# coding: utf-8

# # Interpolation with Radial Basis Functions

# In[1]:

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


# In[2]:

plot_x = np.linspace(-3, 3, 200)


# In[3]:

np.random.seed(20)
centers = np.random.randn(10)*0.05 + np.linspace(-1.5, 1.5, 10)
centers = np.sort(centers)
centers


# In[4]:

radius = 0.3


# In[5]:

def radial_basis_function(x, i):
    return np.exp(-(x-centers[i])**2/radius**2)

pt.plot(plot_x, radial_basis_function(plot_x, 3))


# In[6]:

def f(x): return x**3 - 3*x

pt.plot(plot_x, f(plot_x))


# Let's build a Vandermonde matrix at the centers:

# In[7]:

nodes = centers

V = np.array([
    radial_basis_function(nodes, i)
    for i in range(len(centers))
    ]).T


# Find the coefficients:

# In[8]:

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


# Find the interpolant:

# In[9]:

interpolant = 0
for i in range(len(centers)):
    interpolant += coeffs[i] * radial_basis_function(plot_x, i)

pt.figure(figsize=(8,8))
pt.ylim([-5,5])
pt.plot(plot_x, interpolant, label="Interpolant")
pt.plot(plot_x, f(plot_x), label="$f$")
pt.plot(centers, f(centers), "o")
pt.legend(loc="best")


# * Play around with the radius of the RBFs
# * Play with node placement