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Re: Zero does not Equal Zero is a feature

These difficulties have been in Mathematica since
its first design, and although there have been a
number of redesigns of the arithmetic, it
retains the element of treating floating point
numbers as though they were intervals.
Careful numerical analysis is hindered by this
design, which was intended to be used by non-careful
sloppy people.  But the design has some flaws
which make it necessary for people who know
what they are doing, to re-learn a new and not
very comfortable calculus of numbers, at
least if they want to use Mathematica. It also
interferes with sloppy calculation.
One choice of course is not to use Mathematica
for numerical calculation. There are other
arbitrary precision packages.
Another choice is to test, and reset the
accuracy of numbers at critical points.
Realize, for example, that certain convergent
iterations will not produce more accurate
answers (as is normal), but will produce
vaguer answers at each step because of Mathematica's
arithmetic. They will terminate not when answers
are close, but when they are essentially unknown.

You could also complain about the arithmetic,
but that will probably not have much effect.
It is a "feature".  My (1982) review of Mathematica
made some of the same observations.
Richard Fateman

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