16.5.1 Example: A study on the convergence order of Newton's method

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...