I have some trouble solving a problem in my textbook:
Given the following function: $f(x) = x^{-1} - R$ Assume $R > 0$. Write a short algorithm to find $1/R$ by Newton's method applied to $f$. Do not use division or exponentiation in the algorithm. For positive $R$, what starting points are suitable?
OK, so I've managed to solve the first part of the problem. Using Newton's method and rearranging terms I have gotten:
$x_{n+1} = x_{n}(2 - Rx_{n})$
This is correct according to the book, and I can just use my standard algorithm for Newton's method with this expression. So far so good.
The problem is the second part, where I am supposed to find suitable starting points. I figured that if $x_{1} = -x_{0}$, then the iterations cycle. So then I get:
$\begin{align*} -x_{0} &= x_{0}(2 - Rx_{0})\\ -3x_{0} &= - Rx_{0}^2\\ -3 &= -Rx_{0}\\ x_{0} &= 3/R \end{align*}$
Thus my answer would be that we must have $x_{0} < 3/R$. My book, however, says:
If $R > 1$ then the starting point $x_{0}$ should be close to $0$ but smaller than $1/R$.
So what is wrong with my reasoning here? If anyone can help me out, I would really appreciate it!