Re: (forwarded for Michel Hack): Re: complex intervals
> From: Paul Zimmermann <Paul.Zimmermann@xxxxxxxx>
> To: stds-1788@xxxxxxxxxxxxxxxxx
> Subject: Re: (forwarded for Michel Hack): Re: complex intervals
> Date: Sun, 16 May 2010 10:02:20 +0200
>
> thank you Michel for your detailed answer.
>
> > 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...
>
> I did not have in mind a fixed-precision format, but an arbitrary precision
> one, where usually there is no tight restriction on the exponent range.
> Please make sure the P1788 standard does not forget about variable or
> arbitrary precision!
>
> Paul Zimmermann
Paul,
Even though we have eliminated complex intervals from our
consideration, there are two forms you have not mentioned.
There is a polar form of radius interval together with angle
interval which might be useful for some transformations.
But the most general form for a multi dimentional object of
any kind would be an ellipsoidal form: center (a complex
number), angle of the major axis (a real number in [0,2pi]
or [-pi,pi]), major axis radius (a non-negative real number),
& minor axis (another non-negative real number). This last
has the property that it avoids the interval expansion caused
by distortion of the containment region due to conformal maps.
But Michel's exposition was far better than mine & I would
not post at all were it not for the fact that it reminded me
of something that touches on the mid-rad motion just made.
Michel mentions the problem of limited range. Or, to be more
precise, the problem that a radius represented in a smaller
(fixed) precision usually has a far more limited range than
the larger precision of the midpoint.
I am, therefore, led to ask if the mid-rad community ever
uses a RELATIVE rather than absolute error form. That is,
rather than an (x,eps) form that represents the interval
[x-eps,x+eps], do you use forms in which one stores (x,rel)
to represent [x - x*rel, x + x*rel]? (Special cases for
x = 0, I suppose.)
It would largely solve the range problem Michel cites &
have the property that the inf & sup are obtainable with
a single FMA operation on most modern machines. That is,
it is no worse than an add.
I ask for two reasons. (1) To see if it is in the literature.
And, (2) to ask ourselves if we should consider mid-rad forms
that are made available in 1788 to be of a more general type
than the mid-rad & mid-rad1-rad2 we are considering.
This last would have us rewriting the text of the current
motion when (& if) it works its way into the standard.
Just curious,
Dan