Re: Motion P1788/M0012.01:InnerAdditionAndSubtraction
Juergen et al,
The inner subtraction evidently corresponds to modal arithmetic. However,
I did not originally use it as such, but merely as a way to make
sweeps of the interval Gauss--Seidel method more efficient, by first
adding together n elements of a sum, then using that sum to compute
the sum of just n-1 of the elements. That is, we knew that we had already
added that interval, and we knew that by an inner subtraction we would
get the proper result.
I'll need to rethink its relationship to reverse operations that are
used in constraint propagation, since I may have been mistaken in my
original statement. In particular, suppose we have x and y point values,
and we know
x + y = [-8,13],
where we originally knew x \in [2,3] and y \in [-10,10] and found
the bound [-8,13] by adding the intervals. Suppose now, by independent
means, we find y \in [1,2]. We would have
x \in [-8,13] - [1,2] = [-10,12],
no improvement in x, and
x \in [-8,13] \cancelminus [1,2] = [-9,11], (**)
also no improvement. However, is the conclusion x \in [-9,11]
even correct (for this use of \cancelminus in general)?
What definitely is correct is
x \in ([-8,13] \cancelminus [-10,10]) + [1,2]
= [2,3] + [1,2] = [3,5].
(I just discovered the above thing: GlobSol will run much faster
on certain problems if I implement it. Probably, Neumaier / Schichl
et al, among others, already have it in their systems.) Note that
the programmer here will want to use \cancelminus, but the programmer
will need to know the underlying logic (that is, the history of how the
[-10,12] was obtained). We can provide the operation to the programmer,
but I don't think we should try to track such history with a decoration.
We aren't at the point where the machine can do everything yet.
These constraint propagation operations (e.g. initializing bounds
on x + y, then finding new bounds on x when, say, a new bound on y
is independently found) are very important in numerous codes, both
commercial and non-commercial. As such, I see an important use for reverse
operations.
If, in contrast, we assume y and z are independently varying,
and we have y\in Y, z\in Z,
X = Y \cancelminus Z
is an INNER estimate for the set of possible values for x = y-z, that is,
X is contained in the set of possible values. These can be valuable
in some contexts. In fact, I implemented a module for inner arithmetic
some time ago using "twin arithmetic" (related to Kaucher arithmetic).
However, I haven't used it much.
At the risk of looking naive, I confess I haven't sorted through
all the details of modal arithmetic yet.
Baker
On 3/21/2010 14:56, Jürgen Wolff v Gudenberg wrote:
Nate, P1788
the inner operations are proposed in the Vienna proposal and are
supported by Baker.
Nevertheless I would like to get more information about their
application. I am not convinced by the examples given in the rationale.
Is it correct that they are the first step of Kaucher or modal arithmetic?
Juergen
R. Baker Kearfott schrieb:
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Department of Mathematics, University of Louisiana at Lafayette
(Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
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