# coding: utf-8
# # Using Richardson Extrapolation with Finite Differences
# In[1]:
from math import sin, cos
# Here are a function and its derivative. We also choose a "center" about which we carry out our experiments:
# In[2]:
f = sin
df = cos
x = 2.3
# We then compare the accuracy of:
#
# * First-order (right) differences
# * First-order (right) differences with half the step size
# * An estimate based on these two using Richardson extrapolation
#
# against `true`, the actual derivative
# In[3]:
for k in range(3, 10):
h = 2**(-k)
h1 = 2*h
fd1 = (f(x+h1) - f(x))/(h1)
h2 = h
fd2 = (f(x+h2) - f(x))/h2
p = 1
alpha = - h2**p / (h1**p - h2**p)
beta = 1 - alpha
richardson = alpha*fd1 + beta*fd2
true = df(x)
print("Err FD1: %g\tErr FD: %g\tErr Rich: %g" % (
abs(true-fd1),
abs(true-fd2),
abs(true-richardson)))
# In[4]:
3.39995e-06 / 8.48602e-07