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Re: Numerical accuracy/precision - this is a bug or a feature?

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  • Subject: [mg120268] Re: Numerical accuracy/precision - this is a bug or a feature?
  • From: Andrzej Kozlowski <akoz at>
  • Date: Sat, 16 Jul 2011 05:40:22 -0400 (EDT)
  • References: <ius5op$2g7$> <ius7b6$30t$> <iv9ehj$dct$> <ivjgfp$2b9$> <> <> <> <>

I should have added that actually Mathematica's significance arithmetic 
is considerably more sophisticated than the description as a "first 
order" approximation suggests. This is also explained by Sofroniou and 

Numerical algorithms for computing elementary functions can be written 
in terms of addition and multiplication at some level. However, relying 
on the error propagation rules for these operations would often give 
very pessimistic error bounds in significance arithmetic. Much tighter 
bounds can be obtained by directly imposing error estimates based on 
properties of functions during their numerical computation.

In other words, Mathematica uses special estimates for a variety of 
functions, based on properties of these functions and thus obtains bounds 
are considerably tighter than a simple minded approach would produce.

Andrzej Kozlowski

On 15 Jul 2011, at 10:09, Andrzej Kozlowski wrote:

> You are clearly confusing significant digit arithmetic, which is not
> what Mathematica uses, with significance arithmetic, which is a first
> order approximation to interval arithmetic or distribution based
> approach. Obviously you don't read the posts you reply to and confuse
> both the posters and the contents of what they post. Here is a quote
> from Oleksandr Rasputionov that makes this completely clear:
> Unfortunately so, given that it is severely erroneous: see e.g.
> <>. However, Mathematica's
> approximation of how these uncertainties propagate is first-order, not
> zeroth-order. This does not make it completely reliable, of course, but
> certainly it is not almost always wrong as is the significant digits
> convention. Within the bounds of its own applicability, Mathematica's
> approximation is reasonable, although it would still be a mistake to apply
> it to experimental uncertainty analysis given the much broader scope of
> the latter.
> Note the "first-order not zeroth-order". Also, do take a look at Sofroniou and Spaletta and then you may perhaps understand what "order" means and how "significance arithmetic" differs from the "significant digits" convention. Good grief, did you ever learn about the Taylor series? It must have been a long time ago, I take.
> Andrzej Kozlowski
> On 14 Jul 2011, at 19:55, Richard Fateman wrote:
>> On 7/14/2011 6:27 AM, Andrzej Kozlowski wrote:
>>> On 14 Jul 2011, at 11:21, Richard Fateman wrote:
>>>> On 7/13/2011 12:11 AM, Noqsi wrote:
>>>> ..
>>>>>> see e.g.
>>>>>> <>.
>>>> ..
>>>> Learning mathematics from a physicist is hazardous.
>>>> Learning computer science from a physicist is hazardous too.
>>>> Numbers in a computer are different from experimental measurements.
>>>> nevertheless, I like this article.  It says, among other things,
>>>> 	The technique of propagating the uncertainty from step to step
>>>> throughout the calculation is a very bad technique. It might sometimes
>>>> work for super-simple textbook problems but it is unlikely to work for
>>>> real-world problems.
>>> Well, here is a quote from a very well known book on numerical
> analysis by a mathematician (Henrici, "Elements of Numerical Analysis").
>>>> It is plain that, on a given machine and for a given problem, the local
>>>> rounding errors are not, in fact, random variables. If the same problem
>>>> is run on the same machine a number of times, there will result always the
>>>> same local rounding errors, and therefore also the same accumulated
>>>> error. We may, however, adopt a stochastic model of the propagation of
>>>> rounding error, where the local errors are treated as if they were random
>>>> variables. This stochastic model has been applied in the literature to a
>>>> number of different numerical problems and has produced results that are
>>>> in complete agreement with experimentally observed results in several
>>>> important cases.
>>> The book then describes the statistical method of error propagation of which Mathematica's approach can be regarded as a first order approximation (as pointed out by Oleksandr Rasputinov, who should not be confused with the OP of this thread so:
>> So are we to conclude that Henrici recommends this as a general  numerical computational method?
>> I don't see that here.  I think what he is saying is that if you do some mathematics (see below), then
>> you will get results consistent with what you will get if you actually run the experiment on the computer.
>> This is not surprising. It is a result that says that "theoretical" numerical analysis agrees with
>> "computer experiments" in arithmetic.  It doesn't say Henrici recommends running a computation this way.
>> When Henrici says "adopt a stochastic model ...."  he doesn't mean to write a program. He means to
>> think about each operation like this..  (I show for multiplication of numbers P and Q with errors a b resp.)
>> P*(1+a)  times Q*(1+b)   P*Q *(1+a)*(1+b)* (1+c)   where c is a new "error" bounded by roundoff, e.g. half unit in last place.
>> For each operation in the calculation, make up another error letter... a,b,c,d,e,f,g...
>> assume they are uncorrelated.
