Calculating $\frac{1}{a}$ using Newton-Raphson method I have a computer that doesn't implement division operation (it has only addition, substraction and multiplication). I need to find a method to find the approximate value of $\frac{1}{a}$, where $a\in \mathbb R \setminus\{0\}$. I'm supposed to do that with Newton-Raphson method ($x_{k+1}=x_k-\frac{f(x_k)}{f'(x_k)}$) and there mustn't be any division operation in the final formula.
So far I have tried obvious $f(x)=ax-1$, but then:
$$
x_{k+1}=x_k-\frac{a\cdot x_k-1}{a}=\frac{1}{a}
$$
which obviously haven't brought me any closer to the answer. Do you have any ideas what $f$ function should I take to solve this?

We take function: $f(x)=a-\frac{1}{x}$.
$$
x_{k+1}=x_k-\frac{a-\frac{1}{x_k}}{\frac{1}{x_k^2}}=\frac{\frac{1}{x_k}-a+\frac{1}{x_k}}{\frac{1}{x_k^2}}=2x_k-a\cdot x_k^2
$$
