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")
Out[4]:
[<matplotlib.lines.Line2D at 0x7ffad2e9f370>]

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")
Out[5]:
[<matplotlib.lines.Line2D at 0x7ffad154d6c0>]

Observe Spacing

In [6]:
pt.plot(nodes, 0*nodes, "o")
Out[6]:
[<matplotlib.lines.Line2D at 0x7ffad15ba230>]

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)

else:

    # 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, np.dot(Vfull, coeffs))

pt.plot(nodes, data, "o")
Out[16]:
[<matplotlib.lines.Line2D at 0x7ffad13908b0>]

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)))



la.cond(V)
Out[23]:
1.4142135623730963

Part V: Error Result

Plot the product term from the estimate of truncation error in interpolation for the Chebyshev nodes:

$$\left|\prod_{i=1}^n (x-x_i)\right| $$
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")



pt.legend()
Out[54]:
<matplotlib.legend.Legend at 0x7ffad052d6f0>
In [ ]: