Re: modal intervals
Arnold,
a*x=1 is solvable for all nonzero Kaucher intervals a,
and the solution is alpha*dual (a), where alpha is a real
number:
alpha = 1/(a^-a^+) in inf/sup presentation a=[a^-, a^+]
or
alpha = 1/(a'^2-a"^2) in MR presentation a=(a';a")
here dual [a^-, a^+] = [a^+, a^-], resp. dual (a';a") = (a'; -a")
Svetoslav
PS. i) I do not claim that all interval algebraic problems are
soluble (and same is in real arithmetic). ii) The view
"algebraic interval equations have no meaning in
most applications of interval arithmetic" seems adequate
for the moment. However, it may become inadequate in the
near future.
On 12 Nov 2008 at 11:26, Arnold Neumaier wrote:
> 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
> >
> >
> >
Prof. Svetoslav Markov, DSci, PhD
Head, Dept. "Biomathematics", phone: +359-2-979-3704
Inst. of Mathematics and Informatics, fax: +359-2-971-3649
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