I am looking at a relatively simple problem to reiterate:
$x^4=e^x$
I've found 5 different possible forms
1: $x_{r+1}=\frac{e^x}{x^3}$
2: $x_{r+1}=(\frac{e^x}{x^2})^{0.5}$
3: $x_{r+1}=(\frac{e^x}{x})^{1/3}$
4: $x_{r+1}=e^{x/4}$
5: $x_{r+1}=4Ln(x)$
The two roots are: $x_1=-0.815553, x_2=1.429612$
There are two possible roots that I've found graphically, and equations 2 and 4 both converge to $x_2$ but not $x_1$, and the other three doesn't converge to any. I've done my analytics and used $-1<F'(x)<1$ to verify which of them will actually converge. The question I have is how do I find an iterative process for $x_1$, and is it even possible?
I'm actually facing the same problem with other iterations I'm doing. So any help here will be useful.
Try iterating $(e^x-x)/(x^3-1)$. I think if you start close enough to the negative solution, the iteration will converge to it.
For example, taking the starting point $x=-1$, we produce the sequence, -1, -.6839, -.9005, -.7553, -.8562, -.7870, -.8351, -.8019, -.8250, -.8090, -.8201, -.8124, $\dots$ seems to converge slowly but surely.