# coding: utf-8
# # Finite Differences
# In[1]:
import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt
# ### Part 1: Examining the Differentiation Matrix
# In[2]:
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[3]:
V = np.array([
nodes**i
for i in range(degree+1)
]).T
# In[4]:
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[6]:
diff_mat = Vprime.dot(la.inv(V))
diff_mat
# 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 change if we change $h$?
# * What formula does this amount to, really?
# * What happens if we change the degree?
# * What happens if we shift all nodes?
# In[10]:
# * 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 (with more source nodes)
# * The weights remain the same.
# ### Part 2: Using finite difference formulas
# 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()