Re: why extending numbers by zeros instead of dropping precision
- To: mathgroup at smc.vnet.net
- Subject: [mg118023] Re: why extending numbers by zeros instead of dropping precision
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Sat, 9 Apr 2011 07:11:37 -0400 (EDT)
Here's a result in Mathematica for J-M. Muller's Recurrence, section 5 of
the Kahan paper:
Clear[e]
e[y_, z_] := 108 - (815 - 1500/z)/y
e[{last_, previous_}] := {e[last, previous], last}
digits = 80;
Block[{$MinPrecision = digits, $MaxPricision = digits},
N@First@Nest[e, {SetPrecision[4.25, digits], 4}, 80]
]
5.
(Correct.)
With just ONE less digit, however:
digits = 79;
Block[{$MinPrecision = digits, $MaxPricision = digits},
N@First@Nest[e, {SetPrecision[4.25, digits], 4}, 80]
]
100.005
(The spurious "attractor".)
Bobby
On Fri, 08 Apr 2011 03:14:02 -0500, Richard Fateman
<fateman at cs.berkeley.edu> wrote:
> On 4/7/2011 5:05 AM, Noqsi wrote:
>> 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.
>
> Oh, I see, by "automatic numerical analysis" you mean "numerical
> computation by some mechanical calculation apparatus [automaton??]"
>
> I was assuming you had in mind some kind of "automatic" analysis as
> opposed to the "human" analysis which is commonly used in order to
> invent algorithms and write programs which then do numerical calculation.
>>
>>> and where has it taken us to date?
>>
>> Lots of places. The Moon, for example.
>
> Oh, if you mean scientific numerical digital computing, sure. Though
> you've got to wonder how the Russians, probably WITHOUT much in the way
> of computers, put up an artificial satellite.
>>
>>> 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.
> Oh, so they weren't automatic, but handled in some human way...
>
> Try Wilkinson?
> Computers are stupider, and more capable of propagating error to
>> the point of absurdity, so they require more care from their human
>> tenders.
> more care than what? I think that large numbers of calculations done
> by "hand" or by mechanical calculators (as in pre-electronic days) would
> propagate errors too; they would also include, to the extent that
> humans were involved in transferring numbers to/from paper, and keying
> them in, more checking for blunders.
>
>>
>>>
>>> See, for example http://www.cs.berkeley.edu/~wkahan/Mindless.pdf
>
> I highly recommend you look at the above reference.
>
>
>>>
>>> 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
>
> This was "automatic numerical analysis"?
>>
>> although appreciation of the problem goes back farther (why did the
>> ENIAC have ten digit accumulators?).
>
> You might refer to MANIAC, which had significance arithmetic, and also
> lost out.
>>
>>>
>>> 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.
>
> I recommend you look at the Kahan paper, which would explain to you this
> issue.
>
>
>>
>
>
--
DrMajorBob at yahoo.com