Re: Motion on interval flavors
Dear Arnold,
I do not accept all of what you say.
On 9 Jul 2012 at 14:58, Arnold Neumaier wrote:
Date sent: Mon, 09 Jul 2012 14:58:40 +0200
From: Arnold Neumaier <Arnold.Neumaier@xxxxxxxxxxxx>
Organization: University of Vienna
To: "Corliss, George" <george.corliss@xxxxxxxxxxxxx>
Copies to: Nate Hayes <nh@xxxxxxxxxxxxxxxxx>,
stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
Subject: Re: Motion on interval flavors
> On 07/08/2012 06:42 PM, Corliss, George wrote:
> > Yes, your application to Bezier and b-spline bases for polynomials is a deeper application, but am I on the right track?
> >
> > That is EXACTLY the sort of example I was hoping to hear.
>
> Pros and cons of this are all discussed in my paper on nonstandard
> intervals, which I wrote a few years ago when we started this
> discussion. Nothing has changed in the mean time.
>
>
>
>
> On 07/09/2012 11:35 AM, Svetoslav Markov wrote:
>
> > The historical lesson of the negative numbers teaches us
> > that the correct way is to work in group structures
> > whenever possible. Kaucher/modal intervals provide
> > such structures.
>
> For addition, but the multiplicative structure becomes much poorer.
> In contrast, introducing negative numbers and reals adds structure
> that simplifies the algebra!
Here I disagree ! You probaly have in mind the
table-form multiplication formulae. They can be written
in a simple form of a single expression. A nice
distributive relation exists, etc.
In practice, when manipulating algebraic equations
in real arithmetic the most useful law is cancellation.
It allows you to tranfer terms from one side to another,
to annulate terms like x-x or write x/x = 1. Similar rules
are used in Kaucher arithmetic.
> > Note also that the applications of negative numbers increased
> > only when mathematicians started to use them systematically and got
> > used to the (group-like) properties of real numbers. The history
> > teaches us that soon or later the same will happen with
> > Kaucher intervals.
>
> History teaches that the most useful structures will be used, not the
> most complete ones. otherwise we'd all be using quaternions on a regular
> basis. But they are used only for very special applications where they
> are indeed suitable. Kaucher intervals compare to ordinary intervals
> much more like quaternions to the complex numbers than like reals to
> rationals.
I did not mention the "reals to rationals" relation.
Besides, it seems to me that quaternions find many applications and
are used "on a regular basis".
It may be interesting to recall that Hamilton contributed a lot to the
widespread of negative numbers, just by "marketing" his quaternions.
So, you are right that the ideas behind quaternions and negative numbers
are very close. In both cases one need nice algebraic rules. The price to pay
is the introduction of improper elements.
If one is not prepared to pay this price one needs much more complicated
rules and new operations (like the inner ones). Well, only if one wants to
have the freedom to consider problems outside the scope of problems
treated by "classical" interval arithmetic.
>
> On 07/08/2012 12:52 PM, Svetoslav Markov wrote:>
> > I make the analogy with negative numbers because
> > improper (Kaucher/modal) intervals are just
> > like negative numbers (as their radii are negative).
> >
> > There is a lot in internet about the history of the acceptance
> > of negative numbers, see e. g.:
> >
> > http://en.wikipedia.org/wiki/Negative_number
> > (section "history")
>
> When Kaucher intervals have gained the same widespread use in interval
> method as negative numbers or real numbers have in ordinary numerical
> methods, it is time to standardize their use.
>
> But not earlier. Being more general and/or analogous and/or have
> additional properties is not enough justification for an incorporation
> into standards. Only extensive, widespread use is.
The widespread use of Kaucher arithmetic will certanly depend
on whether or not the IEEE stardard facilitates it.
Svetoslav