Absolute error versus relative error — more suitable?

Question

I understand the differences between relative and absolute error. I can think of examples when relative error is a more suitable error measure but not when absolute error is more suitable than relative error.

So when is absolute error a more suitable measure of error than relative error?

Answer 1

I will give some examples.

1. When computing the range $r = r(\theta)$ of an artillery shell corresponding to the elevation $\theta$, the error $E$ is the difference between the true range $r$ and the computed approximation $\hat{r}$, i.e., $E = r - \hat{r}$. The target will be destroyed if the error is bounded by the kill radius $\rho$ of the exploding shell, i.e. $|E| < \rho$. The relative error given by $R = E/r$ is irrelevant.
2. When computing the elevation $\theta$ necessary to hit a target at the distance $r$, the error $E$ is the difference between the true elevation $\theta$ and the computed approximation $\hat{\theta}$, i.e., $E = \theta - \hat{\theta}$. When the howitzer is aimed it is relevant than the error is less than the resolution afforded by the aiming mechanism. In particular, the relative error $R = E/\theta$ is irrelevant.
3. When cutting steel beams to a certain length the error is irrelevant. Only the relative error matters because the thermal expansion is always proportional to the length of the beam.
4. In chemistry it is vital to have the correct ratio between the reactants regardless of the scale of the reaction. This is straight forward if the concentrations are known with a small relative error.

There are cases where the sign of the error is vital.

5. When firing in support of your advancing infantry, you what the error $E = r - \hat{r}$ to be negative to avoid the possibility of killing your own troops.

6. When computing the strength $S$ of bridge you want to be certain that the computed strength $\hat{S}$ is less than the true strength and greater than the design specification $\tau$, i.e., $$\tau \leq \hat{S} < S.$$

7. Newton's method for the equation $f(x) = 0$ exhibits one sided convergence near a simple root. When the computed residual changes sign you have exhausted the machine's precision and further iterations are pointless.

8. When mailing gifts for your mother, you want to be sure they arrive before her birthday, not after.