max_{\|x\|=1} \|Ax\|_2,$$
# 
# and similarly for different values of $p$.

# --------------------
# We can approximate this by just producing very many random vectors and evaluating the formula:

# In[3]:


xs = np.random.randn(n, 1000)


# First, we need to bring all those vectors to have norm 1. First, compute the norms:

# In[4]:


p = 2
norm_xs = np.sum(np.abs(xs)**p, axis=0)**(1/p)
norm_xs.shape


# Then, divide by the norms and assign to `normalized_xs`:
# 
# Then check the norm of a randomly chosen vector.

# In[16]:


normalized_xs = xs/norm_xs
la.norm(normalized_xs[:, 316], p)


# Let's take a look:

# In[17]:


pt.plot(normalized_xs[0], normalized_xs[1], "o")
pt.gca().set_aspect("equal")


# Now apply $A$ to these normalized vectors:

# In[7]:


A_nxs = A.dot(normalized_xs)


# --------------
# Let's take a look again:

# In[8]:


pt.plot(normalized_xs[0], normalized_xs[1], "o", label="x")
pt.plot(A_nxs[0], A_nxs[1], "o", label="Ax")
pt.legend()
pt.gca().set_aspect("equal")


# Next, compute norms of the $Ax$ vectors:

# In[9]:


norm_Axs = np.sum(np.abs(A_nxs)**p, axis=0)**(1/p)
norm_Axs.shape


# What's the biggest one?

# In[10]:


np.max(norm_Axs)


# Compare that with what `numpy` thinks the matrix norm is:

# In[11]:


la.norm(A, p)


# In[ ]: