accurate summation, wasRe: re motion 50 (Just to you...)

[Richard Fateman <fateman@xxxxxxxxx>]

On 9/26/2013 3:21 AM, Ulrich Kulisch wrote:

...
<snip>

 You would compute the minima and the maxima of the products of the vector components and finally you have to accumulate all the minima and all the maxima.

As you describe it,  this method gives the incorrect answer, (for an interval dot product) so why would you use it?

Let us assume that this accumulation requires computing the sum

10²⁰⁰ + 23456 - 10²⁰⁰.                                                                                                           (1)

If your computer provides an EDP you get the correct answer 23456 and if the EDP is supported by hardware you get it very fast.

The average user (hypothesized below)  probably would not know about EDP and so would not use it.

If your computer does not provide an EDP the average user will accumulate (1) in conventional floating-point arithmetic and he gets the wrong answer 0.

The average user might just add [10^200]+[23456]-[10^200] in the compiler-supplied interval arithmetic mode, and thereby get a correct enclosure,
not zero.  Probably not the tightest enclosure, though.  He might then try multiple precision, if available.

....

The systematic development of a mathematical model requires that the error resulting from the computation can largely be excluded. This requires the best possible arithmetic.

While I agree with some of your points, they tend to  lead to  exact rational arithmetic and/or arbitrary precision floats, as in GMP or MPFR.

This  justification of EDP as a requirement for interval arithmetic hypothesizes one "average"(unskilled) user who
re-implements a 1788 library routine, poorly.   Yet another user, also ignorant of the library routine, re-implements it,
  but is saved because he (not so ignorantly?) decides to use EDP.   Is this worth making the proposal for the
standard longer and more complicated?


RJF