import numpy as np
import matplotlib.pyplot as plt
import pylab
from math import pi, cos, sin

n = 8
x = np.linspace(-1,1,100)
x_nodes = np.linspace(-1,1,n)
x_nodes_ch = [cos(i*pi/n) for i in range(n+1)]

def l_i(i,x,x_nodes):
    li=1
    for j in range(len(x_nodes)):
        if j != i:
            li=li*(x-x_nodes[j])/(x_nodes[i]-x_nodes[j])
    return li

def L(x,x_nodes,y_nodes):
    Lx=0
    for i in range(len(x_nodes)):
        Lx += y_nodes[i]*l_i(i,x,x_nodes)
    return Lx


def f(x):
    return 1/(1+25*x**2)

y_nodes_ch = [f(k) for k in x_nodes_ch]
y_nodes = [f(k) for k in x_nodes]


fig, axes = plt.subplots(nrows=1, ncols=2, figsize=(10, 5))
axes[0].set_xlabel("$x$")
axes[0].set_ylabel("$y$")
axes[0].plot(x,f(x),label="f(x)")
axes[0].plot(x,L(x,x_nodes, y_nodes))
axes[0].plot(x_nodes, y_nodes, 'go', c='pink')
axes[0].grid()

axes[1].set_xlabel("$x$")
axes[1].set_ylabel("$y$")
axes[1].plot(x,f(x),label="f(x)")
axes[1].plot(x,L(x,x_nodes_ch, y_nodes_ch))
axes[1].plot(x_nodes_ch, y_nodes_ch, 'go', c='r')
axes[1].grid()
plt.show()