(forwarded for Michel Hack): Re: complex intervals
Michel: I'm not sure why it was rejected. In any case, thank
you for answering Paul in depth.
P-1788: Please see Michel's comment.
Sincerely,
Baker
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Date: Sat, 15 May 2010 22:34:00 -2000
To: stds-1788 <stds-1788@xxxxxxxxxxxxxxxxx>
From: Michel Hack <hack@xxxxxxxxxxxxxx>
Subject: complex intervals
Paul Zimmermann wrote:
does P1788 plan to say something about "intervals" of complex values,
or only address real intervals?
An early motion decided that the standard was only going to address
Real intervals as sets of Real (not Extended Real) numbers, possibly
empty, and possibly unbounded. (Floating-point Inf may appear as a
bound, but is never a member of an interval.)
Several subsequent posts mention the fact that complex intervals are
beyond the scope of P1788, even though we know that INTLAB supports
them. Complex numbers and complex functions have come up, e.g. in
discussions of how to define the pow() function for (real) intervals
near zero.
There was a discussion in early November 2008 on complex intervals;
Paul actually joined in on Nov 25. We were discussing what the shape
of a complex interval was in the Argand plane: a circle (Euclidean
norm), a diamond (Taxicab norm), or a square (Max abs value norm).
This month (May 2010) has seen a spirited discussion of real midrad
intervals, and the difficulties of trying to treat abstract intervals
in a manner that would accommodate both infsup and midrad formats, and
we concluded that they had to be essentially different types; a motion
has just been put forward to put this on solid grounds.
The arguments for complex intervals would actually be different than
those for real midrad intervals, because (at least for entire functions)
there is relevant mathematical background. Still, in a practical context
the choice of Norm may be an issue (speed vs tight containment).
For more general functions the issue of half bounded intervals may arise;
in the case of complex intervals these might be half-planes: they are
the ones that cannot be represented by a midpoint and a tolerance bound.
In fact, one issue I keep pointing out is the epistemological distinction
between intervals as domains (where half-bounded makes sense and is even
essential) and intervals as uncertain numbers (whether real or complex).
The midrad forms can sensibly handle only the second point of view.
One other minor point (also discussed here recently):
whereas the "mid-rad" representation needs only n+1 values
(where the "+1" can have lower precision for tiny intervals).
With most floating-point formats (not IBM's old HFP) lower precision
brings with it a smaller exponent range. That would limit the exponent
range of the midpoint too: outward rounding would blow up intervals to
obscene widths, possibly the almost useless Entire. Arnold Neumaier's
triplex format, with a separate scaling component, is needed to deal
with this issue. Ok, so it would be n+2 and not n+1...
Michel.
(Resent -- first attempt at 03:17:29 UTC was rejected by listserv
("Unacceptable content" --- let's hope this was a temporary fluke).
---Sent: 2010-05-16 04:31:15 UTC
---Sent: 2010-05-16 04:45:27 UTC