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

# # Newton method

# Consider the function
# $$
# f(x) = \exp(x) - \frac{3}{2} - \arctan(x)
# $$

# In[1]:


get_ipython().run_line_magic('config', "InlineBackend.figure_format = 'svg'")
from numpy import linspace,abs
from sympy import Symbol,exp,atan,diff,lambdify
from matplotlib.pyplot import figure,subplot,plot,grid,title


# We define the function using sympy and automatically compute its derivative. Then we make functions out of these.

# In[2]:


x = Symbol('x')

# Define the function
f = exp(x)-3.0/2.0-atan(x)

# Compute its derivative
df = diff(f,x)
print("f = ", f)
print("df= ", df)

# Make functions
F = lambdify(x, f, modules=['numpy'])
DF = lambdify(x, df, modules=['numpy'])

# Plot the functions
X = linspace(-1,1,100)
figure(figsize=(9,4))
subplot(1,2,1)
plot(X,F(X))
grid(True)
title('Function')
subplot(1,2,2)
plot(X,DF(X))
grid(True)
title('Derivative');


# Now we implement the Newton method. We have to specify the maximum number of iterations, an initial guess and a tolerance to decide when to stop.

# In[3]:


M   = 20    # maximum iterations
x   = 0.5   # initial guess
eps = 1e-14 # relative tolerance on root

f = F(x)
for i in range(M):
    df = DF(x)
    dx = -f/df
    x  = x + dx
    e  = abs(dx)
    f  = F(x)
    print("%6d %22.14e %22.14e %22.14e" % (i,x,e,abs(f)))
    if e < eps * abs(x):
        break


# ## Exercise
# 
# Modify the above Newton method implemention to use `while` loop.