Re: Motion on interval flavors
Dear All:
The discussions appear to get into two questions: 1) whether Kaucher
interval should be included/mentioned in the standard, and 2) whether
Kaucher interval arithmetic is useful.
Mathematicians (and general users of math) had survived for centuries
without using negative numbers. It doesn't mean negative number is
useless. It does simplify calculation and reasoning. Similarly, we
(interval people) have applied the classical interval to solve various
problems without using Kaucher interval. This doesn't mean Kaucher
interval is useless either. It can help to solve "inverse" problems
and simplify the calculation, among others.
Motion 36 says "The standard shall permit different kinds of
interval". This should be enough for future extension. Yet, the
standardization of Kaucher/Modal interval itself is foreseen to take a
very long time, since there are various "dialects" of Kaucher interval
among its limited users. Guess there is plenty of work left for the
Kaucher/Modal interval subgroup.
Best regards,
Yan Wang
>
> On Mon, Jul 9, 2012 at 10:19 AM, Svetoslav Markov <smarkov@xxxxxxxxxx> wrote:
>> 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
>>
>>
>>
>
>
>
> --
> Yan Wang, Ph.D.
> Assistant Professor
> Woodruff School of Mechanical Engineering
> Georgia Institute of Technology
> 813 Ferst Drive, Room MaRC-472, Atlanta, GA 30332-0405
> Tel: +1 404-894-4714 Fax: +1 404-894-9342
> http://www.me.gatech.edu/~ywang
--
Yan Wang, Ph.D.
Assistant Professor
Woodruff School of Mechanical Engineering
Georgia Institute of Technology
813 Ferst Drive, Room MaRC-472, Atlanta, GA 30332-0405
Tel: +1 404-894-4714 Fax: +1 404-894-9342
http://www.me.gatech.edu/~ywang