Chebyshev polynomials

Copyright (C) 2020 Andreas Kloeckner

MIT License

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import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt

Part I: Plotting the Chebyshev polynomials

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?

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 $k$

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")

[<matplotlib.lines.Line2D at 0x7ffad2e9f370>]

Chebyshev Nodes of the First Kind (Roots)

• Generates $n$ nodes

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")

[<matplotlib.lines.Line2D at 0x7ffad154d6c0>]

Observe Spacing

pt.plot(nodes, 0*nodes, "o")

[<matplotlib.lines.Line2D at 0x7ffad15ba230>]

Part III: Chebyshev Interpolation

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)
else:
    # Runge's example
    data = 1/(1+25*nodes**2)
    
coeffs = la.solve(V, data)

x = np.linspace(-1, 1, 1000)
Vfull = np.cos(i*np.arccos(x.reshape(-1, 1)))
pt.plot(x, np.dot(Vfull, coeffs))
pt.plot(nodes, data, "o")

[<matplotlib.lines.Line2D at 0x7ffad13908b0>]

Part IV: Conditioning

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)))

la.cond(V)

1.4142135623730963

Part V: Error Result

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

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)

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")

pt.legend()

<matplotlib.legend.Legend at 0x7ffad052d6f0>