Re: why extending numbers by zeros instead of dropping precision is a good idea
- To: mathgroup at smc.vnet.net
- Subject: [mg117981] Re: why extending numbers by zeros instead of dropping precision is a good idea
- From: Noqsi <noqsiaerospace at gmail.com>
- Date: Thu, 7 Apr 2011 08:05:00 -0400 (EDT)
On Apr 6, 3:12 am, Richard Fateman <fate... at cs.berkeley.edu> wrote: > On 4/4/2011 3:30 AM, Noqsi wrote: > > > On Mar 31, 3:06 am, Richard Fateman<fate... at eecs.berkeley.edu> wrote: > >> It is occasionally stated that subtracting nearly equal quantities from > >> each other is a bad idea and somehow unstable or results in noise. (JT > >> Sardus said it on 3/29/2011, for example.) > > >> This is not always true; in fact it may be true hardly ever. > > > Hardly ever? What a silly assertion. This has been a major concern > > since the dawn of automatic numerical analysis. > > When was this dawn? Oh, if you want a date, February 14, 1946 will do, although anyone who knows this history can argue for earlier or later as they please. > and where has it taken us to date? Lots of places. The Moon, for example. > Do you perhaps mean "automatic ERROR analysis"? No. I mean automatic numerical analysis, as opposed to manual methods (Gauss, Adams, Richardson, ...). But I assume these folks were aware of significance issues, and handled them in informal intelligent human ways. Computers are stupider, and more capable of propagating error to the point of absurdity, so they require more care from their human tenders. > > See, for examplehttp://www.cs.berkeley.edu/~wkahan/Mindless.pdf > > or if you can find it, > W.M. Kahan, "The Regrettable Failure of Automated Error Analysis," > mini-course, <i>Conf. Computers and Mathematics,</i> Massachusetts Inst. > of Technology, 1989. I was thinking more of H. H. Goldstine and J. von Neumann, "Numerical inverting of matrices of high order", Amer. Math. Soc. Bull. 53 (1947), 1021-1099 although appreciation of the problem goes back farther (why did the ENIAC have ten digit accumulators?). > > Oh, as for subtraction being a problem... Typically if you subtract two > nearly equal items you get a small quantity. The small quantity is not > troublesome if you add it to something that is not so small. What > sometimes happens is that you do something ELSE. Like divide by it. > That is more likely to cause problems. Exactly. That's a very common issue in numerical analysis. And that's why your "hardly ever" assertion is silly.