Re: Re: An arithmetic puzzle, equality of numbers.
- To: mathgroup at smc.vnet.net
- Subject: [mg103228] Re: [mg103175] Re: An arithmetic puzzle, equality of numbers.
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Fri, 11 Sep 2009 05:25:19 -0400 (EDT)
- References: <h829m8$3ue$1@smc.vnet.net> <4AA4BA87.3020500@gmail.com>
- Reply-to: drmajorbob at yahoo.com
Is there a mechanism in Mathematica (as it exists today) to Check a computation for 0-precision results? It might be better to have computations throw a Message (which we can turn off), for this, just as it does for division by zero. Bobby On Thu, 10 Sep 2009 06:17:11 -0500, Richard Fateman <fateman at cs.berkeley.edu> wrote: > Andrzej Kozlowski wrote: >> On 8 Sep 2009, at 11:55, Szabolcs Horv=E1t wrote: >> >>> You should have mentioned it in your message then. Otherwise you just >>> confuse beginners/newcomers, and that is not nice. >> >> But, in this particular case, entirely in accordance with long >> tradition on this forum. Google for Richard Fateman, MathGroup, >> Wolfram etc and then look up: >> >> http://en.wikipedia.org/wiki/Troll_(Internet) >> >> Andrzej Kozlowski= >> > > > I think a design that allows one to create objects such as i > that i==0 and i==2 simultaneously is problematical. Do you agree? > > Andrzej thinks this is fine. He thought it was fine in 2000. > > My message was to newcomers, who might not realize the hazards > of significance arithmetic. DanL's message has pointers to some useful > references, including Soufroniou's article, (incidentally, Soufroniou > works for WRI). It says > "There is no substitute for traditional methods of numerical analysis, > such as forward and backward analysis which provide tangible error > estimates." > > The conclusion also suggests significance arithmetic would be useful for > users who "are not experts in the analysis and construction of numerical > methods but are interested in investigating and solving problems, often > against the industrial backdrop of pressing deadlines." > > The question to me is whether significance arithmetic is the best > solution for these naive users or others, given that one consequence is > a system design that seems to produce, perhaps accidentally, and in the > context of those users who "are not experts" a number i such that > 0==i==1. > > (The advice that one should just use machine arithmetic -- which > certainly would avoid this pitfall -- also means that this automatic > error analysis method cannot be used.) > > There's another technique (not trivially foolproof as shown by "Rump's > polynomial" in Sofroniou's article. -- but read the article to make a > better decision via condition numbers.) > > Run your problem in fixed precision arithmetic with N digits. > Run it again in fixed precision with N+m digits [one suggested value is > m=sqrt(N)]. If the results are the same to about N digits, maybe you > have the right answer. You can repeat it for additional confirmation. > This may be faster than significance arithmetic once. Maybe you can > even do single, double, quad, quad-double, using mostly machine > hardware. If you need additional precision, it can be done in software > by simulating fixed precision in Mathematica (via SetPrecision to avoid > significance decay), or by use of a package like MPFR. > > Is this a better strategy for "semi-automated" numerical analysis? > Probably. > > The goal of a fully automated numerical quality error analysis program > is elusive. The use of interval arithmetic (google for Reliable > Computation), provides a rigorous but generally very pessimistic basis. > Many common standard computations in linear algebra, evaluation of > polynomials, etc. have been carefully analyzed and their errors can be > computed using regular arithmetic (Mathematica does this already). > > Does significance arithmetic contribute to improving the situation > for the naive user who is trying to meet a deadline and may not notice > answers or intermediate results with zero Precision? I think there is a > hazard. Hence the message directed to newcomers. > > RJF > -- DrMajorBob at yahoo.com