# coding: utf-8

# # Interpolation with Generalized Vandermonde Matrices

# In[1]:

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


# In[54]:

if True:
    def f(x):
        return np.exp(1.5*x)
    def df(x):
        return 1.5*np.exp(1.5*x)
else:
    def f(x):
        return np.sin(4*x)
    def df(x):
        return 4*np.cos(4*x)


# In[55]:

x = np.linspace(0, 1, 1000)
pt.plot(x, f(x))


# Fix some parameters:

# In[80]:

degree = 2
h = 1

nodes = np.linspace(0, h, degree+1)
nodes


# Build the Vandermonde matrix:

# In[81]:

V = np.array([
    nodes**i
    for i in range(degree+1)
]).T


# In[82]:

V


# Now find the interpolation coefficients as `coeffs`:

# In[83]:

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


# In[84]:

interp = 0*x
for i in range(degree+1):
    interp += coeffs[i] * x**i


# In[85]:

pt.plot(x, f(x), "--", color="gray", label="$f$")
pt.plot(x, interp, color="red", label="Interpolant")
pt.plot(nodes, f(nodes), "o")
pt.legend(loc="best")


# Now evaluate the derivative as `interp_deriv`:

# In[86]:

interp_deriv = 0*x
for i in range(1, degree+1):
    interp_deriv += coeffs[i] * i * x**(i-1)


# In[87]:

pt.plot(x, interp_deriv, color="red", label="Interpolant")
pt.plot(x, df(x), "--", color="gray", label="Derivative")
pt.legend(loc="best")


# In[38]: