Re: modal intervals
On 12 Nov 2008 at 10:16, Arnold Neumaier wrote:
> Svetoslav Markov wrote
> [in ''Edge case conversions, exceptions to IEEE FPA'']:
>
> > standardization of proper interval arithmetic would be
> > equivalent to standardization of nonnegative reals.
>
> In contrast to the reals, which form a field,
> the same is not the case for Kaucher intervals.
> For example, the distributive law fails.
> Kaucher intervals form a complicated algebraic structure,
> not really simpler than that for ordinary intervals.
>
> Thus your analogy completion of R^+ to R vs. completion of
> intervals to Kaucher intervals is poor.
>
Not so poor, as I meant completion with respect to addition. I do not
say that Kaucher structure is simple. However, I would not claim that
a field is a simple structure as well, and that all laws in a field are nice.
For instance, why in a field the product of two negative numbers is positive?
I doubt that there is someone who likes this law. There is
no distributive law in Kaucher arithmetic, but there is a quasidistributive
one, which is not bad at all.
>
> > I believe that a standard for Kaucher interval arithmetic will be simpler
> > than one for proper intervals, as there will be no need to check
> > whether an interval (as input data or computational result) is
> > proper or not.
>
> Input is a tiny part of interval computations, and output will
> be automatically proper if the input is.
how is this translated in the situation with reals?
No algebraic problems like a+ x =b shall be allowed?
Svetoslav
>
> However, computations are more complex, since in most operations
> one needs to check whether an interval is proper or not.
>
> Thus whether or not to include modal intervals or Kaucher arithmetic
> will depend on whether the extra compexity is compensated by
> sufficiently increased practical usefulness.
>
>
> Arnold Neumaier