Re: modal interval
Arnold Neumaier wrote:
> Nate Hayes wrote:
>> Arnold Neumaier wrote:
>>> Modal techniques can possibly evaluate equivalent range enclosures
>>> in a slightly more automatic or faster way, given appropriate
>>> hardware support.
>>
>> I'm glad to hear you admit that much, Arnold.
>
> ... with the stated qualifications;
> - possibly,
> - given appropriate hardware support,
> - it would have to be demonstrated on a realistic benchmark
>
> At the moment, it is only based on hearsay from you, without any
> support by evidence. The evidence quoted below is spurious, since it
> beats only a straw man.
I think it is realistic to estimate that one interval operation can be
computed in about the same time as one floating-point operation. For
example, in the paper
http://grouper.ieee.org/groups/1788/Material/KirchnerKulisch-
HardwareSupport.pdf
Ulrich says:
"The circuits described in this paper show that with modest additional
hardware costs interval arithmetic can be made almost as fast as simple
foating-point arithmetic."
SIMD instructions on modern Intel and AMD chips give us a reasonable guide
of what to expect.
>
>
> Nate Hayes wrote:
>> Arnold Neumaier wrote:
>>> It is impossible to know undisclosed, nowhere detailed methods.
>>> To argue with such confidential knowledge may be good marketing,
>>> but is against the scientific spirit.
>>>
>>> ...
>>>
>>> But to claim this as a scientific fact, it would have to be
>>> demonstrated on a realistic benchmark, or at least on a nontrivial,
>>> realistic example.
>>
>> Chapter 6 of my paper gives one such example. It is simple example,
> but not
>> trivial and not insignificant, since Beziers, b-splines and NURBS
>> form foundation of global industry in fields such as CAD,
>> engineering, desktop publishing and computer graphics.
>
>
> You compare it only with ordinary interval evaluation,
> which is easy to beat. Compared to this, most other methods look good.
> It is like praising a new optimization method because it is faster
> than steepest descent.
For the same computational effort as classical arithmetic, the improvement
is order of magnitude (or more) in terms of quality of the range enclosure
bounds.
>
> But compare it against the use of monotonicity methods, as given in
> my paper, and you'll find no advantage anymore. (I think, the
> monotonicity methods had even a 2% advantage.)
That example in your paper, i.e., the linearInt() function from Vienna
Proposal, is simply the component-wise form of the modal interval arithmetic
operations
A+U*(B-Dual(A)).
> Well, you say, monotonicity methods plus directed rounding are nothing
> else than modal intervals in disguise, but this is in the eye of the
> beholder.
>
>
> With the same right one can say, modal intervals (as used for range
> computations) are nothing but monotonicity methods plus directed
> rounding in disguise, which would mean that modal intervals are
> not needed at all since the traditional methods (monotonicity,
> directed rounding) are sufficient to reproduce the modal results.
From my perspective, it is about computational efficiency and speed,
provided the right hardware is available. For example, (x*y)/(x+y+1) can be
computed with classical interval arithmetic for X=[0,2] and Y=[0,2] in 4
interval operations to obtain the pessimistic range enclosure [0,4].
Endpoint analysis
[(inf(x)*inf(y))/(inf(x)+inf(y)+1),(sup(x)*sup(y))/(sup(x)+sup(y)+1)]
computes the optimal range enclosure [0,0.8] in 8 floating-point operations.
If we assume one interval operation on an interval processor can be computed
in about the same time as one floating-point operation, then the interval
processor computes the range enclosure twice as fast but over 4 times more
pessimistically.
So an end-user can have fast or optimal result, but not both.
A modal interval processor can compute
(X*Y)/(Dual(X)+Dual(Y)+1)=[0,0.8]
in 4 interval operations. So it computes the optimal range enclosure of the
endpoint analysis in the same amount of time as the classical interval
processor (which does not compute an optimal result).
>
> Even with more right - for to apply modal techniques, one needs
> total monotonicity, while the endpoint method can work with simple
> monotonicity. Thus the latter is of broader applicability.
There can still be benefit in these cases. One example in your paper is
(x*y-1)/(x+y+1) for x,y >= 0, which is not totally monotonic. But we can
factor into the equivalent expression (x*y)/(x+y+1)-1/(x+y+1) and compute
the optimal range enclosure
(X*Y)/(Dual(X)+Dual(Y)+1)-1/(X+Y+1)=[-1,0.6]
on a modal interval processor in the same time as 8 floating-point
operations. To compute the optimal range enclosure with endpoint analysis
requires 10 floating-point operations. So computation on a modal processor
is still faster.