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Ulrich Kulisch wrote:
Arnold Neumaier wrote in his mail below:Other comparisons were also mentioned to be needed by other mails.Could you please collect for reference all the uses mentioned in one single mail?
In the mean time, I checked all old mails containing the word comparison in the subject line, and found not a single reference to practical use (except for the proposed use of <= for ordering intervals in a branch and bound process, which is not adequate both since the interval <= is not a linear order, and for other reasons already mentioned in the recent discussion).
This is my answer:I did not collect these and I simply do not have the time to look for them right now. I was at the SCAN meeting last week and have to work through 150 mails now. I hope that somebody else has such a collection. I remember having seen something like this.
Ther is nothing to collect....
The <= relation is the connection link between the set definition of the arithmetic operations for intervals and the explicit formulas for computing the result of the operations by the bounds of the operands, not just for real intervals but also for real interval vectors and interval matrices and again for complex intervals and for complex interval vectors and matrices.This does not yet constitute an algorithmic use. it is a theoretical result, and for this it suffices that the comparison is defined on the theoretical level. The standard, however, is about what needs to be implemented.For the <= relation compatibility relations hold between the algebraic structure and the order stucture similar to the real numbers.. For instance: 1. If a <= b ==> a + c = b + c for all c. 2. If a <= b ==> -b <= -a. 3. If 0 <= a <= b and c >= 0 ==> ac <= bc and ca <= cb. Similar for division.Again, this is a theoretical result only. I do not deny that the <= order has useful theoretical properties that help in understanding the structure of intervals and interval linear algebra. But I haven't seen any algorithmic uses for these.
Thus my statement remains valid that the only comparison operations useful in practice are disjoint, subset (containment), interior (containment in the topological interior).
Arnold Neumaier