Re: Motion 9 - ExactDotProduct
R. Baker Kearfott wrote:
A motion to include an exact dot product in the interval
arithmetic standard has been made and seconded. Therefore,
the three week discussion period begins, to end after
Wednesday, October 27.
1. I strongly support an exact sum, 1-norm, 2-norm, and dot product that
takes real vector arguments and returns the tightest interval result
(as in Section 5.1 the Vienna Proposal, with the norms handled similarly
upon requests in a previous discussion).
2. I find it acceptable to do this via a complete format of very long
floats as proposed in the motion, but would very much prefer if this
is not made mandatory but left to the implementor, since, for actual
applications, the main gain for the user is the form specified in the
Vienna Proposal, and there are lots of algorithms that achieve this
efficiently without a complete format.
3. I see no need at all for corresponding versions of
the optimal dot product etc. for interval input.
The reason is that real sums or inner products easily have lots of
cancellation under addition when the sign differ, while uncertainties in
intervals add up, no matter which signs the operand have.
Thus the accuracy to be gained by a highly accurate summation is usually
only a small fraction of the width that is incuured by summing the
interval widths.
I'd appreciate to know about applications - if there are any - where an
interval dot product would be essential for good performance.
In the spirit of simplicity I therefore suggest that only the real to
interval case of the motion is supported, and not the interval to
interval version, i.e., Section 3 of the motion text is dropped.
Arnold Neumaier