Chebyshev polynomials

Copyright (C) 2020 Andreas Kloeckner

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In [18]:
import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt

Part I: Plotting the Chebyshev polynomials

In [2]:
x = np.linspace(-1, 1, 100)

pt.xlim([-1.2, 1.2])
pt.ylim([-1.2, 1.2])

for k in range(10): # crank up
pt.plot(x, np.cos(k*np.arccos(x)))

Part II: Understanding the Nodes

What if we interpolate random data?

In [3]:
n = 50 # crank up

"Extremal" Chebyshev Nodes (or: Chebyshev Nodes of the Second Kind)

  • Most often used for computation
  • Note: Generates $n+1$ nodes -> drop $k$
In [4]:
k = n-1

i = np.arange(0, k+1)
x = np.linspace(-1, 1, 3000)

def f(x):
return np.cos(k*np.arccos(x))

nodes = np.cos(i/k*np.pi)

pt.plot(x, f(x))
pt.plot(nodes, f(nodes), "o")

Chebyshev Nodes of the First Kind (Roots)

  • Generates $n$ nodes
In [5]:
i = np.arange(1, n+1)
x = np.linspace(-1, 1, 3000)

def f(x):
return np.cos(n*np.arccos(x))

nodes = np.cos((2*i-1)/(2*n)*np.pi)

pt.plot(x, f(x))
pt.plot(nodes, f(nodes), "o")

Observe Spacing

In [6]:
pt.plot(nodes, 0*nodes, "o")

Part III: Chebyshev Interpolation

In [15]:
n = 100

i = np.arange(n, dtype=np.float64)
nodes = np.cos((2*(i+1)-1)/(2*n)*np.pi)

V = np.cos(i*np.arccos(nodes.reshape(-1, 1)))
if 1:
# random data
data = np.random.randn(n)
# Runge's example
data = 1/(1+25*nodes**2)

coeffs = la.solve(V, data)
In [16]:
x = np.linspace(-1, 1, 1000)
Vfull = np.cos(i*np.arccos(x.reshape(-1, 1)))
pt.plot(x,, coeffs))
pt.plot(nodes, data, "o")

Part IV: Conditioning

In [23]:
n = 10 # crank up

i = np.arange(n, dtype=np.float64)
nodes = np.cos((2*(i+1)-1)/(2*n)*np.pi)
V = np.cos(i*np.arccos(nodes.reshape(-1, 1)))


Part V: Error Result

Plot the product term from the estimate of truncation error in interpolation for the Chebyshev nodes: $$\left|\prod_{i=1}^n (x-x_i)\right| $$

In [45]:
def plot_err_prod(nodes, label):
eval_pts = np.linspace(-1, 1, 30000)

product = 1
for xi in nodes:
product = product*(eval_pts-xi)
pt.plot(eval_pts, np.abs(product), label=label)
In [54]:
n = 10 # crank up

i = np.arange(n, dtype=np.float64)
cheb_nodes = np.cos((2*(i+1)-1)/(2*n)*np.pi)
plot_err_prod(cheb_nodes, label="Chebyshev")

if 0:
nodes = np.linspace(-1, 1, n)
plot_err_prod(nodes, label="equispaced")
elif 0:
nodes = cheb_nodes.copy()
nodes[3] += 0.1
plot_err_prod(nodes, label="Perturbed")

In [ ]: