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

# # Arnoldi Iteration
# 
# Copyright (C) 2020 Andreas Kloeckner
# 
# <details>
# <summary>MIT License</summary>
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# 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
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# The above copyright notice and this permission notice shall be included in
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# 
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# THE SOFTWARE.
# </details>

# In[1]:


import numpy as np
import numpy.linalg as la

import matplotlib.pyplot as pt


# Let us make a matrix with a defined set of eigenvalues and eigenvectors, given by `eigvals` and `eigvecs`.

# In[2]:


np.random.seed(40)

# Generate matrix with eigenvalues 1...25
n = 25
eigvals = np.linspace(1., n, n)
eigvecs = np.random.randn(n, n)
print(eigvals)

A = la.solve(eigvecs, np.dot(np.diag(eigvals), eigvecs))
print(la.eig(A)[0])


# ## Initialization

# Set up $Q$ and $H$:

# In[3]:


Q = np.zeros((n, n))
H = np.zeros((n, n))

k = 0


# Pick a starting vector, normalize it

# In[4]:


x0 = np.random.randn(n)
x0 = x0/la.norm(x0)

# Poke it into the first column of Q
Q[:, k] = x0

del x0


# Make a list to save arrays of Ritz values:

# In[5]:


ritz_values = []


# ## Algorithm

# Carry out one iteration of Arnoldi iteration.
# 
# Run this cell in-place (Ctrl-Enter) until H is filled.

# In[30]:


print(k)

u = A @ Q[:, k]

# Carry out Gram-Schmidt on u against Q
for j in range(k+1):
    qj = Q[:, j]
    H[j,k] = qj @ u
    u = u - H[j,k]*qj

if k+1 < n:
    H[k+1, k] = la.norm(u)
    Q[:, k+1] = u/H[k+1, k]

k += 1

pt.spy(H)

ritz_values.append(la.eig(H)[0])


# Check that $Q^T A Q =H$:

# In[31]:


la.norm(Q.T @ A @ Q - H)/ la.norm(A)


# Check that Q is orthogonal:

# In[32]:


la.norm(Q.T @ Q - np.eye(n))


# ## Plot convergence of Ritz values

# Enable the Ritz value collection above to make this work.

# In[33]:


for i, rv in enumerate(ritz_values):
    pt.plot([i] * len(rv), rv, "x")


# In[ ]: