Re: Motion 11 questions & comments...
Dan Zuras Intervals a écrit :
From: "Nate Hayes" <nh@xxxxxxxxxxxxxxxxx>
To: "P1788" <stds-1788@xxxxxxxxxxxxxxxxx>
Subject: Re: Motion 11 questions & comments...
Date: Mon, 8 Feb 2010 12:48:20 -0600
Ah, thank you Nate. I see I was confused about the
nature of reverse operations altogether.
Then, Marco, I am also confused as to the purpose
of these operations.
If they are formally equivalent to the existing
forward operations, why not use those instead?
Well, those together with the intersection
operation in an expression?
If they do not narrow the resulting interval like
the Kaucher/modal operations of which Nate speaks,
how are they 'reverse' in any meaningful sense?
In short, what problems can we solve with these
operations that we cannot solve as well already?
I see in your new document that you have chosen to
define things on the power set of the Reals. While
this may be formally correct for what you have in
mind, I have to say that is it not a very useful
domain of discourse for things to be defined in
the world of our oh so very finite computers.
Further, if these are to be some sort of non-modal
substitute for modal operations, how are we to
generalize them to other operations? Are we to
expect reverse sin, for solving the equation sin(X)
= S, to return infinitely many intervals separated
by 2pi that each have the property that their sines
fall into S?
This business of returning a pair of intervals does
not generalize well beyond the simple pole in divide.
Neither does it play well in expressions.
I think my confusion has graduated to wariness & doubt.
Can you allay my fears by explaining further?
Dan
Dan,
I understand and share you fears and doubt.
IMHO the troubles arise from a too ambitious purpose assigned to the
reverse functions
namely to define at conceptual level 1 a kind of equations solver.
My own understanding of these functions is that they should be defined
at level 2 without any level 1 precursor.
For the shake of the discussion, I only consider the reverse function of
a function f(x) of a single variable
We are trying to find solutions of the interval equation F(X) = Y0
where Y0 are interval in IF not IR
Given Y0, we know a rough approximation of X namely A (of course a
member of IF)
This means that Y0 must be included into F(A)
The reverse functions are tools that should allow us to refine the
initial guess A.
Assuming the A can be written A=[a,b] we search with the following refinment
- Left refinement LR an interval [a,x] such that
- Y0 is included into F(LR)
- LR is included into any interval
U=[a,u] such Y0 is included into F(U)
- Similarly a right refinement RR an interval [y,b] such that
- Y0 is included into F(RR)
- LR is included into any interval
V=[v,b] such Y0 is included into F(V)
-When LR and RR intersect each other the the best refinement
BR can be written BR=[y,x]
-When LR and RR do not intersect they are valid refinement.
A further central refinement CR=[x,y] can be defined whenever
Y0 is included into F(CR)
The previous considerations leads to suggest the definition of four
reverse functions corresponding to the calculation of LR, RR, BR and CR.
These functions should raise flags for the the following events
- The interval A is not a good guess for the solution
- No further refinement can be found
- CR or BR cannot be defined
- LR and RR should be further refined to BR
This approach generalizes the well known Interval Newton's method on
three points
-It allows for a trisection of the initial interval
- It is not dependent on function and derivatives
conditions.
-It deals with an equation Y0 included in F(X) and
not only a 0 \in F(X) condition.
At each step at most three new intervals can be introduced. It is the
responsability of the user to ask further refinements.
Further since we work on IF and not IR the number of solutions is
necessarily finite.
Defining a conceptual model on IR is of course possible. However it
should not be carried since it is misleading.
For example, due the the interval rounding , solution lying in CR at
level 1 may be moved to LR at level 2.
Best regards
Dominique
Dan Zuras Intervals wrote:
I am confused about the definitions of reverse operations
in sections 3 & 4. But this may be due to my confusion
about reverse operations in general.
. . .
But in reverse operations shouldn't the widths get narrower
not wider? Indeed, is that not the whole point of such
operations?
. . .
It seems to me I had this discussion with John & Nate
about a year ago & Nate's solution involved using both
'there exists' & 'for all' quantifiers.
I may have this wrong. Perhaps Nate can help here.
Dan,
My understanding is that reverse mode is different than the
Kaucher/modal operations, and perhaps this is the point to
be clarified. For example, reverse mode does not supply the
additive or multiplicative inverse elements you are speaking
of. Those are supplied by Kaucher/modal operations (of which
there has been no formal motions for, although they have been
discussed at times in this forum).
If you are trying to solve the equation A + B = C, the unique
algebraic solution is obatined by the Kaucher arithmetic,
i.e., B = C - Dual(A). In other words, A - Dual(A) = [0,0]
is an identity. Similarly, the equation A * B = C has unique
algebraic solution B = C / Dual(A), so long as 0 is not an
element of A, i.e., A / Dual(A) = [1,1] is an identity.
The system (IR,+) over closed, bounded and non-empty intervals
IR is a commutative monoid, hence as you notice there are no
inverse elements. Since (IR,+) also has the cancellative
property, it is possible to embed the monoid into a group.
This is where the inverse elements of Kaucher arithmetic
come from, as well as the notion of improper intervals
[b,a] such that a <= b.
Strictly speaking, the "for all" or "there exists" quantifiers
of modal intervals are not necessary. Only the Kaucher
arithmetic (and improper intervals) are required. However,
it is all a separate topic from the motion Marco currently
is putting forward.
Hope this helps.
Nate
--
Dr Dominique LOHEZ
ISEN
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France
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Email: Dominique.Lohez@xxxxxxx