How can I use fixed point iteration for $2x^3-4x^2+x+1=0$ to find the negative root?
The worked solution to the question uses the following iterative formula, it apparently finds a solution at -0.3660 (4sf) but I am unable to retrieve this result:
$$x_{r+1}=\sqrt{\frac{2x_r^3+x_r+1}{4}}$$
I can find positive roots such as 1 and 1.366(4sf) [link], but I am unable to find the negative root for the function using fixed point iteration. I have tried $x_0=-0.5$ which eventually finds the root at $1$. $x_0=-1$ and below seems to result in the square root of a negative number that does not have a real solution.
The (real) graph of $\sqrt{\frac{2x^3+x+1}4}$ does not intersect that of $x$ when $x<0$, but that of $-\sqrt{\frac{2x^3+x+1}4}$ (the negative) does. Try iterating with this negated function from $x=-0.5$.