Given the simple cubic function $$f(x) = x^3 - 1 = 0,$$ You can use the Newton Raphson method to solve and approximate one solution to the function like so:
| n | $x_n$ | Absolute Error |
|---|---|---|
| 1 | 1.41667 | 0.41667 |
| 2 | 1.11053 | 0.11053 |
| 3 | 1.01064 | 0.01064 |
| 4 | 1.00011 | 0.00011 |
| 5 | 1.000000012 | 0.000000012 |
Given this method to solving such a function, I've been asked to show how 'these data suggest a quadratic convergence of Newton’s iteration and demonstrate this result by treating Newton’s method as a fixed-point iteration.'
This has me quite stumped. How can I show a quadratic convergence by using these figures? And how to do so using the fixed point method? Any help would be appreciated.