Re: Multi-precision (was...Please give me advice)
Michel Hack wrote:
This discussion again suffers from the one-sided view that intervals
represent uncertain numbers.
When intervals are used to express ranges of values, the concept of a
midpoint frequently doesn't even make sense. Why should the arithmetic
mean of the endpoints be significant? In many applications (e.g. large
ranges of physical values) the geometric mean makes more sense.
True. Indeed, in our branch and bound codes, we split domain intervals
that do not contain zero at the geometric mean, with much better
results on problems with huge bounds.
On the other hand, multiprecision computations make primarily sense when
many significant digits are wanted, and then the difference is slight.
So it seems as if mid-rad were ok on this account.
However, I encountered in multiprecision interval computations the
following issue for functions with a lot of cancellation:
If the precision is not high enough, cancellation ultimately leads
to very wide intervals. The ratio q of their width divided by the
desired width is indicative (typically within an order of magnitude)
of the factor by which the accuracy must be increased to get the desired
accuracy. In difficult cases, using this heuristics is much preferable
to iteratively increasing the accuracy by one word length at a time
until the accuracy is adequate.
Therefore though the ultimate multiprecision interval calculation only
uses narrow intervals, preliminary multiprecision interval calculations
need to cope well also with wide intervals.
This is why I find the triplex representation for multiprecision
intervals much superior to the mid-rad representation.
Incidentally, the very first multiple precision interval package
(by Brent) also used this format! It is very convenient and has
only very little overhead compared to a mid-rad format.
Arnold Neumaier