Why is a relative forward error enough for computations?

Question

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

Answer 1

Let $T$ denote the target value for our computation and let $A$ denote the computed approximation.

Our objective should always be to produce accurate approximations $A$ as well as an upper bound for the error $E = T- A$ or a reliable error estimate $E \approx E_{\text{est}}$.

We say that $A$ approximates $T$ with at least $k$ significant figures if the relative error $$R = \frac{E}{T} = \frac{T - A}{A}$$ is bounded by $\tau = \frac{1}{2} \times 10^{-k}$, i.e., if $$ |R| \leq \tau.$$

If we have established the relative error bound $$|R| \leq B$$ then we do not need to know $B$ very accurately before we can determine the largest value of $k$ that can be justified.

As an example, let us consider three special cases:

1. Suppose that we have determined $B \approx 4 \times 10^{-7}$. The true value of $B$ must necessarily be in the interval $(3.5,4.5)\times 10^{-7}$. In all cases we can conclude $k=6$, because $B \in (0.35,0.45) \times 10^{-6}$.
2. Suppose that we have determined $B \approx 6 \times 10^{-7}$. The true value of $B$ must necessarily be in the interval $(5.5,6.5)\times 10^{-7}$. We can only conclude that $k \ge 5$, because $B \in (0.055,0.065) \times 10^{-5}$.
3. Suppose that we have determined $B \approx 5 \times 10^{-7}$. The true value of $B$ must necessarily be in the interval $(4.5,5.5)\times 10^{-7}$. As in the previous case we can only say that $k\ge 5$.