Re: IEEEP1788
Dear Ulrich,
On 2015-05-19 12:56:33 +0200, Ulrich Kulisch wrote:
> Arithmetic for R as well as for subsets F of pure floating-point
> numbers is well defined. On this base arithmetic for bounded and
> unbounded intervals of IR and IF easily and clearly can be derived.
> This leads to well known formulas which can be described on a few
> pages.
>
> If division by an interval which includes zero as an interior point
> is excluded, interval arithmetic leads to an exception-free, closed
> calculus, i.e., an operation for two intervals of IR or IF always
> leads to an interval of IR resp. IF again. As an add-on division by
> an interval that includes zero as an interior point also can be
> defined in IR and IF.It leads to two distinct unbounded real
> intervals. These can be used to develop the extended interval Newton
> method which allows computing enclosures of all zeros of a function
> in a given domain.
P1788 does not cover only the basic arithmetic operations, but also
elementary functions. So, you can't just give formulas. P1788 needed
to be generic. And for functions like tan, an exception-free result
is not possible.
> [...] Also the introduction of -0 or +0 does not make sense in
> interval arithmetic. There is only one zero in R. [...]
As already said, there is a single 0 in the bounds of P1788 intervals
(Level 2).
> Please have a look at the penultimate paragraph in your mail above.
> It reads very well. But is it reasonable? Single precision may be
> too short and extended precision too slow.
>
> Interval arithmetic carries the potential to replace floating-point
> arithmetic by some general computing tool where results come with
> highly accurate guarantees. Two things are definitely necessary to
> reach this goal:
>
> A. Fast double precision interval arithmetic and
> B. An exact dot product (EDP).
This is just your opinion. P1788 covers arbitrary number and interval
formats. For many applications, double precision is not sufficient,
in particular when one takes into account that interval arithmetic is
pessimistic.
> My interest in P1788 died when B was kicked out two years ago in 2013.
It has not been kicked out. It is in P1788, just optional. IMHO, this
is too much and it should have been kicked out completely; I'm not
saying that it is useless, just that it shouldn't belong to a standard
on interval arithmetic. Moreover its real utility over other methods
has never been shown.
If there is a real interest in EDP for more than a small community, it
will be implemented everywhere.
> The EDP was available on nearly all the old mechanic calculators as
> a tool for abtaining high accuracy in computing. It can be traced
> back to the early computer by G. W. Leibniz (1685).
We are no longer in the 17th century. The technology and the knowledge
(new algorithms...) has evolved.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <https://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)