Re: Bisection methods: a use-case example
Nate,
I see your point. I am wondering, however, whether or not
defining midpoint([1,+Inf]) = NaN is a necessary consequence
of the midpoint of an infinite interval being undefined at
level 1 or not, or whether or not the definition of midpoint
for such intervals depends on interpreting the upper bound
as Inf or omega. It seems that, if we decide that midpoint
should take on a finite value in such cases, we can, in principle,
regardless of whether or not we view the upper bound as
Inf or omega, set it to any value. In your example, I might
suggest sqrt(1*REALMAX).
If we do specify a particular finite value (which I guess is
somewhat arbitrary, but to a certain extent, isn't standardization
deciding on a common way of doing arbitrary things?) I suspect
there will be users of the standard for whom the standard choice
is not appropriate. Such a user, say, Sarita, would then need to
write her own routine, although Sally might not.
So, which choice do we make to minimize the total inconvenience
over all users?
Baker
On 05/26/2012 09:47 AM, Nate Hayes wrote:
John, Michel, P1788,
To illustrate things from my perspective a little more clearly, here is a
use-case example:
Case A: No Overflow
----------------------
In this scenario, the midpoint of an unbounded interval is undefined at
Level 1, and at Level 2 it returns NaN. So we have, e.g., a P1788 midpoint
operation like this:
midpoint([1,5]) = 3
midpoint([1,+Inf]) = NaN
and the Interval Newton algorithm (14) in section 3.2.2 generates NaNs
instead of finding the solution [1.49998,1.50003].
So our user Sally instead writes a "pragmatic" midpoint operation called
"midpoint2" that is undefined at Level 1 for an unbounded input but returns
REALMAX at Level 2 anyways; Sally's midpoint2 operation gives, e.g.,
midpoint2([1,5]) = 3
midpoint2([1,+Inf]) = REALMAX
Sally likes her midpoint2 better than the P1788 midpoint operation because
hers allows (14) to find the solution, so this is what she uses.
I think everyone agrees Sally did something really pragmatic, but one might
ask the question: is Sally really following the standard or is she just
working around it? Technically, her midpoint2 is undefined at Level 1 for
the input [1,+Inf] just like the P1788 midpoint operation is. If the P1788
midpoint operation returns NaN at Level 2 to avoid contradicting the Level 1
definition, then why is the user's operation not also a contradiction? Just
because the user gives the operation a different name? In other words, does
changing the name of an operation also change the mathematical defintion of
the operation?
Not that I blame Sally. How many other users will re-invent midpoint2 like
Sally did? And who is going to end up using the P1788 midpoint operation in
this scenario, anyway? Anybody?
Case B: Overflow
------------------
In this scenario, the midpoint of an unbounded interval is undefined at
Level 1 and at Level 2. However, the unbounded intervals are re-interpreted
as overflow so we have, e.g., a P1788 midpoint operation defined at Level 1a
and Level 2 like this:
midpoint([1,5]) = 3
midpoint([1,+omega]) = REALMAX
and the Interval Newton algorithm (14) in section 3.2.2 finds the solution
[1.49998,1.50003]. So Sally likes this midpoint; especially since it
might be implemented in a P1788 interval processor and be really fast!
Sally's colleague, Harry, knows that for this particular numerical problem,
the Interval Newton will find the solution in only 14 total iterations if he
uses smedian instead of midpoint. So Harry designs a smedian bisection
routine that, like the P1788 midpoint operation, is defined at both Level 1a
and Level 2 on overflow according to Table 1, i.e.,
smedian([1,5]) = 2.315
smedian([1,+omega]) = h
where h = REALMAX at Level 2, and he uses this instead.
Nate
--
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Ralph Baker Kearfott, rbk@xxxxxxxxxxxxx (337) 482-5346 (fax)
(337) 482-5270 (work) (337) 993-1827 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
(Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
Box 4-1010, Lafayette, LA 70504-1010, USA
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