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

# # Example of fixed point iteration

# Consider finding roots of
# $$
# f(x) = x^2 - a, \qquad a > 0
# $$
# using following fixed point maps
# \begin{eqnarray*}
# x &=& x + c(x^2 - a) \\
# x &=& \frac{a}{x} \\
# x &=& \frac{1}{2}\left( x + \frac{a}{x} \right)
# \end{eqnarray*}

# In[3]:


a = 3.0
c = 0.1
x1, x2, x3 = 2.0, 2.0, 2.0
print("%6d %18.10e %18.10e %18.10e" % (0,x1,x2,x3))
for i in range(10):
    x1 = x1 + c*(x1**2 - a)
    x2 = a/x2
    x3 = 0.5*(x3 + a/x3)
    print("%6d %18.10e %18.10e %18.10e" % (i+1,x1,x2,x3))


# The first iteration diverges, the second oscillates while the last (Newton method) converges.