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Re: why extending numbers by zeros instead of dropping precision is a good idea

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On 4/4/2011 3:30 AM, Noqsi wrote:
> On Mar 31, 3:06 am, Richard Fateman<fate...@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? and where has it taken us to date?
Do you perhaps mean "automatic ERROR analysis"?

See, for example
http://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.


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.




>
>>
>>    It is, for example, the essence of Newton iteration.
>
> Convergence on a fixed point is a special case.

  If the so-called automated analysis only works on programs without 
loops, it would seem that it only works for its own special case, namely 
"programs without loops" or perhaps "programs where each
data element is used no more than once and there are therefore no
dependencies."  If, what you mean by automatic numerical analysis is 
simply interval arithmetic (or significance arithmetic), that would be a 
very grand re-naming of some techniques straining for effective 
application, even after decades of availability.

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Re: why extending numbers by zeros instead of dropping precision is a good idea Richard Fateman <fateman@cs.berkeley.edu> - 2011-04-06 09:12 +0000

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