# 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 [ ]:
f = sin
df = cos

x = 2.3


We then compare the accuracy of:

• First-order (right) differences
• Second-order (centered) differences
• An estimate based on these two using Richardson extrapolation

against true, the actual derivative

In [19]:
for k in range(3, 10):
h = 2**(-k)

fd1 = (f(x+2*h) - f(x))/(2*h)
fd2 = (f(x+h) - f(x))/h

richardson = (-1)*fd1 + 2*fd2

true = df(x)

print "Err FD1: %g\tErr FD2: %g\tErr Rich: %g" % (
abs(true-fd1),
abs(true-fd2),
abs(true-richardson))

Err FD1: 0.08581	Err FD2: 0.0448122	Err Rich: 0.00381441
Err FD1: 0.0448122	Err FD2: 0.022862	Err Rich: 0.000911846
Err FD1: 0.022862	Err FD2: 0.0115423	Err Rich: 0.000222501
Err FD1: 0.0115423	Err FD2: 0.00579859	Err Rich: 5.49282e-05
Err FD1: 0.00579859	Err FD2: 0.00290612	Err Rich: 1.3644e-05
Err FD1: 0.00290612	Err FD2: 0.00145476	Err Rich: 3.39995e-06
Err FD1: 0.00145476	Err FD2: 0.000727804	Err Rich: 8.48602e-07