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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!

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