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Mathematica numerics and... Re: Applying Mathematica to practical problems

Others have commented on most issues raised in this subthread. I
wanted to
touch on just a few details.

On May 31, 2:15 am, Richard Fateman <fate... at> wrote:
> On 5/30/2013 3:09 AM, John Doty wrote:
> > Changing the topic here.
> > On Tuesday, May 28, 2013 1:49:00 AM UTC-6, Richard Fateman wrote:
> >> Learning Mathematica (only) exposes a student to a singularly
> >> erroneous model of computation,

I assume you (as ever) refer to significance arithmetic. If so,
while it is many things, "erroneous" is not one of them. It
operates as designed. You have expressed a couple of reasons
why you find that design not to your liking. The two that most
come to mind: bad behavior in iterations that "should" converge,
and fuzzy equality that you find to be unintuitive 9mostly this
arises at low precision).

> > A personal, subjective judgement. However, I would agree that
> > exposing the student to *any* single model of computation, to the
> > exclusion of others, is destructive.
> Still, there are ones that are well-recognized as standard, a common
> basis for software libraries, shared development environments, etc.
> Others have been found lacking by refereed published articles
> and have failed to gain adherents outside the originators. (Distrust
> of significance arithmetic ala Mathematica is not a personal
> subjective opinion only.).

No, but nor is it one that appears to be widely shared. As best I can
tell, most people in the field either do not write about it, or else
make the observation, correctly, that it is simply  a first-order
approximation to interval arithmetic.

As such, it has most of the qualities of interval arithmetic. Among
these are the issue that results with "large" intervals are sometimes
not very useful. Knowing that one ended up with a large interval,
can be useful: it tells one that either the problem is not well
conditioned, or the method of solving it was not (and specifically, it
may have been a bad idea to use intervals to assess error bounds).

Significance arithmetic brings a few advantages. One is that computations
 are generally faster than their interval counterparts. Another is that it
is "significantly" easier to extend to functions for which interval
methods are out of reach (reason: one can compute derivatives but cannot
always find extrema for every function on every segment). A third is that
it is much easier to extend to complex values.

A drawback, relative to interval arithmetic, is that as a first-order
approximation, in terms of error propagation, significance arithmetic
breaks down at low precision where higher-order terms can cause the
estimates to be off. I will add that, at that point, intervals are going
to give results that are also not terribly useful.

I'm not sure what are the refereed articles referred to above. I suspect
they do not disagree in substantial ways with what I wrote though.  That
is to say, error estimates may be too conservative, and at low precision
results may not be useful.

Fixed precision has its own advantages and disadvantages in terms of speed,
error estimation, and the like. We certainly use it in places, even if
significance arithmetic is the default behavior of bignum arithmetic in
top-level Mathematic. For implementation purposes we use both modes.

> > Mathematica applied to real problems is pretty good here.
> Maybe, but is "pretty good"  the goal, and the occasional identified
> errors be ignored?

We do try to fix many bugs that are brought to our attention. We have a
better track record in some areas than others. I have no reason to
believe that numerics issues have received short shrift though.

Daniel Lichtblau
Wolfram Research

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