# coding: utf-8

# # Jump with Chebyshev Nodes

# In[23]:

import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt
import scipy.special as sps


# In[32]:

n = 50

k = np.arange(1, n+1, dtype=np.float64)

cheb_nodes = np.cos((2*k-1)/(2*n)*np.pi)
pt.plot(cheb_nodes, 0*cheb_nodes, "o")


# Build the Vandermonde matrix for orthogonal polynomials with Chebyshev nodes:

# In[35]:

V = np.array([
    sps.eval_legendre(i, cheb_nodes)
    for i in range(n)
]).T

la.cond(V)


# Notice the condition number of the Vandermonde matrix! How does that compare to our prior ones?

# In[36]:

def f(x):
    return (x>=0).astype(np.float64)


# In[37]:

coeffs = la.solve(V, f(cheb_nodes))


# In[38]:

x = np.linspace(-1, 1, 1000)


# In[39]:

interpolant = 0
for i in range(n):
    interpolant += coeffs[i]*sps.eval_legendre(i, x)


# In[41]:

pt.plot(x, interpolant)
pt.plot(x, f(x), "--", color="gray")


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