import numpy as np
import matplotlib.pyplot as plt

# the function of which the fixed point is computed
def fix_fun(x):
    return np.exp(x)/3

# number of steps used in the fixed point iteration
num_plot_steps = 21

# initial value of the iteration
plt_start_val = 1.5

# create the empty figure and scale the axes correctly
# (axes might have to be scaled differently for different functions)
plt.figure(1)
x_min = -0.2
x_max = 2.5
x_step = 0.01
x_vals = np.arange(x_min,x_max,x_step)
plt.axis([x_min,x_max,x_min,max(x_max,fix_fun(x_max))])
plt.axhline(color='k')
plt.axvline(color='k')
plt.grid(True)

# plot the fixed point function...
plt.plot(x_vals,fix_fun(x_vals),'b-')
# ...and a straight line
plt.plot(x_vals,x_vals,'k-')

# plotting...
x = plt_start_val
y = fix_fun(x)
plt.plot(x,y,'r',marker='8')
for k in np.arange(1,num_plot_steps):
    plt.plot([x,y],[y,y],'r',marker='8')
    x = y
    y = fix_fun(x)
    plt.plot([x,x],[x,y],'r',marker='8')

plt.show()