# -*- coding: utf-8 -*-
"""
Created on Thu Dec 11 15:58:17 2014

@author: grivet

listing 4.1 : Construction du polyôme d'interpolation de Lagrange

Le module 'scipy.interpolate' comporte un programme 'lagrange' qui
forme le polynôme d'interpolation
"""
import numpy as np
import matplotlib.pyplot as plt

npiv = 4
npt = 200

xpiv = np.array([0.5,-0.5,1.5,2.0])
fpiv = np.exp(xpiv)
Lagrange = np.zeros(npt)
L = np.zeros([npiv,npt],float)
x = np.linspace(-2,3,npt)
for i in range(npiv):
    L[i,:] = np.ones(npt)
    for k in range(npiv):
        if k != i:
            L[i,:] = L[i,:]*(x-xpiv[k])/(xpiv[i]-xpiv[k])
    L[i,:] *= fpiv[i]
    plt.plot(x,L[i,:])
    Lagrange = Lagrange + L[i,:]
plt.plot(xpiv,fpiv,'ok')
plt.plot(x,np.exp(x),'--k')
plt.plot(x,Lagrange,'-k')
plt.ylim(-5.0,11.0)
plt.title("polynômes élémentaires et polynôme d'interpolation")
plt.xlabel("$x$")