Re: Constructors & decorations

[Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>]

> Subject: Re: Constructors & decorations 
> From: John Pryce <j.d.pryce@xxxxxxxxxxxx>
> Date: Tue, 9 Feb 2010 22:24:56 +0000
> To: Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>
> 
> Dan
> 
> On 9 Feb 2010, at 21:54, Dan Zuras Intervals wrote:
> > 	This may be a bit of a topic drift but what is the difference
> > 	between Valid & Defined?
> Ah. There are 3 conceptually different groups.
> 
> isValid: The result of a valid constructor call, or of an operation =
> whose inputs were isValid. So recursively, an interval value that makes =
> sense, "not NaI".
> 
> isDefined, isContinuous]: Only make sense in the context of evaluating =
> the interval version yy=ff(xx1, =85, xxn) of a point function y=f(x1, =
> =85, xn). If the original xx's are all isDefined [resp. isContinuous], =
> and the resulting yy is isDefined [resp. isContinuous] then one has =
> proved f is defined [resp. continuous] at each point of the input box =
> (xx1 cross =85 cross xxn) in R^n. (cross = cartesian product).
> 
> isBounded: Unlike the others, not about history. Just says that xx is =
> mathematically bounded even though a bound might have overflowed to oo.
> 
> John=

	To all,

	I apologise for the basenote drift here.  If it goes much
	further I'll take it offline.

	John,

	While all of this makes sense to me I'm not sure it answers
	my questions.  Let me try some examples:

	Let f = sqrt(x^2 - 1)

	So, if I understand your argument correctly, the evaluation
	of yy = ff(xx) is,
	
	(1) Valid (Bounded) iff the input is Valid (Bounded),
	(2) Defined (Continuous) for all Defined (Continuous) inputs
	except for xx such that xx \intersect (-1,1) is non-empty,

	Also ff could return possiblyBounded in the case where the
	input is Bounded but the intermediate expression x^2 overflows.

	Now let f = 1/sqrt(x^2 - 1)

	I believe we have,

	(1) Valid iff the input is Valid,
	(2) Defined for all Defined inputs except for xx such that
	xx \intersect (-1,1) is non-empty,
	(2) Bounded (Continuous) for all inputs except for xx such
	that xx \intersect [-1,1] is non-empty,

	Note that this last is [-1,1] rather than (-1,1).  And that
	overflow cannot happen in this case.

	Do I have that correct?

	Now, can you give me an example of an interval that is Defined
	but not Valid?

	As for Bounded, it seems to me that any result interval with
	an infinite endpoint is notBounded only if it were evaluated
	over a pole of some kind.  Otherwise, in the overflow case,
	I think possiblyBounded is the best answer.

	I think I would like to explore these issues further in a
	discussion of propagation rules.

	But that is a topic for another day...

	Thanks,

				   Dan