RE: RE: Accuracy and Precision

• To: mathgroup at smc.vnet.net
• Subject: [mg36989] RE: [mg36936] RE: Accuracy and Precision
• From: "DrBob" <drbob at bigfoot.com>
• Date: Fri, 4 Oct 2002 05:01:33 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Daniel,

>>The precision/accuracy tracking mechanism will generally let you know,
in some fashion, that you have no trustworthy digits. But it is up to
the user to check that sort of thing.

In this case Mathematica did NOT let us know, in any fashion, that we
had no trustworthy digits.  Precision and Accuracy outputs were
completely misleading.  (16 and -5 respectively.)  Even Andrzej
Kozlowski, who's adept in Mathematica, thought that would be meaningful,
and never came up with a better way to check (other than using infinite
precision for numbers that probably aren't known that exactly).  Peter
Kosta demonstrated that he could get a completely erroneous answer with
Infinite precision.

I blame the problem primarily, and I don't think there's any way to make
the answer meaningful.  That's not Mathematica's fault at all, and users
need to be aware of that old maxim: "garbage in, garbage out".

Still, if the Kernel adds 37-digit numbers with 16-digit precision and
comes up with a 22-digit result, it doesn't take much sophistication to
realize the answer can't have 16-digit precision.

Here's an even more extreme result:

f = SetAccuracy[333.75*b^6 + a^2*(11*a^2*b^2 - b^6 -
121*b^4 - 2) + 5.5*b^8 + a/(2*b), 50];
a = 77617.; b = 33096.;
f
Precision[f]

-1.180591620717411303424`71.0721*^21
71

71.0721 digits of precision?  I don't think so!!

We can do the following instead:

x = Interval[333.75];
y = Interval[5.5];
a = Interval[77617.];
b = Interval[33096.];
x*b^6 + a^2*(11*a^2*b^2 - b^6 - 121*b^4 - 2) + y*b^8 + a/(2*b)

Interval[{-4.486248158726164*^22, 4.2501298345826815*^22}]

and that looks like the right answer, finally!!  I like that method.

However, that doesn't change the fact that Accuracy, Precision, and
SetAccuracy appear to be completely useless.  I haven't seen an example
in which they did what anyone (but you) thought they should do.

Bobby

-----Original Message-----
From: danl at wolfram.com [mailto:danl at wolfram.com]
To: mathgroup at smc.vnet.net
Subject: [mg36989] Re: [mg36936] RE: Accuracy and Precision

DrBob wrote:
>
> I think I'd prefer that Mathematica gave me a clue -- without being
> asked explicitly in JUST the right way that only you know and had
> forgotten -- that there's a precision problem.
>
> Oddly enough, when you DO ask it nicely, you get error messages, but
if
> you're not aware there's a problem, it lets you go on your merry way,
> working with noise.
>
> Bobby

Mathematica is not a mind reader. But the evaluation sequence, while
complicated, is reasonably well documented. If you perform machine
arithmetic, or for that matter significance arithmetic, and there is
massive cancellation error, no use of SetAccuracy after the fact will
fix it.

The precision/accuracy tracking mechanism will generally let you know,
in some fashion, that you have no trustworthy digits. But it is up to
the user to check that sort of thing. It is not obvious to me what sort
of "error" the software might notice to report. If you have a concise
example of input, and expected output, I can look further. I've not seen
anything in this thread that struck me as a failure of the software to
warn the user, but maybe I missed something.

Daniel

```

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