Re: Siegfrieds recent paper
On 2012-01-05 08:29:26 +0100, Siegfried M. Rump wrote:
> In my arithmetic
>
> A = infsup(0,1000); % [0,1000]
> B = exp(A); % [1,inf]
> C = 1/B; % [T,1]
> D = infsup(0,0); % [0,0]
> E = intersect(C,D); % empty
>
> Truely empty, without decoration.
But what if you have:
A = infsup(0,1000);
B = exp(A);
C = 1/B;
D = C-1;
E = infsup(-1,-1);
F = intersect(C,D);
> In P1788
>
> A = infsup(0,1000); % [0,1000]
> B = exp(A); % [1,inf]
> C = 1/B; % [0,1]
> D = infsup(0,0); % [0,0]
> E = intersect(C,D); % [0,0]
IMHO, the main problem here is that only closed intervals are
supported in P1788. But this choice was made on purpose to have
a simpler arithmetic, if I understand correctly.
I think that since we have decorations, two bits could be stored
for free (in the same byte or some other padding byte if any) to
give the information whether the bounds are in the interval or
not. This could be an extension.
> My concern is what happens if P1788 produces
>
> A = infsup(0,1000); % [0,1000]
> B = exp(A); % [1,inf]
> C = 1/B; % [0,1]
> D = 1/C; % [1,inf] with decoration
>
> and D is used forgetting to check the decoration. The conclusion may be
> that an underlying function is well-defined, but it is not.
If you do not assume incorrect conclusions, there would not be any
problem. In a similar way, you shouldn't assume that every element
of the returned interval is in the image of the function.
The default assumption here should be that the function is *not*
necessarily well-defined. If you want more information about the
function, just check the decorations.
> This cannot happen with my definition.
But your definition has drawbacks: your arithmetic can return NaI
due to overestimation in the intermediate computations, while one
would want a more meaningful result.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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