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

# # Normal Equations vs Pseudoinverse
# 
# 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[3]:


import numpy as np
import numpy.linalg as la


# Here's a simple overdetermined linear system, which we'll solve using both the normal equations and the pseudoinverse:

# In[4]:


A = np.random.randn(5, 3)
b = np.random.randn(5)


# ### Normal Equations
# 
# Solve $Ax\cong b$ using the normal equations:

# In[5]:


x1 = la.solve(A.T@A, A.T@b)
x1


# ### Pseudoinverse
# 
# Solve $Ax\cong b$ using the pseudoinverse:

# In[6]:


U, sigma, VT = la.svd(A)
print(U)
print(sigma)
print(VT)


# In[7]:


Sigma_inv = np.zeros_like(A.T)
Sigma_inv[:3,:3] = np.diag(1/sigma)
Sigma_inv


# In[10]:


pinv = VT.T @ Sigma_inv @ U.T
x2 = pinv @ b
x2


# In[9]:


la.norm(x1-x2)


# In[ ]: