Why are the forward error not computable?

Question

I am quite new to Numerical Analysis and I was wondering if someone could guide me into understanding this concept presented below.

Thank you.

Answer 1

Consider the problem of computing $y = f(x)$ where $f : \mathbb{R} \rightarrow \mathbb{R}$. Let $\hat{y}$ denote the computed value, i.e., the value returned by the computer. The forward absolute error $e$ is the difference between the target value and the computed value, i.e., $$e = y - \hat{y}.$$ If $y \not = 0$, then the forward relative error $r$ is defined and is given by $$r = \frac{e}{y} = \frac{y-\hat{y}}{y}.$$ Some texts are only interested in the absolute value of the errors, but from time to time it is useful to know if the computed value is too large or too small. Regardless, we cannot apply the definition and compute $e$ and $r$ unless we know the exact value of $y$.

The backward error is interesting because it allows us to bound the forward error using the relevant condition number. The connection between the relative backward error, the relative forward error and the relative condition number is discussed in this answer to a related question.

It is possible to say substantially more about this as there are several different kinds of condition numbers and the abstract definition of the backward error is not included in the reference. These could be topics for other questions.