Chebyshev1st.py

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

x = sb.Symbol('x')

def Chebyshev1st(N):
    T = [0]*(N+1)
    T[0] = 1
    T[1] = x
    for n in range(1,N):
        T[n+1] = 2 * x*T[n] - T[n-1] 
        T[n+1] = sb.expand(T[n+1])
    return T

N = 5
T = Chebyshev1st(N)

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

YY = []
for n in range(N):
    f = T[n]
    f = sb.lambdify(x,f)
    yy = f(xx)
    YY.append(yy)

plt.close('all')
plt.figure(figsize=(8,4))
for n in range(N):
    if isinstance(YY[n],np.ndarray) is False:
        YY[n] = YY[n]*np.ones_like(xx)
    plt.plot(xx,YY[n],label='$T_%d(x) = %s$' %(n,sb.latex(T[n])))

plt.title('Chebyshev Polynomials (1st)',fontsize=12)
plt.grid()
plt.xlabel('x')
plt.ylabel('y')
plt.legend(fontsize=12,bbox_to_anchor=(1.04,0.8), loc="upper left")
plt.tight_layout()
plt.savefig('ChebyshevPolynomials_1st.png')

Chebyshev1st_2nd.py

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

x = sb.Symbol('x')

def Chebyshev1st(N):
    T = [0]*(N+1)
    T[0] = 1
    T[1] = x
    for n in range(1,N):
        T[n+1] = 2 * x*T[n] - T[n-1] 
        T[n+1] = sb.expand(T[n+1])
    return T

def Chebyshev2nd(N):
    T = [0]*(N+1)
    T[0] = 1
    T[1] = 2*x
    for n in range(1,N):
        T[n+1] = 2 * x*T[n] - T[n-1] 
        T[n+1] = sb.expand(T[n+1])
    return T

N = 5
T1 = Chebyshev1st(N)
T2 = Chebyshev2nd(N)

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

YY1 = []
YY2 = []
for n in range(N):
    f1 = T1[n]
    f1 = sb.lambdify(x,f1)
    yy = f1(xx)
    YY1.append(yy)

    f2 = T2[n]
    f2 = sb.lambdify(x,f2)
    yy = f2(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(T1[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='$U_%d(x) = %s$' %(n,sb.latex(T2[n])))

plt.title('Chebyshev Polynomials (2nd)',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')