Plotting the Chebyshev polynomials of the first kind in Python

The Chebyshev polynomials of the first kind are defined for $x \in [-1, 1]$ as follows

$$\ T_n(x) = \cos(n \cdot \arccos(x))$$

for $n=0,1,2,\dots$

See also:
Chebyshev polynomials of the first kind - MathWorld
Chebyshev polynomials - Wikipedia

In [1]:

import numpy as np
import matplotlib.pyplot as plt

In [2]:

x = np.linspace(-1,1, 2000)

In [3]:

M = 7
c = np.arange(0, M+1)
y = []
for N in c:
    y.append(np.cos(N * np.arccos(x)))

# Here y[n] is the n-th Chebyshev polynomial i.e. it is Tn(x)
# Tn(x) = cos(N * arccos(x))
# for x in the closed interval [-1, 1]

In [4]:

c

Out[4]:

array([0, 1, 2, 3, 4, 5, 6, 7])

In [5]:

fig, ax = plt.subplots(figsize=(11,7))

for i in c:
    ax.plot(x, y[i], linewidth=2.0, label='$T_' + str(i) + '(x)$')
    
ax.set(xlim=(-1.05, 1.5), 
       ylim=(-1.1, 1.1))

plt.grid(True, linestyle='--')
plt.legend(loc='lower right')
plt.show()

In [6]:

# Tn(x) takes values of +1 or -1 in these points
# (n>=1)
def extremum_points(N):
    m = np.arange(N, -1, -1) # N, N-1, N-2, ... , 2, 1, 0
    x = np.cos(m*np.pi / N)
    return x

# Tn(x) takes a value of 0 in these points
# (n>=1)
def root_points(N):
    m = np.arange(N, 0, -1) # N, N-1, N-2, ... , 2, 1
    x = np.cos((2*m-1)*np.pi / (2*N))
    return x

In [7]:

extremum_points(5)

Out[7]:

array([-1.        , -0.80901699, -0.30901699,  0.30901699,  0.80901699,
        1.        ])

In [8]:

root_points(5)

Out[8]:

array([-9.51056516e-01, -5.87785252e-01,  6.12323400e-17,  5.87785252e-01,
        9.51056516e-01])

In [9]:

N = 5
y = np.cos(N*np.arccos(extremum_points(N)))
y

Out[9]:

array([-1.,  1., -1.,  1., -1.,  1.])

In [10]:

N = 5
y = np.cos(N*np.arccos(root_points(N)))
y

Out[10]:

array([ 5.5109106e-16, -4.2862638e-16,  3.0616170e-16, -1.8369702e-16,
       -8.2694608e-16])

In [11]:

N = 10
y = np.cos(N*np.arccos(extremum_points(N)))
y

Out[11]:

array([ 1., -1.,  1., -1.,  1., -1.,  1., -1.,  1., -1.,  1.])

In [12]:

N = 10
y = np.cos(N*np.arccos(root_points(N)))
y

Out[12]:

array([-6.49248498e-15, -7.35407060e-16, -2.69484194e-15, -9.80336420e-16,
       -2.44991258e-15,  5.51091060e-16, -4.28626380e-16,  3.06161700e-16,
       -1.07187544e-15,  2.72576760e-15])