9.2. Exercises on Fixed Point Iteration

Exercise 1

The equation \(x^3 -2x + 1 = 0\) can be written as a fixed point equation in many ways, including

1. \(\displaystyle x = \frac{x^3 + 1}{2}\)
   and

2. \(x = \sqrt[3]{2x-1}\)

For each of these options:

(a) Verify that its fixed points do in fact solve the above cubic equation.

(b) Determine whether fixed point iteration with it will converge to the solution \(r=1\). (assuming a “good enough” initial approximation).

Note: computational experiments can be a useful start, but prove your answers mathematically!