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Re: Re-submission of motion 5: multiple-format arithmetic.

Nate Hayes schrieb:

Arnold Neumaier wrote:

Nate Hayes schrieb:

Moore's theroem remains valid with the proposed definition, since
a real expression defines a real function f(x_1, ..., x_n) only
on domains where no divisor is zero.

Therefore, correctly, for xx=[0,1]
    {1/(x^2-x+1) | x in xx}
       subseteq 1/(xx^2-xx+1) = 1/([0,1]-[0,1]+1) = 1/[0,2]=[1/2,inf],
while with your definition, division by an interval containing zero
gives NaN), and we'd get NaN, violating Moore's law since
      {1/(x^2-x+1) | x in xx} subseteq NaI
does not hold.
The reason it "violates Moore's law" is because x = 0 is not in the domain of the function! That's the whole point of returning NaI... 

???

When I evaluate f(x):=1/(x^2-x+1) at x=0,
I get the perfectly reasonable value f=1.

But 1 is not in NaI, violating Moore's law.

The zero in the interval is solely due to overestimation of the
interval arithmetic.



Thus to hawe Moore's law for not everywhere defined expressions
_requires_ that NaI cannot arise as the result of an arithmetic
operation if the arguments are intervals.




I believe nothing in this motion and rationale hinders the
implementation of various forms of non-standard intervals -- Kahan,
modal, etc. -- as discussed at the end of Vienna/1.2.

I've mentioned before this is simply not true. If traps or flags are
only way to obtain NaI result from an interval operation such as
1/[-2,3], this is hinderance to efficient modal interval
implementations.

This is another eason why modal intervals should not be part of
the standard. It makes the latter unnecessarily complicated,
only to introduce an error-prone technique that can be safely handled
only by a tiny minority of users.
I don't agree at all. It opens 1788 to a wider audience by clarifying and simplifing. 

Modal arithmetic is a very dangerous tool that _easily_ leads to
wrong results without a very good understanding of its theory.
As our off-line discussion last year had revealed, not even you were able to interpret the modal theorems in the literature correctly to 
give always valid results.

Thus the number of users that can safely handle modal arithmetic in
full generality may well be zero.


Arnold Neumaier