TITLE: Interval arithmetic with different definitions of intervals
QUESTION [12 upvotes]: 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.

REPLY [3 votes]: I know of a variant of interval arithmetic called ball arithmetic.  It seems as though it may be like what you're looking for.  Ball arithmetic is currently implemented in the C library, Arb.
Here is a link to a paper about this sort of arithmetic.  Also, it discusses the differences between ball arithmetic and interval arithmetic.
http://www.texmacs.org/joris/ball/ball-abs.html
Also, you may be able to find what you're looking for at http://www.cs.utep.edu/interval-comp/books.html.  Which contains a list of books on interval arithmetic.
Also, take a look at http://fredrikj.net/blog/2012/04/high-precision-ball-arithmetic/.  This is extremely similar to what you wrote.
I can't find any other explicit definitions, but it shouldn't be hard to formulate ball arithmetic over general normed vector spaces.