Legendre_Chebyshev.py

import numpy as np 
import matplotlib.pyplot as plt 
import sympy as sb

def Chebyshev(N):
    P = [0]*(N+1)
    P[0] = 1
    P[1] = x
    for n in range(1,N):
        P[n+1] = 2 * x*P[n] - P[n-1] 
        P[n+1] = sb.expand(P[n+1])

    p = [0]*(N+1)
    for n in range(N+1):
        p[n] = sb.lambdify(x,P[n])

    return P,p

def Legendre(N):
    P = [0]*(N+1)
    P[0] = 1
    P[1] = x

    for n in range(1,N):
        P[n+1] = ( (2*n+1)*x*P[n] - n*P[n-1]) /(n+1) 
        P[n+1] = sb.expand(P[n+1])

    p = [0]*(N+1)
    for n in range(N+1):
        p[n] = sb.lambdify(x,P[n])

    return P,p
    
x = sb.Symbol('x')

N = 6
P1,p1 = Chebyshev(N)
P2,p2 = Legendre(N)

xx = np.linspace(-1,1,1000) 

YY1 = []
YY2 = []
for n in range(N):
    yy = p1[n](xx)
    YY1.append(yy)

    yy = p2[n](xx)
    YY2.append(yy)

plt.close('all')
plt.figure(figsize=(8,8))
plt.subplot(2,1,1)
for n in range(N):
    if isinstance(YY1[n],np.ndarray) is False:
        YY1[n] = YY1[n]*np.ones_like(xx)
    plt.plot(xx,YY1[n],label='$T_%d(x) = %s$' %(n,sb.latex(P1[n])))

plt.title('Chebyshev Polynomials (1st)',fontsize=12)

plt.subplot(2,1,2)
for n in range(N):
    if isinstance(YY2[n],np.ndarray) is False:
        YY2[n] = YY2[n]*np.ones_like(xx)
    plt.plot(xx,YY2[n],label='$P_%d(x) = %s$' %(n,sb.latex(P2[n])))

plt.title('Legendre Polynomials',fontsize=12)

axes = plt.gcf().get_axes()
for axis in axes:
    plt.sca(axis)
    plt.grid()
    plt.xlabel('$x$')
    plt.ylabel('$y$')
    plt.ylim([-1.2,1.2])
    plt.legend(fontsize=12,bbox_to_anchor=(1.04,0.8), loc="upper left")
plt.tight_layout()
plt.savefig('ChebyshevPolynomials2.png')