Chebyshev polynomials

Copyright (C) 2020 Andreas Kloeckner

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In [2]:
import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt

Part I: Plotting the Chebyshev polynomials

In [3]:
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 [ ]:
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 [19]:
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[19]:
[<matplotlib.lines.Line2D at 0x7fb75057fe80>]

Chebyshev Nodes of the First Kind (Roots)

  • Generates $n$ nodes
In [18]:
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[18]:
[<matplotlib.lines.Line2D at 0x7fb75061a668>]

Observe Spacing

In [20]:
pt.plot(nodes, 0*nodes, "o")
Out[20]:
[<matplotlib.lines.Line2D at 0x7fb750520470>]

Part III: Chebyshev Interpolation

In [21]:
V = np.cos(i*np.arccos(nodes.reshape(-1, 1)))
data = np.random.randn(n)
coeffs = la.solve(V, data)
In [22]:
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[22]:
[<matplotlib.lines.Line2D at 0x7fb75066c1d0>]

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
In [ ]: