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Re: Motion P1788/M007.01_NaI

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John Pryce schrieb:

P1788 members

On 24 Aug 2009, at 17:36, Michel Hack wrote:

Vincent Lefevre wrote:

No, you can't use the same reasoning because of the rule on the limits:
  lim_{x -> 0, y -> +oo} x * y
doesn't exist (note that here I don't use the extended real numbers).


The problem is that there are two notions of zero in IA:
  the single point {0}, which is exact, and which is
  idempotent under multiplication with arbitrarily large reals
and
  an interval containing zero, in which case limit processes
  may indeed be relevant.
What is the "meaning" of a FP number x? This was debated long since by two great men, say James W Tukey and James H Wilkinson (it wasn't actually them, someone remind me). JWT's view ("applied numerical analysis") was that x represents some interval, e.g. {all reals that round to x}, whereas JHW's ("pure numerical analysis") was that x is just itself, a single real value. Which view you take can affect what you regard as a good numerical algorithm. For instance, a method that gets all the singular values of a matrix (even the smallest) with small _relative_ error is valuable on the JHW view, but may be seen as pointless on the JWT view. 

Wilkinson's view is the one used universally in numerical analysis.
The number _is_ itself, but may _approximate_ any number close to it
in a vague sense.




Note that interval arithmetic typically involves IEEE operations on
endpoints, and that an endpoint of zero should imply that all points
in the interval have the same sign. 

Except for zero, which as a real number doesn't have a sign.


Nevertheless I agree that when
a product of endpoints is of the form 0*oo the indefiniteness of the
limit may indeed play a role.  Arnold Neumaier however seems to have
good reasons for expecting 0*oo = 0 in this case too.


Only to make the algebra simplify.
I feel there is a risk of confusing levels 1 and 3 here. If I understand Arnold's reasons for the "deviant behavior", see Vienna 7.8-7.11, they have NOTHING to do with the mathematics, but are practical changes that if implemented would simplify level 3 code for interval operations, on machines that use IEEE 754 arithmetic. 


Arnold Neumaier