# coding: utf-8

# # Finite Differences

# In[1]:

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


# In[29]:

degree = 2
h = 0.25

# Assume even degree so that there's a well-defined middle node.
assert degree % 2 == 0

nodes = np.linspace(-h/2, h/2, degree+1)
nodes


# Now construct `V` (the generalized Vandermonde) and `Vprime` (the generalized Vandermonde for the derivatives):

# In[30]:

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


# In[31]:

def monomial_deriv(i, x):
    if i == 0:
        return 0*x
    else:
        return i*nodes**(i-1)

Vprime = np.array([
    monomial_deriv(i, nodes)
    for i in range(degree+1)
]).T


# Combine them to form the derivative matrix:

# In[32]:

diff_mat = Vprime.dot(la.inv(V))


# Let's say we only care about the derivative at the middle node:

# In[33]:

finite_difference_weights = diff_mat[degree//2]
finite_difference_weights


# * What have we learned?
# * What formula does this amount to?
# * How do these weights scale with $h$?
# * What formula does this amount to, really?
# * What happens if we change the degree?
# * What happens if we shift all nodes?
# * What happens if we shift only one of the nodes?
# * How do we apply this formula?

# In[36]:

# * We could have left the middle point out. :)
# * -4*f(x-0.25) + 4*f(x+0.25)
# * They scale with 1/h, as you might expect.
# * (f(x-h/2) + f(x+h/2))/h
# * We get a more complicated (but more accurate) formula.
# * Nothing.
# * We get a different formula (that's valid for those nodes).
# * See below.


# In[55]:

def f(x):
    return np.sin(4*x)
def df(x):
    return 4*np.cos(4*x)


# In[56]:

x = np.arange(10) * 0.125
pt.plot(x, f(x), "o-")


# Now use the weights to compute the finite difference derivative as `deriv`:

# In[57]:

fdw = finite_difference_weights

fx = f(x)
deriv = np.zeros(len(x)-2)
for i in range(1, 1+len(deriv)):
    deriv[i-1] = fx[i-1]*fdw[0] + fx[i]*fdw[1] + fx[i+1]*fdw[2]


# Now plot the finite difference derivative:

# In[59]:

pt.plot(x[1:-1], df(x[1:-1]), label="true")
pt.plot(x[1:-1], deriv, label="FD")
pt.legend()