Re: How to increase Working Precision?

On Mar 8, 11:21 am, rjf <fate...@xxxxxxxxx> wrote:
Mathematica's notions of Precision and Accuracy are
not found in contemporary NA books; they are related
to significance arithmetic

Richard,

You've been criticizing significance arithmetic since
at least your very detailed review of 1990. I don't
really care to join that debate, as I'm not a numerical
analyst. Given that at least two teams with perfect
scores in Trefethen's SIAM 100 digit challenge used
Mathematica exclusively, I'll go ahead and assume that
Mathematica's numerics aren't terrible. (Incidentally,
the second of Trefethen's challenge problems most
definitely required the use of high-precision
arithmetic.)

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
confused user. Here's the thing, though: I've been
teaching undergraduate students to solve numerical
problems in calculus, differential equations, linear
algebra, and more recently numerical analysis using
Mathematica since version 1.2 and I don't believe I've
ever seen the types of problems you describe arise in
that setting.

That's not to say that these problems can't arise in
that setting but it's easy to see why they would be
rare. The simple fact of the matter is that exact
computations (like (Pi^2)/6) or machine precision
computations (like (Pi^2.0)/6.0) are simply much more
common. High precision computation is typically used
when one needs, well, high precision. It's not
unreasonable to expect a bit more of the user at that
point. As even you admit in your posts to this group
on July 12 of 2007, significance arithmetic "isn't an
entirely silly idea" in that context.

I wouldn't say that your complaints are completely
baseless. Personally, I'd like the user to be
able to control when arbitrary precision kicks in.
But I find your overall arguments against significance
arithmetic and indeed against Mathematica in general
to be uncompelling.

Mark
.

Relevant Pages

  • Re: Help Please: Transferspreadsheet brings in odd format
    ... Rounding of numerics in digital computers is often an issue. ... store numerics only to the required precision (Integer, ... possibility of giving an "e to the x format", ...
    (microsoft.public.access.conversion)
  • Re: very high order ODE solver
    ... So I guess I need an ODE solver of a high order, ... precision, but perhaps there are software for increased precision, like 256 ... several free software libraries extending numerics to high precision, ... Windows applications. ...
    (sci.math.num-analysis)
  • Re: very high order ODE solver
    ... So I guess I need an ODE solver of a high order, ... precision, but perhaps there are software for increased precision, like 256 ... several free software libraries extending numerics to high precision, ... used numerical scheme. ...
    (sci.math.num-analysis)
  • Re: How to increase Working Precision?
    ... arbitrary precision in applied situations is quite ... not 10 problems, each 10 digits. ... the machine-arithmetic subroutine library of Mathematica, ... iterations I've suggested than the 1-2 or 3 arithmetic ...
    (sci.math.symbolic)
  • Re: 0^0 oh no!
    ... you'd use SetPrecision to tell the system to ... in Mathematica one would typically use SetPrecision if coding ... For languages that work in fixed precision (I think ... think typically one would, at the finish of an iterative algorithm, ...
    (sci.math.symbolic)
Loading