import numpy as np
import matplotlib.pyplot as plt

start = 1 # intitial value
iterations = 10 # number of iterations

print(start)

graphmin = 0
graphmax = 4

def f(x): # model
    return x - (np.tan(x/4)-1)*(4*np.cos(x/4)**2)

xv = np.linspace(0.0,10.0,2*iterations+1) # create array for points xvalue (initial values do not matter)
yv = np.linspace(0.0,10.0,2*iterations+1) # create array for points yvalue (initial values do not matter)

def plot_graphical(x0,n):
    
    x = x0
    xv[0] = x0
    yv[0] = 0
    for i in range(0,n): #iterate
        xv[2*i+1] = x # first point is (x,f(x))
        x = f(x)
        yv[2*i+1] = x
        xv[2*i+2] = x # second point is (f(x),f(x))
        yv[2*i+2] = x
        print(x) # print sequence (optional)
    plt.plot(xv,yv) # connect up all these points blue

plt.xlabel('x')
plt.ylabel('f(x)')
plot_graphical(start,iterations)
xcon = np.arange(graphmin,graphmax, 0.01)
plt.plot(xcon,xcon, 'g') # y = x plotted green
ycon = f(xcon)
plt.plot(xcon,ycon, 'r') # function plotted red

plt.show()