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RE: [Caml-list] Efficient and canonical set representation?
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 Date: 2003-11-13 (15:49) From: Eric Dahlman Subject: Re: [Caml-list] Rounding mode
```Hi Diego,

I looked at that page thanks a bunch!  That was just what I was hoping
for and it answered many of my questions.

-Eric

Diego Olivier Fernandez Pons wrote:

>     Bonjour,
>
>
>>Somewhat off topic but why is this necessary from a numerical math
>>type of perspective. I am honestly curious as I don't see how this
>>would interact with the calculation in a meaningful way.
>
>
> You are right when you say that there are many sources of errors in
> numerical computations and that rounding errors are usually
> insignificant with respect to them.
>
> The point is that stochastic arithmetic (and its deterministic variant
> interval arithmetic) are useful to find where the accurancy of your
> computation is falling drastically (e.g. cancellations)
>
> I really haven't the place to explain extensively how CESTAC works but
> there are a few explanations in the ANP website
>
>
> (CADNA for C/C++ source codes, user's guide. Chapter 4. Survey of the
> CESTAC method. Many examples also on the homepages).
>
> The main idea is that in a first order approximation, the number of
> significant digits of a result can be estimated with respects to the
> dispersion of the different values it can take using several rounding
> modes.
>
> Then, you can avoid doing unstable computations like dividing by a
> small number (epsilon) very noised which makes you believe it is a
> good 'pivot' in a gaussian resolution, etc. The whole computation will
> then give a more accurate value.
>
> The website gives an example where usual gauss method finds
>
>    x1 = 60 x2 = - 8.9 x3 = 0.0 and x4 = 1.0
>
> when you estimate the errors, you find
>
>    x1 = 1.0 x2 = 1.0 x3 = 0.1 e-07 and x4 = 1.0
>
> exact values are
>
>    x1 = 1 x2 = 1 x3 = 0.1 e-07 x4 = 1
>
> The difference is only due to a 'bad' pivot succesfully detected and
> therefore avoided.
>
>
>         Diego Olivier
>
>
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