#!/usr/bin/env python
# coding: utf-8

# # Chebyshev polynomials

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

# ## Part I: Plotting the Chebyshev polynomials

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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?

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n = 50 # crank up

i = np.arange(n, dtype=np.float64)

# Chebyshev nodes:
#nodes = np.cos((2*(i+1)-1)/(2*n)*np.pi)

# Equispace nodes:
nodes = np.linspace(-1, 1, n)

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

# ## Part III: Chebyshev Interpolation

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V = np.cos(i*np.arccos(nodes.reshape(-1, 1)))
data = np.random.randn(n)
coeffs = la.solve(V, data)

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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")

# ## Part IV: Conditioning

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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)

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