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

# # Lebesgue Constant
# 
# Copyright (C) 2024 Andreas Kloeckner
# 
# <details>
# <summary>MIT License</summary>
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# THE SOFTWARE.
# </details>

# In[1]:


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


# In[2]:


n = 10
nodes = np.linspace(-1, 1, n)


# Create a Vandermonde matrix (as `vdm`) for monomial interpolation with `nodes`.

# In[3]:


vdm = np.array([
    nodes**i for i in range(n)
]).T


# Create a "fine" set of nodes (as `nodes_fine`) and a corresponding Vandermonde matrix with the monomials (as `vdm_fine`):

# In[4]:


nodes_fine = np.linspace(-1, 1, 10_000)

vdm_fine = np.array([
    nodes_fine**i for i in range(n)
]).T


# Generate 300 random functions with nodal data $\|\boldsymbol y\|_\infty \le 1$ and plot their interpolants on the fine grid:

# In[10]:


for _ in range(300):
    y = 2*np.random.rand(n) - 1
    coeffs = la.solve(vdm, y)
    yfine = vdm_fine @ coeffs
    plt.plot(nodes_fine, yfine)


# Systematically estimate the Lebesgue constant:

# In[11]:


la.norm(vdm_fine @ la.inv(vdm), np.inf)


# Plug in all-ones data, disregarding signs:

# In[12]:


interp = vdm_fine @ la.inv(vdm)
worst = np.sum(np.abs(interp), axis=1)
plt.plot(nodes_fine, worst)


# Now experiment:
# 
# - Increase `n`.
# - Can you come up with better `nodes`?

# In[46]:


n = 20
nodes = np.linspace(-1, 1, n)
plt.plot(nodes, "o")
nodes = np.sign(nodes)*np.abs(nodes)**0.5
plt.plot(nodes, "o")


# ### Chebyshev nodes
# 
# Extremal/second kind:

# In[7]:


k = n - 1
i = np.arange(0, k+1)

def f(x):
    return np.cos(k*np.arccos(x))

nodes = np.cos(i/k*np.pi)


# Roots/first kind:

# In[14]:


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)


# In[ ]: