An iterative method that iterates 
is said to converge with order 
 with 
, if there exists a positive constant 
 such that
Newton's method, when started with a good initial value, has order 
, and for certain problems, even 
. Newton's method when applied to the problem 
 gives the following iteration scheme:
Which converges cubically; that is, q = 3.
This implies that the number of correct digits triples from iteration to iteration. To demonstrate cubic convergence and to numerically determine the constant 
is hardly possible with the standard 16-digit float data type.
The following code uses SymPy together with high-precision evaluation instead and takes the study on cubic convergence to the extreme:
import sympy as sym
x = sym.Rational(1,2)
xns=[x]
for i in range(1,9):
x = (x - sym.atan(x)*(1+x**2)).evalf(3000)
xns.append(x)
The result...
