Re: How to increase Working Precision?

On Mar 9, 1:36 pm, mcmcc...@xxxxxxxx wrote:

There is one point I'd like to make however. Most of
your writings on significance arithmetic include
examples where the user enters a number as a very long
string of digits or iterates a function many times or
performs some such trick to trigger Mathematica's
arbitrary precision routines, evidently baffling the...


In freshman physics class I learned to do signficance arithmetic when
combining measurements done with a meter stick, a thermometer, and
other inherently low precision devices. Maybe two adds and a
multiply, and you get an answer whose number of digits is
appropriate; you get penalized on the lab if you write down too many
digits. That is probably what you have observed with your calculus
(or other) undergraduate students.

If CAS are to be used for long sequences of calculations typical of
some kinds of scientific calculation, which are, I contend, more like
the iterations I've suggested than the 1-2 or 3 arithmetic ops I've
done in physics labs, then the arithmetic must be appropriate for such
calculations. That's where we disagree on what is appropriate.

As far as providing adequate arithmetic for Trefethen's challenges, I
suspect it is an additional credit to the providers of solutions in
Mathematica that they were not tripped up by Mathematica's
arithmetic; of course exact rational arithmetic is unaffected by the
issues here.

Furthermore I should point out that it is not impossible to simulate
better arithmetic in Mathematica: after each relevant arithmetic
operation one can re-set the accuracy of the result. This is the way
one can induce a convergent iterative process to converge when it
otherwise would not, using Mathematica. Thus anything that can be
phrased as a fixed-point problem or root-finding can be fed into a
clever program that resets accuracy, which presumably clever people
(like Dan L) would do, but J. Random User would not. Which is why
significance arithmetic is , in my view, hazardous for J. Random User.

RJF
.

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