12
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Trivial observation: did you mean $+$ in your first rule example? $\endgroup$
    – Colm Bhandal
    Sep 5 '15 at 21:13
  • $\begingroup$ @ColmBhandal: Yes, I'll fix it... Thanks! $\endgroup$ Sep 8 '15 at 17:57
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.