On behalf of Jim Demmel -- Re: back to the roots
Please see the following communication from Jim.
Baker
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Regarding the justification of using high precision computation
when the inputs are known only to low precision, suppose you are
simulating a physical system that should conserve energy,
momentum or other quantities to many digits. To be able to
interpret the output as physically meaningful for some
approximate input, maintaining conservation accurately is
probably necessary. Consider climate modeling.
Whether an exact dot product is the right building block or not
is a different question.
We have been talking to one person running a climate model (Pat
Worley) who had a problem because global (literally) summations
he was computing were not accurate enough, in particular not
reproducible on his parallel machine. This led him to use a very
high accuracy summation, although it turned out that
reproducibility (despite different parallel summation orders at
different points in the code) with the "usual" double precision
accuracy was enough. We showed him how to get this
reproducibility and "standard" accuracy much faster, which seemed
to be enough for his example.
Whether or not an exact dot product was needed for this example,
the point is that (much) higher accuracy was needed for
intermediate results than the accuracy to which the input is
known.
Jim
PS I'm not sure I'm allowed to post to
sts-1788@xxxxxxxxxxxxxxxxx, so feel free to forward this if you
think it is relevant to the discussion.
On 7/1/13 5:44 AM, Vincent Lefevre wrote:
> On 2013-07-01 09:47:56 +0200, Ulrich Kulisch wrote:
>> <Assume you somehow got an approximate solution of a linear system
and you
>> want to improve it by a defect <correction. Then you assume of
course that
>> the data of the approximate solution are infinitely precise. Even if <no
>> digit of the "approximate" solution is correct you can with Rump's
method
>> compute a highly accurate <enclosure. Also here you assume that the
>> "approximate" solution is infinitely precise.
> You can assume that the approximate solution is infinitely precise
> because you can do this for free: this is just an assumption. But
> this doesn't mean that all the following computations need to be
> exact.
--
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Ralph Baker Kearfott, rbk@xxxxxxxxxxxxx (337) 482-5346 (fax)
(337) 482-5270 (work) (337) 993-1827 (home)
URL: http://interval.louisiana.edu/kearfott.html
Department of Mathematics, University of Louisiana at Lafayette
(Room 217 Maxim D. Doucet Hall, 1403 Johnston Street)
Box 4-1010, Lafayette, LA 70504-1010, USA
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