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Integration capabilities

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Protagoras

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Sep 4, 2003, 6:23:40 PM9/4/03
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I would be interested to find out which major CAS has the best
capabilities for solving problems associated with integration. For
example, Axiom is supposed to have a complete implementation of Risch's
algorithm, which would make it the leader, I guess, when it comes to computing
antidifferentiations. What about integration over intervals of the real line
with singular points?

Richard Fateman

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Sep 5, 2003, 2:14:49 AM9/5/03
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I think you would be better off if you had some
specific examples to post here.

Your belief about Axiom is, I think, mistaken.

There are also problems which can be integrated, but not
in terms of elementary functions, which the usual Risch
algorithm will not find.

You can try out mathematica's integration program at
integrals.wolfram.com

There are probably demo sites for other CAS that you can
access if you ask around.

RJF

Protagoras

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Sep 5, 2003, 10:47:01 AM9/5/03
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On Fri, 05 Sep 2003 06:14:49 +0000, Richard Fateman wrote:

> I think you would be better off if you had some
> specific examples to post here.

I have no specific examples. I am just interested to learn what each
system can, or cannot, do in principle.

> Your belief about Axiom is, I think, mistaken.
>
> There are also problems which can be integrated, but not
> in terms of elementary functions, which the usual Risch
> algorithm will not find.

That's what I had in mind. The rumor I heard is that if you give to
Axiom some elementary function for antidifferentiation, and it is unable to
return a closed answer in terms of elementary functions, then it has
effectively proved that it can't be done. Which is certainly not the case
with, say, Macsyma or Reduce.

> You can try out mathematica's integration program at
> integrals.wolfram.com

Thanks for the suggestion. As you will no doubt now understand, it is not
my goal to try and do integrations galore, but to find out what is
implemented in each system, in this respect.

Manuel Bronstein

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Sep 5, 2003, 12:12:33 PM9/5/03
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Protagoras wrote:
>
> That's what I had in mind. The rumor I heard is that if you give to
> Axiom some elementary function for antidifferentiation, and it is unable to
> return a closed answer in terms of elementary functions, then it has
> effectively proved that it can't be done. Which is certainly not the case
> with, say, Macsyma or Reduce.

That rumor is almost correct: if Axiom returns an unevaluated integral,
then
it has proven that no elementary antiderivative exists. There are
however
some cases where Axiom can return an error message saying that you've
hit
an unimplemented branch of the algorithm, in which case it cannot
conclude.
So Richard was right in pointing out that the Risch algorithm is not
fully implemented there either. Axiom is unique in making the difference
between unimplemented branches and proofs of non-integrability, and also
in actually proving the algebraic independence of the building blocks
of the integrand before concluding nonintegrability (others typically
assume this independence after performing some heuristic dependence
checking).

-- Manuel Bronstein
-- Manuel.B...@sophia.inria.fr
-- http://www.inria.fr/cafe/Manuel.Bronstein/

Richard Fateman

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Sep 5, 2003, 7:41:44 PM9/5/03
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Protagoras wrote:
>
>The rumor I heard is that if you give to
> Axiom some elementary function for antidifferentiation, and it is unable to
> return a closed answer in terms of elementary functions, then it has
> effectively proved that it can't be done. Which is certainly not the case
> with, say, Macsyma or Reduce.
>

Depending on the class of functions in the integrand, and subject
to certain computational difficulties (not proving that something
is zero), any implementation, including that in Maple, Macsyma, Reduce,
... uses a decision procedure that will, in effect, find the answer
or show that none exists, assuming the implementation has no bugs.

Any program can have bugs, so a failure of the program
to produce a suitable answer may only indicate the existence of a
bug.

I think you should find plenty to read about (e.g. Bronstein's book
on symbolic integration); finding bugs in some of the computer
algebra systems is like shooting fish in a barrel.

RJF

Protagoras

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Sep 5, 2003, 8:40:57 PM9/5/03
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Your feedback is very much appreciated.

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