# Interval arithmetic with different definitions of intervals

Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]+[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as $$\{a\}_\delta=[a(1-\delta);a(1+\delta)]$$ or $$[a]_\epsilon := \left[ae^{-\epsilon};ae^\epsilon\right]$$ which base on some kind of relative error around a number $a$. For example for the later definition one has the rule $$[a]_\epsilon\cdot[b]_\delta=[ab]_{\epsilon+\delta}$$

My Question: Can you point me to textbooks/papers where interval arithmetic with different interval definitions such as the two above is discussed? I am looking for those interval definitions which base on some kind of relative error so that multiplication with those intervals is easy.

Note: It may help, that \begin{align} [a]_\epsilon &= \left\{ y \in \mathbb R^{+}: \left\|\frac{y}{a}\right\|_{\log} = |\log(y)-\log(a)| < \epsilon \right\} \\ &= \left[ae^{-\epsilon};ae^{\epsilon}\right] \end{align} whereby $\left\|\frac{y}{a}\right\|_{\log}$ is the so called log ratio distance.

• Trivial observation: did you mean $+$ in your first rule example?
– Colm Bhandal
Sep 5 '15 at 21:13
• @ColmBhandal: Yes, I'll fix it... Thanks! Sep 8 '15 at 17:57