I just have a very brief question regarding the formula for error bounds in Newton's method. Depending on where you look, this will either be written as:
$e_{n+1} \approx \frac{f^{\prime \prime}(r)}{2 f^{\prime}(r)}e_{n}^2$
or:
$e_{n+1} \approx -\frac{f^{\prime \prime}(r)}{2 f^{\prime}(r)}e_{n}^2$
In other words, the sign preceeding the fraction term will differ. Why is this so? I have encountered both versions in different places, and I just wondered if it is arbitrary what you choose to use. After all, if you take the absolute value, the magnitude of the error bound will be the same, but if you leave the answer as it is, without taking the absolute value, then, naturally, one answer will be positive and another will be negative.
If anyone can clear this up for me, I would greatly appreciate it!