 Solution

Choose (iii), $|x_{n+1}-x_n|<2000ϵ$. Here the root has magnitude between $1000$ and $2000$. Machine epsilon measures relative precision near unity; the spacing of representable floating-point numbers near the root is therefore of order $10^3ϵ$. A step tolerance scaled to that magnitude is consistent with the attainable absolute positional accuracy. For IEEE binary64, using $ϵ=2^{-52}$, $2000ϵ\approx 4.44\times 10^{-13}$, comparable to the local spacing. A fixed numerical constant cannot promise an exact correctly rounded root, but this is the most sensible of the three proposals.

Condition (ii) asks for a step below $2ϵ$, a scale appropriate near one rather than near this root. A nonzero representable change near the root is normally much larger. The test may trigger only after exact stagnation, and rounding-induced oscillation can prevent that; it asks for unnecessarily fine absolute resolution.

Condition (i) measures a residual, not a position error. It changes if the same root problem is multiplied by a constant. Moreover, evaluating $f$ can involve cancellation among quantities of order $1000$, so an absolute residual of order $ϵ$ need not be meaningful or attainable. Here $1/2<f^{\prime }(x^{\ast })\le 1$, making residual and position errors comparable in exact arithmetic, but their requested absolute scale is still too small. A practical scale-aware stopping criterion for Newton iteration should also detect stagnation or a cycle rather than assuming exact arithmetic persists indefinitely.