Daniel Lemire wrote:
>
>
> I'm sure there are bugs in Maple, Mathematica and Maxima, but I don't
> tend to believe that Maple will often give a wrong answer. The chances
> of that are much lower than me getting the wrong sign or something.
It is hard to know how accurate you are at calculation, but
the chance that one of these programs will make a blunder in
the middle of a long computation --- much too long to check by
hand --- is not zero.
I dug out of my files an old problem that has been examined in
the literature and is no stranger to the bug lists of Macsyma, Maple,
Mathematica. It dates back to 1989 if not earlier, and I think
may even have come out of a calculus textbook.
Integrate 1/sqrt(2-2*cos(x)) from x=-pi/2 to pi/2 .
Mathematica 4.1, without any warnings, gives a certain expression,
\!\(2\ Log[4] - 2\ Log[Cos[\[Pi]\/8]] + 2\ Log[Sin[\[Pi]\/8]]\)
which N[%] evaluates numerically to give 1.00984
Maxima 5.6 returns the integral unevaluated.
Macsyma (commercial version) says "integral is divergent"
Maple 7 says infinity.
Ten years ago these systems got other (wrong) answers.
(The integral is divergent, I believe).
Does this mean that Maple is always right even when
Mathematica is wrong? I doubt it, but I haven't played
that much with Maple 7.
While it is possible to argue that "not complete as possible"
and "wrong" are different, my guess is that the analysis done
on the singularities of the integral in Mathematica was, as you say,
not as complete as possible. The next step in
the solution resulted in an answer that was totally wrong. Thus
the distinction between incomplete and wrong evaporates
when you use the incomplete results in the next step. Since
these computer algebra systems are supposed to be used
for long chains of calculations in which humans do not
(cannot) check each result, correctness of the highest
order is important.
RJF