Which of the following iterative formulae could be used to solve the equation $x^3-2x^2-x+1=0$ using the fixed point iteration method?

Question

Which of the following iterative formulae could be used to solve the equation $x^3-2x^2-x+1=0$ using the fixed point iteration method? $$x=x^3-3x^2-1$$ $$x=\sqrt[3]{3x^2+x-1}$$ $$x=\pm\sqrt{\frac{x^3-x-1}{3}}$$ $$x=\frac{x+1}{x^2-3x}$$ I have tried rearranging and expanding out all of the formulae but none of them seem to result in $x^3-2x^2-x+1=0$.

Answer 1

When you are trying to find a solution of the equation $f(x)=0$ using the fixed point method, i.e., rewriting your equation as $g(x)=x$ and using the sequence $x_{n+1} = g(x_n)$, the fixed point theorem gives conditions for the convergence of this sequence $x_n$ to a solution of $f(x)=0$.

In particular, given enough regularity, if $|g'(\alpha)|<1$, where $\alpha$ is the solution, then, for $x_0$ sufficiently close to $\alpha$, we have that $x_n \to \alpha$.

Conversely, if $|g'(\alpha)|>1$, the fixed point iteration cannot converge, unless, by some miracle, $x_p = \alpha$, for some $p$.

If you test all the proposed iteration functions $g$, you'll see that:

1. The first function does not work for any of the roots.
2. The second function works for the larger root, but not for the other two.
3. The third function does not work for any of the roots.
4. The forth function works for the smaller root but not for the other two.