Re: MidRad to/from InfSup (was: the "set paradigm" is harmful)
On Wed, 11 Feb 2009 21:47:20 -0100, Michel Hack <hack@xxxxxxxxxxxxxx>
wrote:
> | If xx=[l,u] is compact, m and r shall have the values defined by
> | set round up
> | r = 0.5*(u-l);m=l+r.
That formula is due to Prof. Oishi from Waseda university, Tokyo. See
http://www.ti3.tu-harburg.de/paper/rump/OiRu02.pdf
I am sorry, I misread the text above. Oishi's method, as it is in
the above paper, is
setround up
m = (l+u)/2
r = m-l
It follows that m-r <= l AND u <= m+r.
So we what do make of John Pryce's example?
Input: [0.9993, 1.001] (format: 4-digit Decimal FP)
r = 0.5 * (1.001 - 0.9993) = 0.0008500 (exactly)
m = 0.9993 + 0.0008500 = 1.0001500 (exactly)
= 1.001 (rounded up)
Result: 1.001+-0.0008500 (decimal precision 4 digits)
Hovever, 1.001 +- 0.00085 does not cover the original lower bound,
as the exact range is 1.00015 to 1.00185 -- it is slightly shifted.
With the correct formula it looks as follows:
Input: [0.9993, 1.001] (format: 4-digit Decimal FP)
m = 0.5 * (0.9993 + 1.001) = 0.5 * 2.0003 (exactly)
-> 0.5 * 2.001 (rounded up)
= 1.0005 (exactly)
-> 1.001 (rounded up)
r = 1.001 - 0.9993 = 0.0017 (exactly)
= 0.0017 (rounded up)
Result: 1.001+-0.0017 (decimal precision 4 digits)
A correct inclusion, though overestimated.
Sometimes the following formula, though a little more expensive, yields
a better result. In the example above there seems no difference.
setround up
m = l + (u-l)/2
r = m-l
Now, the cited paper (after a very quick glance, I have to admit)
does not actually claim that: m-r <= l AND u <= m+r
It claims that a certain quantity X contained in [l, u] is also
contained in [m-r, m+r], and that the MidRad form is easier to
compute and yet as good (or nearly as good) as the InfSup form.
That is an entirely different matter.
No, the quantity X is an arbitrary quantity within [l,u], thus
X in [m-r,m+r] implies m-r <= X AND X <= m+r , and therefore
m-r <= l AND u <= m+r .
Best wishes
Siegfried M. Rump
--
=====================================================
Prof. Dr. Siegfried M. Rump
Institute for Reliable Computing
Hamburg University of Technology
Schwarzenbergstr. 95
21071 Hamburg
Germany
phone +49 40 42878 3027
fax +49 40 42878 2489
http://www.ti3.tu-harburg.de
and
Visiting Professor at Waseda University
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