>> The fact that this theoretical approach and numerically running "several important cases" is a statement
>> about correlation of roundoff in these cases, not a statement of advisability of whatever for a model of
>> how to write a computer system.
>> By the way, I think that Henrici was an extremely fine theoretical numerical analyst, and a fine writer too.
>>>> Clearly Rasputinov thinks that if they are not equal they should not be
>>>> Equal.  Thus the answer is False.
>>> is *clearly* False. In fact Oleksander expressed something closer to the opposite view.)
>> This thread is too long.   I don't know at this point if you are agreeing that it is false or contradicting that "is False" is false.
>>> And if one quote is not enough, here is another, from another text on numerical analysis. (Conte, de Boor, ""Elementary Numerical Analysis).
>>> It describes 4 approaches to error analysis, interval arithmetic, significant-digit arithmetic, the "statistical approach" and backward error analysis. Here is what it says about the second and the third one:
>> Huh, if we are talking about the second and third one, why does he say third and fourth?
>> Are you using 0-based indexing and deBoor is using 1-based indexing????
>>>> A third approach is significant-digit arithmetic. As pointed out earlier, whenever two nearly equal machine numbers are subtracted, there is a danger that some significant digits will be lost. In significant-digit arithmetic an attempt is made to keep track of digits so lost. In one version
>>>> only the significant digits in any number are retained, all others being discarded. At the end of a computation we will thus be assured that all digits retained are significant. The main objection to this method is that some information is lost whenever digits are discarded, and that the results obtained are likely to be much too conservative. Experimentation with this technique is still going on, although the experience to date is not too promising.
>> OK, so deBoor (who is retired and therefore not likely to revise the "experience to date")  says this method  "is not too promising".
>> This sounds to me like he is not endorsing what Mathematica does.
>>>> A fourth method which gives considerable promise of providing an adequate mathematical theory of round-off-error propagation is based on a statistical approach. It begins with the assumption that round-off errors are independent. This assumption is, of course, not valid, because if the same problem is run on the same machine several times, the answers will always be the same. We can, however, adopt a stochastic model of the propagation of round-off errors in which the local errors are treated as if they were random variables. Thus we can assume that the local round-off errors are either uniformly or normally distributed between their extreme values. Using statistical methods, we can then obtain the standard devia- tion, the variance of distribution, and estimates of the accumulated round- off error. The statistical approach is considered in some detail by Ham- ming [1] and Henrici [2]. The method does involve substantial analysis and additional computer time, but in the 
> periments conducted to date it has obtained error estimates which are in remarkable agreement with experimentally available evidence.
>> deBoor is essentially quoting Henrici, and this statistical approach is to say that all those error terms I mentioned above,  a,b,c,d,e,f,...
>> can be chosen from some distribution   (the way I've written it,  a ,....,z ....   would essentially be chosen from {-u,u} where u  == 2^(-W) where the fraction part of the floating-point number is W bits.  )   and you can compute the final expression as ANSWER+<somehorrendousfunctionof>(a,b,c,....).
>> What deBoor says is that this (theoretical numerical analysis) "method" promises to provide
>> a theory of round-off error propagation.   He is not saying this is a practical method for scientific computing.  When he uses the work "method"
>> he means a mathematical method for analyzing roundoff.
>> In any case, Mathematica does not do this.  I would further argue that Mathematica makes it hard to carry out the experiments that might be done to demonstrate that  this theory applied in any particular sample computation.
>>> The fundamental paper of Mathematica's error propagation is "Precise numerical computation" by Mark Sofroniou and  Giulia Spaletta in The Journal of Logic and Algebraic Programming 64 (2005) 1139=6134. This paper describes Mathematica's "significance arithmetic" as a first order approximation to Interval Arithmetic. It makes no mention of distributions.
>> I thought I read this paper in some Mathematica documentation or conference proceedings.
>>> Oleksandr Rasputionov, in an earlier post here, interpreted  "significance arithmetic" as a first order approximation to the fourth method above.
>> Huh? First of all, the original poster was slawek.  Rasputinov seems to think that Mathematica numbers are like Intervals  (basically a good intuition until you think about equality.) and refers to them as distributions.  This is not deBoor's 4th "method" of theoretically analyzing round-off.
>> In fact it is deBoor's 1st method, interval arithmetic.  This has the advantage of being maybe 4 to 8 times slower than regular arithmetic, and also has a huge literature  (see "Reliable Computation") describing variations, advantages, disadvantages, etc.
>>> I have not considered this very carefully, but it seems pretty clear that he is right, and that the two "first order" approximations are in fact isomorphic.
