# -*- coding: utf-8 -*-
"""
Created on Wed Jun 21 15:49:48 2017

Exercice 2-2 : Calcul du polynômes de Hermite par la relation 
de récurrence H_{n+1} = 2xH_n - 2nH_{n-1} 

@author: grivet
"""
import numpy as np
import matplotlib.pyplot as plt

def H(n,x):
   if n == 0:
      fn = 1
   if n == 1:
      fn = 2*x
   if n > 1:
      k = 1; Havder = 1; Hder = 2*x
      while k < n:
        H = 2*x*Hder - 2*k*Havder
        Havder = Hder
        Hder = H
        k = k+1
      fn = H;
   return fn
xmin = -2.0; xmax = 2.0
x = np.linspace(xmin,xmax,200)
fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(x,H(2,x),label = "n = 2")
ax.plot(x,H(3,x),label = "n = 3")
ax.plot(x,H(4,x),label = "n = 4")
ax.plot(x,H(5,x),label = "n = 5") 
ax.spines['top'].set_color('none')
ax.spines['left'].set_position('center')
ax.spines['right'].set_color('none')
ax.spines['bottom'].set_position('center')
ax.spines['left'].set_smart_bounds(True)
ax.spines['bottom'].set_smart_bounds(True)
plt.legend()
plt.xlabel("$x$"); plt.ylabel("$H_n(x)$");
plt.title("polynômes de Hermite, n = 2,3,4,5 ")