# What if the relative error is undefined?

The relative error is defined by the simple formula:

$$\text{Rel. Error} = \frac{|v_\text{approx}-v_\text{analytical}|}{v_\text{analytical}}$$

but what if the theoretical value $v_\text{analytical}$ should be $0$? then our relative error is undefined.... this is also quite a common occurs. If our analytical function is $x^2$ then at its $x=0$ we have a problem.

I'm trying to program this on a computer. How do I make sure that I don't have any problems with this formula?

• I'd say that in such cases the relative error is not a good error measurement technique. Nov 3, 2017 at 13:56

You could argue that in that particular case, the relative error is not a good measure.

Note that usually the relative error is defined a the ratio of the absolute error and the absolute true value, i.e. $$\mathrm{Rel. Error} = \frac{ |v_{\mathrm{approx}} - v_{\mathrm{analytical}}| }{ |v_\mathrm{analytical}| }.$$

An alternative would be to use just the absolute error or to define the relative error as follows:

$$\mathrm{Rel. Error} = \frac{ |v_{\mathrm{approx}} - v_{\mathrm{analytical}}| }{ 1+|v_\mathrm{analytical}| }.$$

Relative error is generally only used for quantities that can only be non-negative.

Unfortunately, there is no good answer.

There is of course no relative error to a zero quantity and you need to use an absolute error. Then two embarrasing questions:

• what absolute error can you choose ?

• when do you switch from relative to absolute ?

These are application-dependent and can't be "hard-coded" in a program.