>> Uh, this is unclear, unless you mean that Mathematica's number system is essentially interval arithmetic, but with a confusing front end.
>>> The first order approach is, of course, justified on grounds of performance. It is perfectly "rigorous" in the same sense as any "first order" approach is (i.e. taking a linear approximation to the Taylor series of a non-linear function). It works fine under certain conditions and will produce nonsense when these conditions do not hold.
>>> The fact that significance arithmetic is "useful" needs no justification other than the fact that it is used successfully by NSolve and Reduce in achieving validated symbolic results by numerical methods which are vastly faster than purely symbolic ones.
>> Really?
>> 1. This does not mean that other methods, e.g. validated methods for accurately evaluating polynomials (etc)  NOT based on significance arithmetic would not be faster and better.  DanL claims it is used and useful there, so that's nice.  BUT..
>> 2. This does not mean that this arithmetic should be used by default by user arithmetic.
>>> It is also useful, for users such as myself, who sometimes need fast first order error anlysis.
>> OK, If you find it useful yourself, fine. I daresay you are not a typical user.
>>> I have lots of posts by Richard on this topic (or perhaps it was the same post lots of time, it's so hard to tell), but I have never understood what his main point is.
>> Main points:  Mathematica's number system is non-standard, peculiar, hard to explain,  capable of returning mysterious non-answers without indications of error to innocent users, a bad foundation for building higher-level functionality.
>> I have other criticisms of Mathematica, but I think that the sentence above is enough for you to process today.
>>> It seems to me that is because he himself has not yet decided this, although he has been posting on this topic for over 20 years (I think).
>>> Sometimes he seems to be disparaging significance arithmetic itself.
>> I think deBoor did that.
>>> When Daniel points out how effective it is in his implementation of numerical Groebner basis, or in Adam Strzebonski's work on Reduce, he either ignores this altogether or claims that Groebner bases, etc.  are themselves not "useful".
>> I think Reduce is a very nice program when it works. If I am not mistaken, all work on numerical Groebner basis should be reducible to the evaluation of polynomials, for which there are faster and more accurate methods available not using significance arithmetic.  On the other hand, I might be mischaracterizing DanL work, since I admit to not having studied it.
>>> On other occasions he takes on the role of the defender of the interest of the "naive user" (presumably like the OP, who however would be better described as "intentionally naive") who is going to be confused by the "quirky" nature of significance arithmetic (at low =
>> No, I don't think that slawek was a "troll".  I think he was =
genuinely confused, as might anyone be who has some prior computer =
arithmetic exposure and for the first time encounters a system with =
exact rational arithmetic.
>>> In doing so he conveniently ignores that fact of the existence of =
thousands of "naive users" who never become confused (sometimes because =
they always work with machine precision numbers and only use =
significance arrhythmic unknowingly, e.g. when using Reduce).
>> Most naive users don't tread near the dangerous spots. Some naive =
users never notice that their answers are nonsense.
>> Every so often we get a naive user who DOES notice, and he/she sends =
email to this newsgroup.
>>> And moreover, for those who do find a need for some sort of error =
analysis he offers no alternative, except perhaps to learn backward =
error analysis.
>> Actually, there's a whole bunch of libraries of methods with error =
estimates for common computations, where backward error analysis or some =
other method was used to provide extra information.
>>> Except, of course, that should they do so they would no longer be =
"naive" and thus outside of Richard's area of concern.
>> They would probably not be writing in Mathematica.
>>> And in any case, anyone who needs and understands backward error =
analysis can use it now, and can't imagine that even Richard would claim =
that reducing users' options is a good thing.
>> I have no problem with Mathematica providing as an option, some other =
arithmetic.  It is WRI that has made it rather hard for the user to
>> figure out how to  "use the arithmetic of the rest of the world".
>>> Finally, perhaps all that Richard is so upset about is simply =
Mathematica's habit of defining numbers as "fuzz balls" or =
>> I'm not sure that "upset" is the right term.  After all, I don't have =
to use Mathematica for numerical computation. And I rarely do.
>>> In other words, if Mathematica used a more usual "definition" of =
number, and used significance arithmetic for "error" propagation, that =
would be done by applying some separate function or turning on an =
option,  than everything would be fine?
>> Actually, that's pretty close to correct.
>>> If that is all, than it seems to me that Richard has for years been =
making mountains of molehills.
>> So why are you (and WRI)  so opposed to this notion?  Note that it =
would also have to affect other parts of the system including of course,
>> Equal and friends.
>>> Andrzej Kozlowski

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