Accuracy and error analysis (was Re: [Maxima] primes)

[C Y , Fri, 13 May 2005 08:34:53 -0700 (PDT)]:
> True.  If an answer is given at all, it needs to be reliable.  But I
> still fail to understand why the correct procedure (whatever it may be)
> cannot be an automated one.

Of course it can be automated to a degree.

However, I really believe that you have to understand where your
errors/uncertainties come from: in order to be correct, an error
analysis must take into account what terms are independent, how
uncertainties are correlated, and what kind of distribution describes
each term.  Once you have made all that explicit, automation is (in
principle) no problem any more.  I just don't see the user actually
taking the time to make all that explicit from the outset, and it is
certainly not even possible in the scheme you have in Mma.

> It might involve defining error procedures for every function in
> Maxima, but I just can't believe it's impossible unless proper error
> analysis is also impossible.

Ah, but there we run into problems: a correct error analysis of
f(g(x), h(x)), where x is the independent variable, must take into
account the correlations between the two arguments of f(), or else you
only end up with the kind of worst case bounds that Mma offers you.

BTW, the failures in Mma that persist at least to version 5 - the last
one I knew and checked - seem to imply that those error procedures are
not so easy to define in all cases.

> And since taking derivatives IS well defined, the computer can
> verify the correctness of this solution.

But if you are talking about derivatives, you are probably thinking
just of significance arithmetic where errors are all assumed to be
small and independent and everything is linearized.

One can certainly implement that - Mma does it, and it is not
difficult conceptually -, but what use is it?  Linearization means
that you often don't even get rigorous bounds; and neglect of
correlations leads to often dramatic loss of significance that has
nothing to do with the significance of the computation.  All this
comes at the prize of introducing a number of ugly work-arounds for
those limitations, and at a cost in computing time and memory that is
sufficiently high for the various means of circumventing significance
arithmetic to be a constant topic in MathGroup at the time I read it.

I am sure it is good for marketing, though.

At any rate, I would hope that any error-propagating scheme someone
might come up with is not integrated as tightly into Maxima as is the
case for Mma.

Regards,

Albert.