The goal is to solve the root finding problem using fixed point equations.
I am somewhat stuck here. I have derived several forms of the following equation: $$ f(x) = x^3 - 2x + 1 $$ The forms I was meant to derive include:
$g(x) = 2/x - 1/x^2$ which I was able to simplify to $(2x - 1)/x^2$
$g(x) = \sqrt{2 - 1/x}$
$g(x) = -(1-2x)^{1/3}$
The initial point $p_o$ is $0.5$. However, I cannot seem to get the equations to iterate to show convergence as no matter what I have tried for manipulations has resulted in the same equation. I have multiplied by conjugates and attempted to remove denominators. Trying the simplify function in wolfram alpha was of no help. I am supposed to derive the fixed point equations above and then determine if they converge. I am not a terrific mathematician but thought I could at least manipulate the above equations. Am I missing or not understanding something? How can I manipulate the equations to find roots or divergence instead of $0$ and then an undefined result?
My first attempt was on a simpler problem and worked easily so I think I understand fixed point iteration to a certain extent at least.
Any help is appreciated. Thanks.
Perhaps this comment is not very pertinent but, as I understand, the fixed point theorem is relevant in the uniqueness. But your function $f(x)=x^3-2x+1$ admits three fixed points. You have (approximately) $$f(x)=x$$ for the three values $$x\approx -1.879\\x\approx 0.347\\x\approx 1.532$$