Re: Up-to date Interval Arithmetic
On 2015-03-27 12:16:51 +0100, Ulrich Kulisch wrote:
> Am 26.03.2015 um 10:10 schrieb Vincent Lefevre:
> >About Unum: "It gets mathematical rigor that even conventional
> >interval arithmetic is not able to attain."
> >
> >In what sense conventional interval arithmetic is not rigorous???
> >
> An answer to the three question marks is attached.
In addition to the problem of the meaning of "rigor", I don't
understand why:
1. The open interval J is valid.
For instance, take a = [1,1] and b = [2,2].
Then I = [RNDD(1+2),RNDU(1+2)] = [3,3]
and J = (RNDD(1+2),RNDU(1+2)) = (3,3) = Empty
2. The closed interval I is included in the open interval J.
In any case, though I agree than a system with closed/open intervals
may give tighter results than a system with just closed ones, the
standard doesn't prevent from writing a flavor with closed/open
intervals, which could be based on conventional FP arithmetic.
Concerning the dot products, computing them exactly and doing a single
rounding at the end will indeed give you a tighter interval than raw
interval arithmetic, but it will also be much slower, at least with
the current hardware. So, the user should select what is best for him.
This doesn't make floating-point based interval arithmetic worse than
some other arithmetic. And the unum format certainly doesn't solve
this cost/accuracy problem.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <https://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)