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Re: What is your philosophy? Tracking or Static?



George,
IMHO your examples are not illustrative of argument in favor of some philosophy. They show how problems must be reformulated when interval arithmetic is aimed
Corliss, George a écrit :
Dan,

Thank you for the useful characterization.  I, too, began assuming tracking, but I am coming more to the static view.

Early in AD, a frequently cited difficulty was scaling:

    x is a vector
    s = max(|x|)
    x = x / s  // Normalize
    y = lots of computations
    y = y * s  // Rescale back

Is y a differentiable function of x?  AD is forced to assume not.  Intervals can help decide this question more precisely, but the analogy is whether we care about the history or only the result.
When x is replaced with a box X, the scaling factor must be calculated for the box as a whole, so it a constant for all the vector in the box.
So the problematic dependence of s with a particular vector is removed.

Or, suppose
     x is a scalar
     f(x) = sign(x)
     y = 0 * f(x)

Is y a continuous function of x?
This problem is ill-posed.
The interval arithmetic provide a correct enclosure of the exact result.
Since sign is discontinuous , the continuity is definitely lost for future calculation exactly in the same way a the result of a interval arithmetic calculation may provide a heavy overestimation of the the exact result.

The rule must be
the system must not lie
Nothing more.

Best regards,

Dominique

I am becoming a static-ist.

George

On May 25, 2011, at 6:17 AM, Dan Zuras Intervals wrote:

Date: Tue, 24 May 2011 19:05:38 -0500
From: Ralph Baker Kearfott <rbk@xxxxxxxxxxxx>
To: Nate Hayes <nh@xxxxxxxxxxxxxxxxx>
CC: John Pryce <j.d.pryce@xxxxxxxxxxxx>,
stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
Subject: Re: Ar we succeeding?

All,

Does someone else also have an opinion concerning this (please)?

Baker

	Baker, et al,

	I find I do have an opinion on this.  And as long as Nate's
	motion is in its discussion period, now is as good a time
	as any to discuss it.

	Let me put it in the form of a question I put to Nate:

	Is your philosophy about decorations a tracking approach or
	a static approach?

	There seem to be two schools of thought about the meaning
	of decorations.

	There is the TRACKING school in which decorations are the
	maximal (most pessimistic) result of the tree of evaluations
	that led up to the result to which they are attached.  That
	is, every exceptional or noteworthy incident in that tree is
	recorded for all to see whether it is relevant to the final
	result or not.

	Then there is the STATIC school in which decorations are
	information concerning the current result only.  Earlier
	decorations may pass through to this result if they still
	apply & may be discarded if they do not.  In this case the
	decoration must be able to be interpreted in the context
	of the final result whatever happened before.

	(In either school, the decoration must be ordered WRT to
	subsets of arguments.  I believe this is both necessary &
	sufficient for an FTDIA to be proved.)

	I asked this question of Nate because his motion seemed to
	be primarily of the tracking school but with some static
	features thrown in.

	I think we need to be consistent on this point.  As much
	for our own understanding as to explain the meaning of
	decorations to the rest of the world.

	I will admit that I started out in the Tracking school.
	But some remarks I've heard in this forum & privately have
	suggested to me that the Static school might serve us better
	as a standard.

	So I ask of all of you: Which philosophy should we espouse?
	Tracking or Static?

	I believe that once we decide this many of our more
	difficult questions will fall out as obvious.

	Yours,

				Dan

Dr. George F. Corliss
Electrical and Computer Engineering
Marquette University
P.O. Box 1881
1515 W. Wisconsin Ave
Milwaukee WI 53201-1881 USA
414-288-6599; GasDay: 288-4400; Fax 288-5579
George.Corliss@xxxxxxxxxxxxx
www.eng.mu.edu/corlissg




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