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

# # Jump with Chebyshev Nodes

# In[1]:


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

# In[2]:


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[3]:


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[4]:


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

# In[5]:


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

# In[6]:


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

# In[7]:


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

# In[8]:


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

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