Re: Testable Predictions by HdB

In article <87k6h111sl.fsf@xxxxxxxxxxxxx> "Jesse F. Hughes" <jesse@xxxxxxxxxxxxx> writes:
> Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:
....
> > Maybe I suggested that. But since modern philosophy doesn't understand
> > common speech very well, we have to be _extremely_ careful. Therefore:
> >
> > The numbers in the Theory are real numbers: r \in R .
> > The numbers in the Tests are: floating point numbers.
> > But:
> > Floating point numbers <> Real numbers
....
> You mean that the fact that you test a theorem about real numbers by
> doing a calculation involving floating point numbers is a good thing?

It is indeed the case that some people think that is a good thing.
They do not realise that different floating point systems have
different properties, so what may be refuted by one system, will not
be refuted by another system. Moreover, to refute a theorem in a
floating point system you need to know how it works exactly, and
that gives you the ability to construct examples that contradict a
theorem in reals when applied to floating point systems. With random
searching you will almost never get a wrong result.

Instructive in this is the triangle inequality which was referred to
by Han. See the thread:
<http://groups.google.nl/group/sci.math.num-analysis/browse_thread/thread/9a9c72c1513e8711/d8bc07b9b8bb6003>
for a reasonably short discussion (only 21 articles). Random search
will in general *not* show failure in a particular f-p system, and
constructed counter-examples in some f-p system are *not* in general
counter-examples in other f-p systems. (And, of course, compiler
optimisation can also play a role.)

Han's paper has a theorem that there are two stable numbers in his
iteration x = 1/(x - 1). We all know that that is mathematically
true. On the other hand, when you try to test it on a floating
point system you will find that either 0, 1 or 2 of those numbers
are stable in the floating point system, depending on hardware and
software used. (I think I can construct valid floating point
systems for either of the three possibilities.) Darn, there are
floating point systems where Newton-Raphson has an extremely large
number of fixed points for the square root calculation (see the
thread above). Perhaps I can even construct a floating point
system where more than 2 fixed points can be shown on his iteration.
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