MathGroup Archive 2005

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: Some bugs in Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg59606] Re: Some bugs in Mathematica
  • From: Bill Rowe <readnewsciv at earthlink.net>
  • Date: Sat, 13 Aug 2005 03:26:46 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

On 8/12/05 at 12:08 AM, akhmel at hotmail.com (Alex) wrote:

> This is what Mathematica does.

>\!\(Integrate[1\/\(r \(\@\( r\^2 - a\^2\)\) \@\(r\^2 - b\^2\)\),
>r]\)

>\!\(\(-\(\(\@\(1 - a\^2\/r\^2\)\ \@\(1 - b\^2\/r\^2\)\ AppellF1[1,
>1\/2,
>1\/2, 2, a\^2\/r\^2, b\^2\/r\^2]\)\/\(2\ \@\(\(-a\^2\) + r\^2\)\
>\@\(\(-b\^2\) +
>\ r\^2\)\)\)\)\)

>Now I claim that Mathematica fails here to do the simplification.
>After all, integration is the process of finding the
>anti-derivative. I show below that anti-derivative is:

>\!\(Simplify[
>D[\(1\/\(a\ b\)\) ArcTanh[\(b \@\( r\^2 - a\^2\)\)\/\(a \@\( r\^2 -
>b\^2\)\)],
>r]]\)

>\!\(1\/\(r\ \@\(\(-a\^2\) + r\^2\)\ \@\(\(-b\^2\) + r\^2\)\)\)

>Isn't my result much simpler than that of Mathematica? How come
>Mathematica can't figure it out?

In essence, for the same reason a human misses a particular simplification, i.e., Mathematica simply hasn't been told about that particular simplification.

Simplfy and FullSiimplfy have a finite set of transforms to apply to achieve simplification. If none of these result in a simplification, the result is returned in its original form. This is not one bit different than a human who tries to simplify an expression by every means he knows and stops when none of his attempts work.

>Whatever example I take, it seems like Mathematica gives inadequate
>results almost always. Here is another example:

>\!\(Integrate[x\/\(\(\@\(x\^2 - a\^2\)\) \@\(x\^2 - b\^2\)\), x]\)

>\!\(1\/2\ Log[\(-a\^2\) - b\^2 + 2\ x\^2 +
>2\ \@\(\(-a\^2\) + x\^2\)\ \@\(\(-b\^2\) + x\^2\)]\)

>As you see, in this case, Mathematica understood that it is
>logarithm and not an Appel function. Funny thing though is that the
>previous integral is exactly the same as this one. If we make a
>substitution, x = a b / r.

Do you really want Mathematica to try every possible gouping of variables in a complex expression that might result in a simpler expression? For arbitrary expressions, I suspect an attempt to do this would cause the execution time of FullSimplify to increase exponentially with complexity and it would be trivial to come up with examples that would lead to unacceptable execution times on even the fastest hardware.

Mathematica is simply a very useful tool. Like any tool, it has limitations and isn't perfect. Used intelligently, Mathematica produces useful results in reasonable time. Used blindly, you get pretty much what you deserve.
--
To reply via email subtract one hundred and four


  • Prev by Date: Re: Re: Mathematica goes Bad
  • Next by Date: Re: Strange statistics function integration
  • Previous by thread: Re: Some bugs in Mathematica
  • Next by thread: Re: Some bugs in Mathematica