The equation $f(x)=x^3+x-1=0$ admites a unique solution $\alpha\in[0;1]$. I want to approximate the solution $\alpha$ using fixed-point method, for that, sitting $x=g(x)=x+wf(x)$ an equivalent of $f(x)=0$. So we get for $g(0), g(1)\in[0;1]$, $w\in[-1;0]$.
My question: $w$ belongs to which interval in order to apply the fixed-point method.
You can visualize this iteration using phase plots for the iteration $x_{k+1}=g(x_k)$.
The gallery shows behavior from periodic cycles and then convergence to the fixed point, the root of $f$, with fastest convergence around $w=-0.6$. You can quantify this by considering the contraction factor $$q=\max_{x\in[0,1]}|g'(x)|=\max|1-wf'(x)|.$$ Then if $|2g(0.5)-1|=|2wf(0.5)|<1-q$ one has also $$g([0,1])\subset[g(0.5)-0.5q,\, g(0.5)+0.5q]\subset[0,1]$$ by the mean value theorem, and thus a fixed point.