Re: modal intervals
Svetoslav Markov schrieb:
On 12 Nov 2008 at 10:16, Arnold Neumaier wrote:
Svetoslav Markov wrote
>>> standardization of proper interval arithmetic would be
>>> equivalent to standardization of nonnegative reals.
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 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?
If you think algebraic problems should be solvable then
what about the algebraic problem a*x=1? It is not solvable
for all nonzero Kaucher intervals a,
So you should regard standardization of Kaucher intervals
to be equivalent to standardization of integers.
But we need reals....
Unlike algebraic equations for reals, which arise everywhere in
applications, algebraic interval equations have no meaning
in most applications of interval arithmetic. One can study them
for purely mathematical reasons, but this has no computational
consequences.
Modal intervals provide a semantics for Kaucher intervals,
but their algebraic use is restricted to expressions with
strongly restricted monotonicity properties.
To solve an equation like (x+y)^2*(x-y)=z for x given
modal intervals y and z still has no meaning.
Anyway, actual widespread use or at least demonstrated usefulness
in significant real-life applications should be the primary
criterion for basic decisions that significantly affect the
complexity and speed of implementations.
Arnold Neumaier
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