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Bill, 10²⁰⁰
+
23456 - 10²⁰⁰. (1) If your
computer provides an EDP you get the correct answer 23456 and
if the EDP is
supported by hardware you get it very fast. If your
computer does not provide an EDP the average user will
accumulate (1) in
conventional floating-point arithmetic and he gets the wrong
answer 0. I still
remember the old days of computing when every computer had its
own arithmetic
which often was not best possible. Computer users, vendors,
and even
mathematicians often argued that a more accurate arithmetic is
not needed since
the underlying data are imprecise, or discretization errors
are much larger
than rounding errors. This kind
of justification is rather dubious. One source of error does
not justify adding
another one. Even an imprecise mathematical model or imprecise
data will suffer
from the use of an imprecise or sloppy arithmetic. The
systematic development
of a mathematical model requires that the error resulting from
the computation
can largely be excluded. This requires the best possible
arithmetic. What
happens outside the computer is the responsibility of the
user. As soon as the
data are in the computer treating them as exact is a must. The
vendor is
responsible for the arithmetic that is used in the computer.
To be as accurate
as possible is also a must. External and internal error
sources must be
separately identifiable. The dot product is a very frequent
arithmetic
operation which easily can be computed exactly. Not providing
this operation on
computers is the source of many avoidable errors.
Am 20.09.2013 01:15, schrieb G. William (Bill) Walster: Great, Ulrich. When you do answer, you might want to explain how it is that computing EDPs will help computing with intervals to address the following problem <http://www.scientificamerican.com/article.cfm?id=puzzling-measurement-of-big-g-gravitational-constant-ignites-debate-slide-show&WT.mc_id=SA_CAT_SPCPHYS_20130919> It seems to me that this is a prime example of the kind of problem on which the interval computational and research communities should concentrate their efforts. We have a unique opportunity in situations like this. Cheers, Bill On 9/18/13 8:37 PM, Ulrich Kulisch wrote:Bill, I shall answer your question if I have more time. For the moment take the present discussion which started with computing a dot product for vectors with interval data. I have to leave for a meeting in one hour. Best regards Ulrich Am 18.09.2013 20:02, schrieb G. William (Bill) Walster:Ulrich, Lets assume everything you say about EDP is correct. Just because EDP is a fundamental mathematical operation does not mean that it should be required in an interval arithmetic standard. Requiring EDP in an interval arithmetic standard is a separate issue. As I have repeatedly written and received no answer from you or anybody else, please provide a single example of an interval computation based on non-degenerate interval inputs that substantially benefits from the availability of EDP. Thanks in advance, Respectfully, Bill -- Karlsruher Institut für Technologie (KIT) Institut für Angewandte und Numerische Mathematik D-76128 Karlsruhe, Germany Prof. Ulrich Kulisch Telefon: +49 721 608-42680 Fax: +49 721 608-46679 E-Mail: ulrich.kulisch@xxxxxxx www.kit.edu www.math.kit.edu/ianm2/~kulisch/ KIT - Universität des Landes Baden-Württemberg und nationales Großforschungszentrum in der Helmholtz-Gesellschaft |