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

[C Y , Fri, 13 May 2005 10:04:29 -0700 (PDT)]:
> I think I might have stated this wrong - what I want is for Maxima,
> given all relevant informtion about a data/equation set, to be able to
> do error analysis on the well specified input.  I'm guessing the
> question then comes down to what is "well specified", and if that's
> more than +-d I'm quite willing to have Maxima inquire about whatever
> it needs to know when accuracy is essential.  If I can't answer the
> questions, then it is implicit that I can't do proper error analysis
> either, since I don't understand my system fully.  That's what I'm
> after.

OK, that is certainly much more ambitious than what Mma gives you with
its bigfloats.  Moreover, if it can be made to work, this would
actually be useful to many.

And given the need to actually specify the uncertainty associated with
someinput, this analysis is certainly something the user has to ask
for explicitly, by calling some command; Mma instead tries to read the
user's mind by counting digits upon input.

This more ambitous task might also reduce some of the problems when
mixing numerical and symbolical calculations in Mma.  It seems that
you are after correctness, whereas Mma (in, e.g., the N... functions)
often just decides to pretend that some quantity is known with greater
accuracy than it is, which cannot possibly be correct.

> The user should be able to denote the correlations between the
> arguements.  I have no good vision for a syntax for this, probably
> because I have a ways to go to understand the problem properly, but
> I like the idea.

Well the obvious notation is to just take the joint probability
distribution of all independent quantities (including those that do
not enter the ``pure'' computation) as an additional argument to the
error-propagating function.

Of course this places quite a burden on the user, but it also means
that the user still has to make the difficult choices, and in
particular that she has to decide on the precise distribution to use:
Maxima will then only (try to) perform the routine part of the
analysis.  And for this, the uncertainty-affected quantities need not
even be numeric at all.

Whether to call this automated error analysis is another question - I
would view it more as providing a set of definitions for doing some
specific tasks that happen to be useful for error analysis.

Albert.