Date: Thu, 10 Sep 2009 13:25:08 +0200
From: Arnold Neumaier <Arnold.Neumaier@xxxxxxxxxxxx>
To: Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>
CC: Ulrich Kulisch <Ulrich.Kulisch@xxxxxxxxxxx>,
stds-1788@xxxxxxxxxxxxxxxxx, Nathalie Revol <Nathalie.Revol@xxxxxxxxxxx>,
vladic@xxxxxxxx, Ronald Boisvert <boisvert@xxxxxxxx>
Subject: Re: Request for motion [Fwd: Input from IFIP WG 2.5 to IEEE Interval
Standards Working Group]
Dan Zuras Intervals wrote:
If we are to pursue this at this time, please include accurate
versions of all of sum, dot product, sum of squares, & sum of
absolute values for all supported precisions.
That the sum is needed is obvious.
The sum of squares is needed for an accurate 2-norm & is
different than calling dot product with the same argument
twice in ways that are important in our context.
The sum of absolute values is needed for a 1-norm & is pretty
cheap once you have an accurate sum.
Yes, these are useful. Perhaps also the 2-norm itself!
Yes.
Finally, let me caution you that how these things behave on
empty & NaI elements may turn out to be important to this
group.
I think only interval-valued results for noninterval inputs should be
provided by the standard. Then there are no problems.
Well, I am concerned about the elements chosen from empty
or NaI intervals that end up being elements of the vectors
in one of these operations.
As we have not even touched on the issue of representation
of empty & NaI yet, I thought it important.
If some element is NaN or two terms in the sum are +inf and -inf,
the result should be the empty set; otherwise the tightest enclosing
interval of the exact result should be returned.
Arnold Neumaier
Arnold,
Careful here. Please look at clause 9.4 in 754-2008.
In the case of sum or dot product you are quite correct.
But in the case of the norm operations (sum of squares &
sum of absolute value) the existence of an infinity determines
the value of the norm even if a NaN element is to be found
elsewhere in the vector.
The rules are complicated & vary a bit from operation to
operation. They are also controversial & I will not bother
justifying them again here.
Let me simply urge caution in defining the exceptional cases.
Accuracy in the unexceptional case is not the only interesting
thing about these operations.
Dan