Re: General pow(xx,yy) function in Motion 10

[Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>]

> From: "Corliss, George" <george.corliss@xxxxxxxxxxxxx>
> To: Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>, Nate Hayes
> 	<nh@xxxxxxxxxxxxxxxxx>
> CC: "stds-1788@xxxxxxxxxxxxxxxxx" <stds-1788@xxxxxxxxxxxxxxxxx>
> Date: Sun, 6 Dec 2009 20:05:13 -0600
> Subject: Re: General pow(xx,yy) function in Motion 10
> 
> Dan, Nate, and all,
> 
> Do I understand the issue is, "If xx includes negative values, what is the
> correct answer?"  If yy is rational with an odd denominator, the answer is
> fundamentally different from yy rational with an even denominator?
> 
> If I miss the point, please get me back on track.
> 
> If that is essentially the issue, I wonder whether it is a question worth
> answering at all?  Whenever my engineering colleagues want to irritate me,
> they observe that, outside of pure mathematics (implying irrelevance), NO
> number is accurate to single precision.  In most applications, we are lucky
> to know 3 digits.  By implication, all intervals are thick.
> 
> If we accept their "all intervals are thick" assertion, all yy include both
> odd and even denominators.  What does that imply about the proper
> interpretation?  Of have you told me, but I did not get it?
> 
> Dr. George F. Corliss

	No, I believe that Nate's concern is more serious than that.

	You see, when xx is negative & both xx & yy are singletons,
	there are really two 'correct' answers, +/-e^(yy*ln(|xx|)).
	The pow function returns the interval with those endpoints
	in that case.  But the interior contains no answers, just
	the endpoints.

	Now, if either xx or yy is of finite width, the set of
	valid solutions extends inwards from the endpoints
	towards the center of the interval.  Probably still two
	disjoint sets but they eventually do meet up.

	Nate's concern is that any branch & bound method that
	bifurcates one operand will leave one half with a smaller
	interval (that part of the interior) & the other half
	with the entire original interval or very nearly so.

	He is correct, this IS a danger.

	But the danger is in the search method rather than any
	fault of the library function.

	Back when I first proposed this sort of pow I said that
	there should be auxiliary functions, say powPos & posNeg,
	that map to one manifold of the full power function &
	not the other.  In the course of the discussion leading
	up to this motion, those functions were lost.

	Nate is exposing the need for them.

	It is easily fixed with some later motion so I haven't
	given it much thought for now.

	I guess that was a mistake. :-)

	So, a search method that decides to bifurcate along the
	two manifolds of the pow function when it becomes
	necessary is an easy fix.

	There is a more serious problem in such search methods,
	however.  I briefly outlined that problem in my previous
	posting.

	But as it has nothing to do with the merits of motion 10,
	I wanted to limit my discussion of it.

	I see I have failed in that as well. :-)

	Let me just say that, IMHO, none of this has anything to
	do with the merits of motion 10 or pow as the only interval
	function that assures containment.  We should add the
	single manifold functions at some later date but it need
	not form part of our discussion today.

	Well, no more than it has already, that is. :-)

	Again, sorry for the digression, folks.


				Dan