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Re: Re: re: Accuracy and Precision
*To*: mathgroup at smc.vnet.net
*Subject*: [mg37198] Re: [mg37146] Re: [mg37130] re: Accuracy and Precision
*From*: Daniel Lichtblau <danl at wolfram.com>
*Date*: Wed, 16 Oct 2002 14:26:07 -0400 (EDT)
*References*: <200210110758.DAA01962@smc.vnet.net> <200210120904.FAA11982@smc.vnet.net> <200210130956.FAA18230@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Mark Coleman wrote:
>
> Greetings,
>
> I have read with great interest this lively debate on numerical prcesion and
> accuracy. As I work in the fields of finance and economics, where we feel
> ourselves blessed if we get three digits of accuracy, I'm curious as to what
> scientific endeavors require 50+ digits of precision? As I recall there are
> some areas, such as high energy physics and some elements of astronomy, that
> might require so many digits in some circumstances. Are there others?
>
> Thanks
>
> -Mark
>
> <snip>
I'm not sure what scientific endeavors might directly benefit. I can say
that alot of computational endeavors need high precision, and of course
some of these are used behind the scenes in scientific computation.
Below I list a few.
(i) Exact computation that uses approximation behind the scenes. Even
someting so mundane as integer division can fall into this category.
(ii) Finding relations among real or integer values can be done with
high precision arithmetic. For example, one can implement LLL lattice
reduction via approximate arithmetic. it's cousin, PSLQ, is entirely an
approximate arithmetic procedure and at times it requires high
precision.
Applications of these would include cryptography and the like, hence
this lies at least partly in the realm of applied math.
(iii) Some algorithms may have low precision input but require higher
precision at intermediate steps. An example is the method we use for
solving systems of multivariate polynomial equations. You can regard the
input as representing a "family" of problems (parametrized by the range
of fuzz implied by the low precision input). Clearly the raising of
precision in such circumstances is in some sense "artificial", insofar
as what we obtain is a solution to a particular member of the family
(actually to a narrow subfamily). However we also make some attempt to
detect ill conditioning; if the problem is well conditioned then
solutions to all members of the family will be reasonably near to the
one we obtain.
(iv) Some statistical functions may require fairly high precision behind
the scenes in order to obtain reasonable results for inputs that are not
outrageous. This can often be mollified by changing the algorithm used
but sometimes high precision is the simplest way to proceed.
(v) Computational geometry problems frequently become nongeneric
(collinear triples, coplanar quadruples, etc.). High precision
arithmetic can help to handle cases wherein nongenericity makes the
problem pathological (often perturbation or similar devices are also
needed).
(vi) High precision arithmetic can be used to advantage in numeric
integration to counter various ills such as cancellation error.
I'm sure there are oodles of other computational examples wherein high
precision saves the day. The moral is that, while scientific examples
rarely provide high precison input, methods of computation required by
scientists may well still require high precision arithmetic.
Also note that while financial forecasting may be blessed to get three
digits, other aspects of the financial world require much more. Around
10 years ago a bank investigated purchasing Mathematica. Apparently they
wanted to be certain they had amounts figured to better than the nearest
penny (or so I heard). When working with exchange rates I suppose this
could be important; crude rounding might allow for those weird "secrets
of the rich in currency trading" e-mails I keep getting to actually
work. Wanna buy a Euro from me?
Daniel Lichtblau
Wolfram Research
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